tutorial: exact numerical computation in algebra and geometry · many problems in computational...
TRANSCRIPT
![Page 1: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/1.jpg)
Tutorial:Exact Numerical Computation
in Algebra and Geometry
Chee K. Yap
Courant Institute of Mathematical SciencesNew York University
andKorea Institute of Advanced Study (KIAS)
Seoul, Korea
34th ISSAC, July 28–31, 2009
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 1 / 113
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Tutorial: Exact Numerical Computation in Algebraand Geometry
Many problems in Computational Science & Engineering (CS&E) aredefined on the continuum. Standard algorithms for these problems arenumerical and approximate. Their computational techniques includeiteration, subdivision, and approximation. Such techniques are rarelyseen in exact or algebraic algorithms. In this tutorial, we discuss a modeof computation called Exact Numerical Computation (ENC) thatachieves exactness through numerical approximation. Through ENC,we naturally incorporate iteration, subdivision and approximation intoour design of algorithms for computer algebra and computationalgeometry. Such algorithms are both novel and practical. This tutorial onENC is divided into three equal parts:(a) ENC and Zero Problems(b) Explicitization and Subdivision Algorithms(c) Complexity Analysis of Adaptivity
![Page 3: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/3.jpg)
Overview of Tutorial
Background is algebraic and geometric computation
Motivation: much of computing world (CS&E) is continuous
But Theoretical Computer Science has gone completely discrete
The discrete view alone is inadequate for CS&E.
What role for exact computation in the continuum?
Geometric insights holds the key
Exact Numerical Computation (ENC) is a proposed synthesis
Lecture in 3 parts (a) ENC and Zero Problems (b) Explicitization and Subdivision Algorithms (c) Complexity Analysis of Adaptivity
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 3 / 113
![Page 4: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/4.jpg)
Overview of Tutorial
Background is algebraic and geometric computation
Motivation: much of computing world (CS&E) is continuous
But Theoretical Computer Science has gone completely discrete
The discrete view alone is inadequate for CS&E.
What role for exact computation in the continuum?
Geometric insights holds the key
Exact Numerical Computation (ENC) is a proposed synthesis
Lecture in 3 parts (a) ENC and Zero Problems (b) Explicitization and Subdivision Algorithms (c) Complexity Analysis of Adaptivity
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 3 / 113
![Page 5: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/5.jpg)
Overview of Tutorial
Background is algebraic and geometric computation
Motivation: much of computing world (CS&E) is continuous
But Theoretical Computer Science has gone completely discrete
The discrete view alone is inadequate for CS&E.
What role for exact computation in the continuum?
Geometric insights holds the key
Exact Numerical Computation (ENC) is a proposed synthesis
Lecture in 3 parts (a) ENC and Zero Problems (b) Explicitization and Subdivision Algorithms (c) Complexity Analysis of Adaptivity
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 3 / 113
![Page 6: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/6.jpg)
Overview of Tutorial
Background is algebraic and geometric computation
Motivation: much of computing world (CS&E) is continuous
But Theoretical Computer Science has gone completely discrete
The discrete view alone is inadequate for CS&E.
What role for exact computation in the continuum?
Geometric insights holds the key
Exact Numerical Computation (ENC) is a proposed synthesis
Lecture in 3 parts (a) ENC and Zero Problems (b) Explicitization and Subdivision Algorithms (c) Complexity Analysis of Adaptivity
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 3 / 113
![Page 7: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/7.jpg)
Overview of Tutorial
Background is algebraic and geometric computation
Motivation: much of computing world (CS&E) is continuous
But Theoretical Computer Science has gone completely discrete
The discrete view alone is inadequate for CS&E.
What role for exact computation in the continuum?
Geometric insights holds the key
Exact Numerical Computation (ENC) is a proposed synthesis
Lecture in 3 parts (a) ENC and Zero Problems (b) Explicitization and Subdivision Algorithms (c) Complexity Analysis of Adaptivity
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 3 / 113
![Page 8: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/8.jpg)
Overview of Tutorial
Background is algebraic and geometric computation
Motivation: much of computing world (CS&E) is continuous
But Theoretical Computer Science has gone completely discrete
The discrete view alone is inadequate for CS&E.
What role for exact computation in the continuum?
Geometric insights holds the key
Exact Numerical Computation (ENC) is a proposed synthesis
Lecture in 3 parts (a) ENC and Zero Problems (b) Explicitization and Subdivision Algorithms (c) Complexity Analysis of Adaptivity
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 3 / 113
![Page 9: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/9.jpg)
Overview of Tutorial
Background is algebraic and geometric computation
Motivation: much of computing world (CS&E) is continuous
But Theoretical Computer Science has gone completely discrete
The discrete view alone is inadequate for CS&E.
What role for exact computation in the continuum?
Geometric insights holds the key
Exact Numerical Computation (ENC) is a proposed synthesis
Lecture in 3 parts (a) ENC and Zero Problems (b) Explicitization and Subdivision Algorithms (c) Complexity Analysis of Adaptivity
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 3 / 113
![Page 10: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/10.jpg)
Overview of Tutorial
Background is algebraic and geometric computation
Motivation: much of computing world (CS&E) is continuous
But Theoretical Computer Science has gone completely discrete
The discrete view alone is inadequate for CS&E.
What role for exact computation in the continuum?
Geometric insights holds the key
Exact Numerical Computation (ENC) is a proposed synthesis
Lecture in 3 parts (a) ENC and Zero Problems (b) Explicitization and Subdivision Algorithms (c) Complexity Analysis of Adaptivity
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 3 / 113
![Page 11: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/11.jpg)
Overview of Tutorial
Background is algebraic and geometric computation
Motivation: much of computing world (CS&E) is continuous
But Theoretical Computer Science has gone completely discrete
The discrete view alone is inadequate for CS&E.
What role for exact computation in the continuum?
Geometric insights holds the key
Exact Numerical Computation (ENC) is a proposed synthesis
Lecture in 3 parts (a) ENC and Zero Problems (b) Explicitization and Subdivision Algorithms (c) Complexity Analysis of Adaptivity
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 3 / 113
![Page 12: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/12.jpg)
Overview of Tutorial
Background is algebraic and geometric computation
Motivation: much of computing world (CS&E) is continuous
But Theoretical Computer Science has gone completely discrete
The discrete view alone is inadequate for CS&E.
What role for exact computation in the continuum?
Geometric insights holds the key
Exact Numerical Computation (ENC) is a proposed synthesis
Lecture in 3 parts (a) ENC and Zero Problems (b) Explicitization and Subdivision Algorithms (c) Complexity Analysis of Adaptivity
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 3 / 113
![Page 13: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/13.jpg)
Overview of Tutorial
Background is algebraic and geometric computation
Motivation: much of computing world (CS&E) is continuous
But Theoretical Computer Science has gone completely discrete
The discrete view alone is inadequate for CS&E.
What role for exact computation in the continuum?
Geometric insights holds the key
Exact Numerical Computation (ENC) is a proposed synthesis
Lecture in 3 parts (a) ENC and Zero Problems (b) Explicitization and Subdivision Algorithms (c) Complexity Analysis of Adaptivity
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 3 / 113
![Page 14: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/14.jpg)
PART 1
Exact Numeric Computation andthe Zero Problem
“The history of the zero recognition problem is somewhat confused bythe fact that many people do not recognize it as a problem at all.”
— DANIEL RICHARDSON (1996)
“Algebra is generous, she often gives more than is asked of her.”
— JEAN LE ROND D’ALEMBERT (1717-83)Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 4 / 113
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Introduction: What is Geometry?
Coming Up Next
1 Introduction: What is Geometric Computation?
2 Five Examples of Geometric Computation
3 Exact Numeric Computation – A Synthesis
4 Exact Geometric Computation
5 Constructive Zero Bounds
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 5 / 113
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Introduction: What is Geometry?
Introduction to Geometric Computation
PUZZLE 1:Is Geometry discrete or continuous?
PUZZLE 2:How come numbers do not arise in Computational Geometry?
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 6 / 113
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Introduction: What is Geometry?
Introduction to Geometric Computation
PUZZLE 1:Is Geometry discrete or continuous?
PUZZLE 2:How come numbers do not arise in Computational Geometry?
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 6 / 113
![Page 18: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/18.jpg)
Introduction: What is Geometry?
Introduction to Geometric Computation
PUZZLE 1:Is Geometry discrete or continuous?
PUZZLE 2:How come numbers do not arise in Computational Geometry?
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 6 / 113
![Page 19: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/19.jpg)
Introduction: What is Geometry?
Introduction to Geometric Computation
PUZZLE 1:Is Geometry discrete or continuous?
PUZZLE 2:How come numbers do not arise in Computational Geometry?
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 6 / 113
![Page 20: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/20.jpg)
Introduction: What is Geometry?
What is Computational Geometry?
Geometric Objects
Prototype: Points, Lines, Circles (Euclidean Geometry)
Arrangement of hyperplanes and hypersurfaces
Zero sets and their Singularities
Semi-algebraic sets
Non-algebraic sets
Geometric complexes
Geometric Problems
Constructing geometric objects
Searching in geometric complexes or structures
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 7 / 113
![Page 21: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/21.jpg)
Introduction: What is Geometry?
What is Computational Geometry?
Geometric Objects
Prototype: Points, Lines, Circles (Euclidean Geometry)
Arrangement of hyperplanes and hypersurfaces
Zero sets and their Singularities
Semi-algebraic sets
Non-algebraic sets
Geometric complexes
Geometric Problems
Constructing geometric objects
Searching in geometric complexes or structures
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 7 / 113
![Page 22: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/22.jpg)
Introduction: What is Geometry?
What is Computational Geometry?
Geometric Objects
Prototype: Points, Lines, Circles (Euclidean Geometry)
Arrangement of hyperplanes and hypersurfaces
Zero sets and their Singularities
Semi-algebraic sets
Non-algebraic sets
Geometric complexes
Geometric Problems
Constructing geometric objects
Searching in geometric complexes or structures
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 7 / 113
![Page 23: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/23.jpg)
Introduction: What is Geometry?
What is Computational Geometry?
Geometric Objects
Prototype: Points, Lines, Circles (Euclidean Geometry)
Arrangement of hyperplanes and hypersurfaces
Zero sets and their Singularities
Semi-algebraic sets
Non-algebraic sets
Geometric complexes
Geometric Problems
Constructing geometric objects
Searching in geometric complexes or structures
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 7 / 113
![Page 24: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/24.jpg)
Introduction: What is Geometry?
What is Computational Geometry?
Geometric Objects
Prototype: Points, Lines, Circles (Euclidean Geometry)
Arrangement of hyperplanes and hypersurfaces
Zero sets and their Singularities
Semi-algebraic sets
Non-algebraic sets
Geometric complexes
Geometric Problems
Constructing geometric objects
Searching in geometric complexes or structures
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 7 / 113
![Page 25: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/25.jpg)
Introduction: What is Geometry?
What is Computational Geometry?
Geometric Objects
Prototype: Points, Lines, Circles (Euclidean Geometry)
Arrangement of hyperplanes and hypersurfaces
Zero sets and their Singularities
Semi-algebraic sets
Non-algebraic sets
Geometric complexes
Geometric Problems
Constructing geometric objects
Searching in geometric complexes or structures
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 7 / 113
![Page 26: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/26.jpg)
Introduction: What is Geometry?
What is Computational Geometry?
Geometric Objects
Prototype: Points, Lines, Circles (Euclidean Geometry)
Arrangement of hyperplanes and hypersurfaces
Zero sets and their Singularities
Semi-algebraic sets
Non-algebraic sets
Geometric complexes
Geometric Problems
Constructing geometric objects
Searching in geometric complexes or structures
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 7 / 113
![Page 27: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/27.jpg)
Introduction: What is Geometry?
What is Computational Geometry?
Geometric Objects
Prototype: Points, Lines, Circles (Euclidean Geometry)
Arrangement of hyperplanes and hypersurfaces
Zero sets and their Singularities
Semi-algebraic sets
Non-algebraic sets
Geometric complexes
Geometric Problems
Constructing geometric objects
Searching in geometric complexes or structures
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 7 / 113
![Page 28: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/28.jpg)
Introduction: What is Geometry?
What is Computational Geometry?
Geometric Objects
Prototype: Points, Lines, Circles (Euclidean Geometry)
Arrangement of hyperplanes and hypersurfaces
Zero sets and their Singularities
Semi-algebraic sets
Non-algebraic sets
Geometric complexes
Geometric Problems
Constructing geometric objects
Searching in geometric complexes or structures
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 7 / 113
![Page 29: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/29.jpg)
Introduction: What is Geometry?
What is Computational Geometry?
Geometric Objects
Prototype: Points, Lines, Circles (Euclidean Geometry)
Arrangement of hyperplanes and hypersurfaces
Zero sets and their Singularities
Semi-algebraic sets
Non-algebraic sets
Geometric complexes
Geometric Problems
Constructing geometric objects
Searching in geometric complexes or structures
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 7 / 113
![Page 30: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/30.jpg)
Introduction: What is Geometry?
What is Computational Geometry?
Geometric Objects
Prototype: Points, Lines, Circles (Euclidean Geometry)
Arrangement of hyperplanes and hypersurfaces
Zero sets and their Singularities
Semi-algebraic sets
Non-algebraic sets
Geometric complexes
Geometric Problems
Constructing geometric objects
Searching in geometric complexes or structures
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 7 / 113
![Page 31: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/31.jpg)
Introduction: What is Geometry?
Dual Descriptions of Geometry
Where do Geometric Objects Live?
As Points in Parametric Space P E.g., for lines given by L(a,b,c) := aX +bY + c = 0,
the space is P :=(a,b,c) : a2 +b2 > 0
⊆R
3.
As Loci in Ambient Space A E.g., Locus of the Line(1,−2,0) is
the set(x ,y) ∈R
2 : x−2y = 0⊆A= R
2.
More involved example:Cell Complexes (in the sense of algebraic topology)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 8 / 113
![Page 32: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/32.jpg)
Introduction: What is Geometry?
Dual Descriptions of Geometry
Where do Geometric Objects Live?
As Points in Parametric Space P E.g., for lines given by L(a,b,c) := aX +bY + c = 0,
the space is P :=(a,b,c) : a2 +b2 > 0
⊆R
3.
As Loci in Ambient Space A E.g., Locus of the Line(1,−2,0) is
the set(x ,y) ∈R
2 : x−2y = 0⊆A= R
2.
More involved example:Cell Complexes (in the sense of algebraic topology)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 8 / 113
![Page 33: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/33.jpg)
Introduction: What is Geometry?
Dual Descriptions of Geometry
Where do Geometric Objects Live?
As Points in Parametric Space P E.g., for lines given by L(a,b,c) := aX +bY + c = 0,
the space is P :=(a,b,c) : a2 +b2 > 0
⊆R
3.
As Loci in Ambient Space A E.g., Locus of the Line(1,−2,0) is
the set(x ,y) ∈R
2 : x−2y = 0⊆A= R
2.
More involved example:Cell Complexes (in the sense of algebraic topology)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 8 / 113
![Page 34: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/34.jpg)
Introduction: What is Geometry?
Dual Descriptions of Geometry
Where do Geometric Objects Live?
As Points in Parametric Space P E.g., for lines given by L(a,b,c) := aX +bY + c = 0,
the space is P :=(a,b,c) : a2 +b2 > 0
⊆R
3.
As Loci in Ambient Space A E.g., Locus of the Line(1,−2,0) is
the set(x ,y) ∈R
2 : x−2y = 0⊆A= R
2.
More involved example:Cell Complexes (in the sense of algebraic topology)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 8 / 113
![Page 35: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/35.jpg)
Introduction: What is Geometry?
Dual Descriptions of Geometry
Where do Geometric Objects Live?
As Points in Parametric Space P E.g., for lines given by L(a,b,c) := aX +bY + c = 0,
the space is P :=(a,b,c) : a2 +b2 > 0
⊆R
3.
As Loci in Ambient Space A E.g., Locus of the Line(1,−2,0) is
the set(x ,y) ∈R
2 : x−2y = 0⊆A= R
2.
More involved example:Cell Complexes (in the sense of algebraic topology)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 8 / 113
![Page 36: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/36.jpg)
Introduction: What is Geometry?
Dual Descriptions of Geometry
Where do Geometric Objects Live?
As Points in Parametric Space P E.g., for lines given by L(a,b,c) := aX +bY + c = 0,
the space is P :=(a,b,c) : a2 +b2 > 0
⊆R
3.
As Loci in Ambient Space A E.g., Locus of the Line(1,−2,0) is
the set(x ,y) ∈R
2 : x−2y = 0⊆A= R
2.
More involved example:Cell Complexes (in the sense of algebraic topology)
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Introduction: What is Geometry?
Dual Descriptions of Geometry
Where do Geometric Objects Live?
As Points in Parametric Space P E.g., for lines given by L(a,b,c) := aX +bY + c = 0,
the space is P :=(a,b,c) : a2 +b2 > 0
⊆R
3.
As Loci in Ambient Space A E.g., Locus of the Line(1,−2,0) is
the set(x ,y) ∈R
2 : x−2y = 0⊆A= R
2.
More involved example:Cell Complexes (in the sense of algebraic topology)
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Introduction: What is Geometry?
Computation: Geometric vs. Algebraic
Where is the Computation?
Algebraic Computation: in parameter space P E.g., Grobner bases Polynomial manipulation, Expensive (double exponential time)
Geometric Computation: in ambient space A E.g., Finding Zeros of Polynomials in R
n
Numerical, Combinatorial, Adaptive (single exponential time)
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Introduction: What is Geometry?
Computation: Geometric vs. Algebraic
Where is the Computation?
Algebraic Computation: in parameter space P E.g., Grobner bases Polynomial manipulation, Expensive (double exponential time)
Geometric Computation: in ambient space A E.g., Finding Zeros of Polynomials in R
n
Numerical, Combinatorial, Adaptive (single exponential time)
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Introduction: What is Geometry?
Computation: Geometric vs. Algebraic
Where is the Computation?
Algebraic Computation: in parameter space P E.g., Grobner bases Polynomial manipulation, Expensive (double exponential time)
Geometric Computation: in ambient space A E.g., Finding Zeros of Polynomials in R
n
Numerical, Combinatorial, Adaptive (single exponential time)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 9 / 113
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Introduction: What is Geometry?
Computation: Geometric vs. Algebraic
Where is the Computation?
Algebraic Computation: in parameter space P E.g., Grobner bases Polynomial manipulation, Expensive (double exponential time)
Geometric Computation: in ambient space A E.g., Finding Zeros of Polynomials in R
n
Numerical, Combinatorial, Adaptive (single exponential time)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 9 / 113
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Introduction: What is Geometry?
Computation: Geometric vs. Algebraic
Where is the Computation?
Algebraic Computation: in parameter space P E.g., Grobner bases Polynomial manipulation, Expensive (double exponential time)
Geometric Computation: in ambient space A E.g., Finding Zeros of Polynomials in R
n
Numerical, Combinatorial, Adaptive (single exponential time)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 9 / 113
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Introduction: What is Geometry?
Computation: Geometric vs. Algebraic
Where is the Computation?
Algebraic Computation: in parameter space P E.g., Grobner bases Polynomial manipulation, Expensive (double exponential time)
Geometric Computation: in ambient space A E.g., Finding Zeros of Polynomials in R
n
Numerical, Combinatorial, Adaptive (single exponential time)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 9 / 113
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(Contd.) Computation: Geometric vs. Algebraic
Answer to PUZZLE 1: “BOTH”Geometry is discrete (in P) (algebraic computation)
Geometry is continuous (in A) (analytic computation)
Actions in the Ambient Space
Geometric Relationships on different Object types arise in A E.g., Point is ON/LEFT/RIGHT of a Line
Analytic properties of Objects comes from their loci
E.g., Proximity, Approximations, Isotopy, etc
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(Contd.) Computation: Geometric vs. Algebraic
Answer to PUZZLE 1: “BOTH”Geometry is discrete (in P) (algebraic computation)
Geometry is continuous (in A) (analytic computation)
Actions in the Ambient Space
Geometric Relationships on different Object types arise in A E.g., Point is ON/LEFT/RIGHT of a Line
Analytic properties of Objects comes from their loci
E.g., Proximity, Approximations, Isotopy, etc
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(Contd.) Computation: Geometric vs. Algebraic
Answer to PUZZLE 1: “BOTH”Geometry is discrete (in P) (algebraic computation)
Geometry is continuous (in A) (analytic computation)
Actions in the Ambient Space
Geometric Relationships on different Object types arise in A E.g., Point is ON/LEFT/RIGHT of a Line
Analytic properties of Objects comes from their loci
E.g., Proximity, Approximations, Isotopy, etc
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(Contd.) Computation: Geometric vs. Algebraic
Answer to PUZZLE 1: “BOTH”Geometry is discrete (in P) (algebraic computation)
Geometry is continuous (in A) (analytic computation)
Actions in the Ambient Space
Geometric Relationships on different Object types arise in A E.g., Point is ON/LEFT/RIGHT of a Line
Analytic properties of Objects comes from their loci
E.g., Proximity, Approximations, Isotopy, etc
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(Contd.) Computation: Geometric vs. Algebraic
Answer to PUZZLE 1: “BOTH”Geometry is discrete (in P) (algebraic computation)
Geometry is continuous (in A) (analytic computation)
Actions in the Ambient Space
Geometric Relationships on different Object types arise in A E.g., Point is ON/LEFT/RIGHT of a Line
Analytic properties of Objects comes from their loci
E.g., Proximity, Approximations, Isotopy, etc
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(Contd.) Computation: Geometric vs. Algebraic
Answer to PUZZLE 1: “BOTH”Geometry is discrete (in P) (algebraic computation)
Geometry is continuous (in A) (analytic computation)
Actions in the Ambient Space
Geometric Relationships on different Object types arise in A E.g., Point is ON/LEFT/RIGHT of a Line
Analytic properties of Objects comes from their loci
E.g., Proximity, Approximations, Isotopy, etc
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(Contd.) Computation: Geometric vs. Algebraic
Answer to PUZZLE 1: “BOTH”Geometry is discrete (in P) (algebraic computation)
Geometry is continuous (in A) (analytic computation)
Actions in the Ambient Space
Geometric Relationships on different Object types arise in A E.g., Point is ON/LEFT/RIGHT of a Line
Analytic properties of Objects comes from their loci
E.g., Proximity, Approximations, Isotopy, etc
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(Contd.) Computation: Geometric vs. Algebraic
Answer to PUZZLE 1: “BOTH”Geometry is discrete (in P) (algebraic computation)
Geometry is continuous (in A) (analytic computation)
Actions in the Ambient Space
Geometric Relationships on different Object types arise in A E.g., Point is ON/LEFT/RIGHT of a Line
Analytic properties of Objects comes from their loci
E.g., Proximity, Approximations, Isotopy, etc
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Mini Summary
Geometry is discrete (algebraic view)
Geometry is continuous (analytic view)
Up Next : What do Computational Geometers think?
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Mini Summary
Geometry is discrete (algebraic view)
Geometry is continuous (analytic view)
Up Next : What do Computational Geometers think?
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Mini Summary
Geometry is discrete (algebraic view)
Geometry is continuous (analytic view)
Up Next : What do Computational Geometers think?
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Mini Summary
Geometry is discrete (algebraic view)
Geometry is continuous (analytic view)
Up Next : What do Computational Geometers think?
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Mini Summary
Geometry is discrete (algebraic view)
Geometry is continuous (analytic view)
Up Next : What do Computational Geometers think?
![Page 57: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/57.jpg)
Five Examples of Geometric Computation
Coming Up Next
1 Introduction: What is Geometric Computation?
2 Five Examples of Geometric Computation
3 Exact Numeric Computation – A Synthesis
4 Exact Geometric Computation
5 Constructive Zero Bounds
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Five Examples of Geometric Computation
Five Examples of Geometric Computation
(I) Convex Hulls
(II) Euclidean Shortest Path
(III) Disc Avoiding Shortest Path
(IV) Mesh Generation
(V) Discrete Morse Theory
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Five Examples of Geometric Computation
Five Examples of Geometric Computation
(I) Convex Hulls
(II) Euclidean Shortest Path
(III) Disc Avoiding Shortest Path
(IV) Mesh Generation
(V) Discrete Morse Theory
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 13 / 113
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Five Examples of Geometric Computation
Five Examples of Geometric Computation
(I) Convex Hulls
(II) Euclidean Shortest Path
(III) Disc Avoiding Shortest Path
(IV) Mesh Generation
(V) Discrete Morse Theory
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 13 / 113
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Five Examples of Geometric Computation
Five Examples of Geometric Computation
(I) Convex Hulls
(II) Euclidean Shortest Path
(III) Disc Avoiding Shortest Path
(IV) Mesh Generation
(V) Discrete Morse Theory
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 13 / 113
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Five Examples of Geometric Computation
Five Examples of Geometric Computation
(I) Convex Hulls
(II) Euclidean Shortest Path
(III) Disc Avoiding Shortest Path
(IV) Mesh Generation
(V) Discrete Morse Theory
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 13 / 113
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Five Examples of Geometric Computation
Five Examples of Geometric Computation
(I) Convex Hulls
(II) Euclidean Shortest Path
(III) Disc Avoiding Shortest Path
(IV) Mesh Generation
(V) Discrete Morse Theory
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 13 / 113
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Five Examples of Geometric Computation
Five Examples of Geometric Computation
(I) Convex Hulls
(II) Euclidean Shortest Path
(III) Disc Avoiding Shortest Path
(IV) Mesh Generation
(V) Discrete Morse Theory
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 13 / 113
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Five Examples of Geometric Computation
(I) Convex Hulls
Convex Hull of Points in Rn
n = 1: finding max and min
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Five Examples of Geometric Computation
(I) Convex Hulls
Convex Hull of Points in Rn
n = 1: finding max and min
n = 2,3: find a convex polygon or polytope
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 14 / 113
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Five Examples of Geometric Computation
(I) Convex Hulls
Convex Hull of Points in Rn
n = 1: finding max and min
n = 2,3: find a convex polygon or polytope
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 14 / 113
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Five Examples of Geometric Computation
(I) Convex Hulls
Convex Hull of Points in Rn
n = 1: finding max and min
n = 2,3: find a convex polygon or polytope
8
23
4
5
6
7
1
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Five Examples of Geometric Computation
(I) Convex Hulls
Convex Hull of Points in Rn
n = 1: finding max and min
n = 2,3: find a convex polygon or polytope
8
23
4
5
6
7
1 Can be reduced to a single predicateOrientation(P0,P1, . . . ,Pn)
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Five Examples of Geometric Computation
(I) Convex Hulls
Convex Hull of Points in Rn
n = 1: finding max and min
n = 2,3: find a convex polygon or polytope
8
23
4
5
6
7
1 Can be reduced to a single predicateOrientation(P0,P1, . . . ,Pn)
Main issue is combinatorial in nature
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Five Examples of Geometric Computation
(I) Convex Hulls
Convex Hull of Points in Rn
n = 1: finding max and min
n = 2,3: find a convex polygon or polytope
8
23
4
5
6
7
1 Can be reduced to a single predicateOrientation(P0,P1, . . . ,Pn)
Main issue is combinatorial in nature
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Five Examples of Geometric Computation
(I) Convex Hulls
Convex Hull of Points in Rn
n = 1: finding max and min
n = 2,3: find a convex polygon or polytope
8
23
4
5
6
7
1 Can be reduced to a single predicateOrientation(P0,P1, . . . ,Pn)
Main issue is combinatorial in nature
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Five Examples of Geometric Computation
(II) Euclidean Shortest Path (ESP)
Shortest Path amidst Polygonal ObstaclesShortest path from p to q avoiding A,B,C
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Five Examples of Geometric Computation
(II) Euclidean Shortest Path (ESP)
Shortest Path amidst Polygonal ObstaclesShortest path from p to q avoiding A,B,C
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 15 / 113
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Five Examples of Geometric Computation
(II) Euclidean Shortest Path (ESP)
Shortest Path amidst Polygonal ObstaclesShortest path from p to q avoiding A,B,C
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 15 / 113
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Five Examples of Geometric Computation
(II) Euclidean Shortest Path (ESP)
Shortest Path amidst Polygonal ObstaclesShortest path from p to q avoiding A,B,C
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 15 / 113
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Five Examples of Geometric Computation
(II) Euclidean Shortest Path (ESP)
Shortest Path amidst Polygonal ObstaclesShortest path from p to q avoiding A,B,C
p
q
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Five Examples of Geometric Computation
(II) Euclidean Shortest Path (ESP)
Shortest Path amidst Polygonal ObstaclesShortest path from p to q avoiding A,B,C
p
q
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Five Examples of Geometric Computation
(II) Euclidean Shortest Path (ESP)
Shortest Path amidst Polygonal ObstaclesShortest path from p to q avoiding A,B,C
p
q
C
B
A
B
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Five Examples of Geometric Computation
(II) Euclidean Shortest Path (ESP)
Shortest Path amidst Polygonal ObstaclesShortest path from p to q avoiding A,B,C
p
q
C
B
A
B
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 15 / 113
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Five Examples of Geometric Computation
(II) Euclidean Shortest Path (ESP)
Shortest Path amidst Polygonal ObstaclesShortest path from p to q avoiding A,B,C
p
q
C
B
A
B
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 15 / 113
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Five Examples of Geometric Computation
(II) Euclidean Shortest Path (ESP)
Shortest Path amidst Polygonal ObstaclesShortest path from p to q avoiding A,B,C
p
q
C
B
A
B
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 15 / 113
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Five Examples of Geometric Computation
(II) Euclidean Shortest Path (ESP)
Shortest Path amidst Polygonal ObstaclesShortest path from p to q avoiding A,B,C
p
q
C
B
A
B
Segment length is asquare-root
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 15 / 113
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Five Examples of Geometric Computation
(II) Euclidean Shortest Path (ESP)
Shortest Path amidst Polygonal ObstaclesShortest path from p to q avoiding A,B,C
p
q
C
B
A
B
Segment length is asquare-root
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 15 / 113
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Five Examples of Geometric Computation
(II) Euclidean Shortest Path (ESP)
Shortest Path amidst Polygonal ObstaclesShortest path from p to q avoiding A,B,C
p
q
C
B
A
B
Segment length is asquare-root
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 15 / 113
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ESP, contd.
Reduction to Dijkstra’s Algorithm
Combinatorial complexity: O(n2 logn) (negligible)
Sum of Square-roots Problem: Is ∑mi=1 ai
√bi = 0?
Not known to be polynomial-time!
Algebraic Approach: Repeated Squaring Method (Nontrivial forInequalites!)
Ω(2m) (slow, unless you are lucky! (Illustrate))
Numerical Approach: Zero Bound Method O(log(1/|e|)) (fast, unless you are unlucky! (Illustrate))
Luck deals differently for the two approaches
![Page 87: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/87.jpg)
ESP, contd.
Reduction to Dijkstra’s Algorithm
Combinatorial complexity: O(n2 logn) (negligible)
Sum of Square-roots Problem: Is ∑mi=1 ai
√bi = 0?
Not known to be polynomial-time!
Algebraic Approach: Repeated Squaring Method (Nontrivial forInequalites!)
Ω(2m) (slow, unless you are lucky! (Illustrate))
Numerical Approach: Zero Bound Method O(log(1/|e|)) (fast, unless you are unlucky! (Illustrate))
Luck deals differently for the two approaches
![Page 88: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/88.jpg)
ESP, contd.
Reduction to Dijkstra’s Algorithm
Combinatorial complexity: O(n2 logn) (negligible)
Sum of Square-roots Problem: Is ∑mi=1 ai
√bi = 0?
Not known to be polynomial-time!
Algebraic Approach: Repeated Squaring Method (Nontrivial forInequalites!)
Ω(2m) (slow, unless you are lucky! (Illustrate))
Numerical Approach: Zero Bound Method O(log(1/|e|)) (fast, unless you are unlucky! (Illustrate))
Luck deals differently for the two approaches
![Page 89: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/89.jpg)
ESP, contd.
Reduction to Dijkstra’s Algorithm
Combinatorial complexity: O(n2 logn) (negligible)
Sum of Square-roots Problem: Is ∑mi=1 ai
√bi = 0?
Not known to be polynomial-time!
Algebraic Approach: Repeated Squaring Method (Nontrivial forInequalites!)
Ω(2m) (slow, unless you are lucky! (Illustrate))
Numerical Approach: Zero Bound Method O(log(1/|e|)) (fast, unless you are unlucky! (Illustrate))
Luck deals differently for the two approaches
![Page 90: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/90.jpg)
ESP, contd.
Reduction to Dijkstra’s Algorithm
Combinatorial complexity: O(n2 logn) (negligible)
Sum of Square-roots Problem: Is ∑mi=1 ai
√bi = 0?
Not known to be polynomial-time!
Algebraic Approach: Repeated Squaring Method (Nontrivial forInequalites!)
Ω(2m) (slow, unless you are lucky! (Illustrate))
Numerical Approach: Zero Bound Method O(log(1/|e|)) (fast, unless you are unlucky! (Illustrate))
Luck deals differently for the two approaches
![Page 91: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/91.jpg)
ESP, contd.
Reduction to Dijkstra’s Algorithm
Combinatorial complexity: O(n2 logn) (negligible)
Sum of Square-roots Problem: Is ∑mi=1 ai
√bi = 0?
Not known to be polynomial-time!
Algebraic Approach: Repeated Squaring Method (Nontrivial forInequalites!)
Ω(2m) (slow, unless you are lucky! (Illustrate))
Numerical Approach: Zero Bound Method O(log(1/|e|)) (fast, unless you are unlucky! (Illustrate))
Luck deals differently for the two approaches
![Page 92: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/92.jpg)
ESP, contd.
Reduction to Dijkstra’s Algorithm
Combinatorial complexity: O(n2 logn) (negligible)
Sum of Square-roots Problem: Is ∑mi=1 ai
√bi = 0?
Not known to be polynomial-time!
Algebraic Approach: Repeated Squaring Method (Nontrivial forInequalites!)
Ω(2m) (slow, unless you are lucky! (Illustrate))
Numerical Approach: Zero Bound Method O(log(1/|e|)) (fast, unless you are unlucky! (Illustrate))
Luck deals differently for the two approaches
![Page 93: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/93.jpg)
ESP, contd.
Reduction to Dijkstra’s Algorithm
Combinatorial complexity: O(n2 logn) (negligible)
Sum of Square-roots Problem: Is ∑mi=1 ai
√bi = 0?
Not known to be polynomial-time!
Algebraic Approach: Repeated Squaring Method (Nontrivial forInequalites!)
Ω(2m) (slow, unless you are lucky! (Illustrate))
Numerical Approach: Zero Bound Method O(log(1/|e|)) (fast, unless you are unlucky! (Illustrate))
Luck deals differently for the two approaches
![Page 94: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/94.jpg)
Five Examples of Geometric Computation
(III) Shortest Path Amidst Discs
Shortest Path amidst DiscsShortest path from p to q avoiding discs A,B
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 17 / 113
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Five Examples of Geometric Computation
(III) Shortest Path Amidst Discs
Shortest Path amidst DiscsShortest path from p to q avoiding discs A,B
p −p
q
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 17 / 113
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Five Examples of Geometric Computation
(III) Shortest Path Amidst Discs
Shortest Path amidst DiscsShortest path from p to q avoiding discs A,B
p −p
q
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 17 / 113
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Five Examples of Geometric Computation
(III) Shortest Path Amidst Discs
Shortest Path amidst DiscsShortest path from p to q avoiding discs A,B
p −p
q
A
B
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 17 / 113
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Five Examples of Geometric Computation
(III) Shortest Path Amidst Discs
Shortest Path amidst DiscsShortest path from p to q avoiding discs A,B
p −p
q
A
B
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 17 / 113
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Five Examples of Geometric Computation
(III) Shortest Path Amidst Discs
Shortest Path amidst DiscsShortest path from p to q avoiding discs A,B
p −p
q
A
B
Segment length is asquare-root of an algebraicnumber
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 17 / 113
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Five Examples of Geometric Computation
(III) Shortest Path Amidst Discs
Shortest Path amidst DiscsShortest path from p to q avoiding discs A,B
p −p
q
A
B
Segment length is asquare-root of an algebraicnumber
Arc lengh is rθ
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 17 / 113
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Five Examples of Geometric Computation
(III) Shortest Path Amidst Discs
Shortest Path amidst DiscsShortest path from p to q avoiding discs A,B
p −p
q
A
B
Segment length is asquare-root of an algebraicnumber
Arc lengh is rθ
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 17 / 113
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Five Examples of Geometric Computation
(III) Shortest Path Amidst Discs
Shortest Path amidst DiscsShortest path from p to q avoiding discs A,B
p −p
q
A
B
Segment length is asquare-root of an algebraicnumber
Arc lengh is rθ
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 17 / 113
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Five Examples of Geometric Computation
(III) Shortest Path Amidst Discs
Shortest Path amidst DiscsShortest path from p to q avoiding discs A,B
p −p
q
A
B
Segment length is asquare-root of an algebraicnumber
Arc lengh is rθ
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 17 / 113
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Disc Obstacles, contd.
Reduction to Dijkstra’s Algorithm (Again?)
Combinatorial complexity: O(n2 logn) (negligible, exercise)
Path length = γ +αwhere γ is algebraic , but α is transcendental
Not even clear that we can compute this!
Why? Numerical Halting Problem
Analogue of the Turing Halting Problem
Also semi-decidable
![Page 105: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/105.jpg)
Disc Obstacles, contd.
Reduction to Dijkstra’s Algorithm (Again?)
Combinatorial complexity: O(n2 logn) (negligible, exercise)
Path length = γ +αwhere γ is algebraic , but α is transcendental
Not even clear that we can compute this!
Why? Numerical Halting Problem
Analogue of the Turing Halting Problem
Also semi-decidable
![Page 106: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/106.jpg)
Disc Obstacles, contd.
Reduction to Dijkstra’s Algorithm (Again?)
Combinatorial complexity: O(n2 logn) (negligible, exercise)
Path length = γ +αwhere γ is algebraic , but α is transcendental
Not even clear that we can compute this!
Why? Numerical Halting Problem
Analogue of the Turing Halting Problem
Also semi-decidable
![Page 107: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/107.jpg)
Disc Obstacles, contd.
Reduction to Dijkstra’s Algorithm (Again?)
Combinatorial complexity: O(n2 logn) (negligible, exercise)
Path length = γ +αwhere γ is algebraic , but α is transcendental
Not even clear that we can compute this!
Why? Numerical Halting Problem
Analogue of the Turing Halting Problem
Also semi-decidable
![Page 108: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/108.jpg)
Disc Obstacles, contd.
Reduction to Dijkstra’s Algorithm (Again?)
Combinatorial complexity: O(n2 logn) (negligible, exercise)
Path length = γ +αwhere γ is algebraic , but α is transcendental
Not even clear that we can compute this!
Why? Numerical Halting Problem
Analogue of the Turing Halting Problem
Also semi-decidable
![Page 109: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/109.jpg)
Disc Obstacles, contd.
Reduction to Dijkstra’s Algorithm (Again?)
Combinatorial complexity: O(n2 logn) (negligible, exercise)
Path length = γ +αwhere γ is algebraic , but α is transcendental
Not even clear that we can compute this!
Why? Numerical Halting Problem
Analogue of the Turing Halting Problem
Also semi-decidable
![Page 110: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/110.jpg)
Disc Obstacles, contd.
Reduction to Dijkstra’s Algorithm (Again?)
Combinatorial complexity: O(n2 logn) (negligible, exercise)
Path length = γ +αwhere γ is algebraic , but α is transcendental
Not even clear that we can compute this!
Why? Numerical Halting Problem
Analogue of the Turing Halting Problem
Also semi-decidable
![Page 111: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/111.jpg)
Disc Obstacles, contd.
Reduction to Dijkstra’s Algorithm (Again?)
Combinatorial complexity: O(n2 logn) (negligible, exercise)
Path length = γ +αwhere γ is algebraic , but α is transcendental
Not even clear that we can compute this!
Why? Numerical Halting Problem
Analogue of the Turing Halting Problem
Also semi-decidable
![Page 112: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/112.jpg)
Five Examples of Geometric Computation
Addition/Subtraction of Arc Lengths
Simple Case: Unit DiscsLet A = [C,p,q,n] and A′ = [C′,p′,q′,n′] encode two arc lengths.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 19 / 113
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Five Examples of Geometric Computation
Addition/Subtraction of Arc Lengths
Simple Case: Unit DiscsLet A = [C,p,q,n] and A′ = [C′,p′,q′,n′] encode two arc lengths.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 19 / 113
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Five Examples of Geometric Computation
Addition/Subtraction of Arc Lengths
Simple Case: Unit DiscsLet A = [C,p,q,n] and A′ = [C′,p′,q′,n′] encode two arc lengths.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 19 / 113
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Five Examples of Geometric Computation
Addition/Subtraction of Arc Lengths
Simple Case: Unit DiscsLet A = [C,p,q,n] and A′ = [C′,p′,q′,n′] encode two arc lengths.
θ
θ′
p
q
p′
q′
C
C ′
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 19 / 113
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Five Examples of Geometric Computation
Addition/Subtraction of Arc Lengths
Simple Case: Unit DiscsLet A = [C,p,q,n] and A′ = [C′,p′,q′,n′] encode two arc lengths.
θ
θ′
p
q
p′
q′
C
C ′
θ′
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 19 / 113
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Five Examples of Geometric Computation
Addition/Subtraction of Arc Lengths
Simple Case: Unit DiscsLet A = [C,p,q,n] and A′ = [C′,p′,q′,n′] encode two arc lengths.
θ
θ′
p
q
p′
q′
C
C ′
θ′
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 19 / 113
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Five Examples of Geometric Computation
Addition/Subtraction of Arc Lengths
Simple Case: Unit DiscsLet A = [C,p,q,n] and A′ = [C′,p′,q′,n′] encode two arc lengths.
θ
θ′
p
q
p′
q′
C
C ′
θ′
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 19 / 113
![Page 119: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/119.jpg)
Five Examples of Geometric Computation
Addition/Subtraction of Arc Lengths
Simple Case: Unit DiscsLet A = [C,p,q,n] and A′ = [C′,p′,q′,n′] encode two arc lengths.
θ
θ′
p
q
p′
q′
C
C ′
θ′
CC ′
θ′
θ
p
q
q′
p′
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 19 / 113
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Five Examples of Geometric Computation
Addition/Subtraction of Arc Lengths
Simple Case: Unit DiscsLet A = [C,p,q,n] and A′ = [C′,p′,q′,n′] encode two arc lengths.
θ
θ′
p
q
p′
q′
C
C ′
θ′
CC ′
θ′
θ
p
q
q′
p′
θ′
q′′
C ′′
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 19 / 113
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Five Examples of Geometric Computation
Addition/Subtraction of Arc Lengths
Simple Case: Unit DiscsLet A = [C,p,q,n] and A′ = [C′,p′,q′,n′] encode two arc lengths.
θ
θ′
p
q
p′
q′
C
C ′
θ′
CC ′
θ′
θ
p
q
q′
p′
θ′
q′′
C ′′
q′′′
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 19 / 113
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Five Examples of Geometric Computation
Computability of Shortest Paths[Chang/Choi/Kwon/Park/Y. (2005)]
Is it really Transcendental?LEMMA: cosθi is algebraic.COROLLARY (Lindemann 1882): θi is transcendental.
Theorem (Unit Disc)Shortest Path for unit disc obstacles is computable.
Rational CaseMuch harder – use Chebyshev functions of first kind.Main issue: how to transfer arc lengths to circles with different radii.
Theorem (Commensurable Radii)Shortest Path for commensurable radii discs is computable.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 20 / 113
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Five Examples of Geometric Computation
Computability of Shortest Paths[Chang/Choi/Kwon/Park/Y. (2005)]
Is it really Transcendental?LEMMA: cosθi is algebraic.COROLLARY (Lindemann 1882): θi is transcendental.
Theorem (Unit Disc)Shortest Path for unit disc obstacles is computable.
Rational CaseMuch harder – use Chebyshev functions of first kind.Main issue: how to transfer arc lengths to circles with different radii.
Theorem (Commensurable Radii)Shortest Path for commensurable radii discs is computable.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 20 / 113
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Five Examples of Geometric Computation
Computability of Shortest Paths[Chang/Choi/Kwon/Park/Y. (2005)]
Is it really Transcendental?LEMMA: cosθi is algebraic.COROLLARY (Lindemann 1882): θi is transcendental.
Theorem (Unit Disc)Shortest Path for unit disc obstacles is computable.
Rational CaseMuch harder – use Chebyshev functions of first kind.Main issue: how to transfer arc lengths to circles with different radii.
Theorem (Commensurable Radii)Shortest Path for commensurable radii discs is computable.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 20 / 113
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Five Examples of Geometric Computation
Computability of Shortest Paths[Chang/Choi/Kwon/Park/Y. (2005)]
Is it really Transcendental?LEMMA: cosθi is algebraic.COROLLARY (Lindemann 1882): θi is transcendental.
Theorem (Unit Disc)Shortest Path for unit disc obstacles is computable.
Rational CaseMuch harder – use Chebyshev functions of first kind.Main issue: how to transfer arc lengths to circles with different radii.
Theorem (Commensurable Radii)Shortest Path for commensurable radii discs is computable.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 20 / 113
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Five Examples of Geometric Computation
Computability of Shortest Paths (contd.)
No complexity Bounds!Elementary methods fail us.Appeal to Baker’s Linear Form in Logarithms: |α0 +∑n
i=1 αi logβi |> B
Theorem (Commensurable Radii Complexity)Shortest Paths for rational discs is in single-exponential time.
Rare positive result from Transcendental Number Theory
First transcendental geometric problem shown computable
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 21 / 113
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Five Examples of Geometric Computation
Computability of Shortest Paths (contd.)
No complexity Bounds!Elementary methods fail us.Appeal to Baker’s Linear Form in Logarithms: |α0 +∑n
i=1 αi logβi |> B
Theorem (Commensurable Radii Complexity)Shortest Paths for rational discs is in single-exponential time.
Rare positive result from Transcendental Number Theory
First transcendental geometric problem shown computable
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 21 / 113
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Five Examples of Geometric Computation
Computability of Shortest Paths (contd.)
No complexity Bounds!Elementary methods fail us.Appeal to Baker’s Linear Form in Logarithms: |α0 +∑n
i=1 αi logβi |> B
Theorem (Commensurable Radii Complexity)Shortest Paths for rational discs is in single-exponential time.
Rare positive result from Transcendental Number Theory
First transcendental geometric problem shown computable
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 21 / 113
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Five Examples of Geometric Computation
Computability of Shortest Paths (contd.)
No complexity Bounds!Elementary methods fail us.Appeal to Baker’s Linear Form in Logarithms: |α0 +∑n
i=1 αi logβi |> B
Theorem (Commensurable Radii Complexity)Shortest Paths for rational discs is in single-exponential time.
Rare positive result from Transcendental Number Theory
First transcendental geometric problem shown computable
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 21 / 113
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Five Examples of Geometric Computation
(IV) Mesh GenerationMeshing of Surfaces
Surface S = f−1(0) where f : Rn→R (n = 1,2,3)
Wants a triangulated surface S that is isotopic to S
“Tangled Cube” “Chair”
Case n = 1 is root isolation !
Return to meshing in Lecture 2
Applications
Visualization, Graphics, Simulation, Modeling: prerequisite
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 22 / 113
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Five Examples of Geometric Computation
(IV) Mesh GenerationMeshing of Surfaces
Surface S = f−1(0) where f : Rn→R (n = 1,2,3)
Wants a triangulated surface S that is isotopic to S
“Tangled Cube” “Chair”
Case n = 1 is root isolation !
Return to meshing in Lecture 2
Applications
Visualization, Graphics, Simulation, Modeling: prerequisite
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 22 / 113
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Five Examples of Geometric Computation
(IV) Mesh GenerationMeshing of Surfaces
Surface S = f−1(0) where f : Rn→R (n = 1,2,3)
Wants a triangulated surface S that is isotopic to S
“Tangled Cube” “Chair”
Case n = 1 is root isolation !
Return to meshing in Lecture 2
Applications
Visualization, Graphics, Simulation, Modeling: prerequisite
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 22 / 113
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Five Examples of Geometric Computation
(IV) Mesh GenerationMeshing of Surfaces
Surface S = f−1(0) where f : Rn→R (n = 1,2,3)
Wants a triangulated surface S that is isotopic to S
“Tangled Cube” “Chair”
Case n = 1 is root isolation !
Return to meshing in Lecture 2
Applications
Visualization, Graphics, Simulation, Modeling: prerequisite
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 22 / 113
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Five Examples of Geometric Computation
(IV) Mesh GenerationMeshing of Surfaces
Surface S = f−1(0) where f : Rn→R (n = 1,2,3)
Wants a triangulated surface S that is isotopic to S
“Tangled Cube” “Chair”
Case n = 1 is root isolation !
Return to meshing in Lecture 2
Applications
Visualization, Graphics, Simulation, Modeling: prerequisite
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 22 / 113
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Five Examples of Geometric Computation
(IV) Mesh GenerationMeshing of Surfaces
Surface S = f−1(0) where f : Rn→R (n = 1,2,3)
Wants a triangulated surface S that is isotopic to S
“Tangled Cube” “Chair”
Case n = 1 is root isolation !
Return to meshing in Lecture 2
Applications
Visualization, Graphics, Simulation, Modeling: prerequisite
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 22 / 113
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Five Examples of Geometric Computation
(IV) Mesh GenerationMeshing of Surfaces
Surface S = f−1(0) where f : Rn→R (n = 1,2,3)
Wants a triangulated surface S that is isotopic to S
“Tangled Cube” “Chair”
Case n = 1 is root isolation !
Return to meshing in Lecture 2
Applications
Visualization, Graphics, Simulation, Modeling: prerequisite
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 22 / 113
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Five Examples of Geometric Computation
(IV) Mesh GenerationMeshing of Surfaces
Surface S = f−1(0) where f : Rn→R (n = 1,2,3)
Wants a triangulated surface S that is isotopic to S
“Tangled Cube” “Chair”
Case n = 1 is root isolation !
Return to meshing in Lecture 2
Applications
Visualization, Graphics, Simulation, Modeling: prerequisite
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 22 / 113
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Five Examples of Geometric Computation
(IV) Mesh GenerationMeshing of Surfaces
Surface S = f−1(0) where f : Rn→R (n = 1,2,3)
Wants a triangulated surface S that is isotopic to S
“Tangled Cube” “Chair”
Case n = 1 is root isolation !
Return to meshing in Lecture 2
Applications
Visualization, Graphics, Simulation, Modeling: prerequisite
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 22 / 113
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Five Examples of Geometric Computation
(IV) Mesh GenerationMeshing of Surfaces
Surface S = f−1(0) where f : Rn→R (n = 1,2,3)
Wants a triangulated surface S that is isotopic to S
“Tangled Cube” “Chair”
Case n = 1 is root isolation !
Return to meshing in Lecture 2
Applications
Visualization, Graphics, Simulation, Modeling: prerequisite
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 22 / 113
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Five Examples of Geometric Computation
(V) Discrete Morse Theory
Edelsbrunner, Harer, Zomorodian (2003)
Methodology: discrete analogues of continuous concepts
Differential geometry, Ricci flows, etc
Morse-Smale Complex of a surface S = f−1(0):
Critical Points (max/min/saddle)
Integral Lines
OPEN: How to connect saddle to itsmaximas
Exactness Bottleneck: this “Continuous-to-Discrete” transformation
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 23 / 113
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Five Examples of Geometric Computation
(V) Discrete Morse Theory
Edelsbrunner, Harer, Zomorodian (2003)
Methodology: discrete analogues of continuous concepts
Differential geometry, Ricci flows, etc
Morse-Smale Complex of a surface S = f−1(0):
Critical Points (max/min/saddle)
Integral Lines
OPEN: How to connect saddle to itsmaximas
Exactness Bottleneck: this “Continuous-to-Discrete” transformation
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 23 / 113
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Five Examples of Geometric Computation
(V) Discrete Morse Theory
Edelsbrunner, Harer, Zomorodian (2003)
Methodology: discrete analogues of continuous concepts
Differential geometry, Ricci flows, etc
Morse-Smale Complex of a surface S = f−1(0):
Critical Points (max/min/saddle)
Integral Lines
OPEN: How to connect saddle to itsmaximas
Exactness Bottleneck: this “Continuous-to-Discrete” transformation
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 23 / 113
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Five Examples of Geometric Computation
(V) Discrete Morse Theory
Edelsbrunner, Harer, Zomorodian (2003)
Methodology: discrete analogues of continuous concepts
Differential geometry, Ricci flows, etc
Morse-Smale Complex of a surface S = f−1(0):
Critical Points (max/min/saddle)
Integral Lines
OPEN: How to connect saddle to itsmaximas
Exactness Bottleneck: this “Continuous-to-Discrete” transformation
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 23 / 113
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Five Examples of Geometric Computation
(V) Discrete Morse Theory
Edelsbrunner, Harer, Zomorodian (2003)
Methodology: discrete analogues of continuous concepts
Differential geometry, Ricci flows, etc
Morse-Smale Complex of a surface S = f−1(0):
Critical Points (max/min/saddle)
Integral Lines
OPEN: How to connect saddle to itsmaximas
Exactness Bottleneck: this “Continuous-to-Discrete” transformation
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 23 / 113
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Five Examples of Geometric Computation
(V) Discrete Morse Theory
Edelsbrunner, Harer, Zomorodian (2003)
Methodology: discrete analogues of continuous concepts
Differential geometry, Ricci flows, etc
Morse-Smale Complex of a surface S = f−1(0):
Critical Points (max/min/saddle)
Integral Lines
OPEN: How to connect saddle to itsmaximas
Exactness Bottleneck: this “Continuous-to-Discrete” transformation
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 23 / 113
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Five Examples of Geometric Computation
(V) Discrete Morse Theory
Edelsbrunner, Harer, Zomorodian (2003)
Methodology: discrete analogues of continuous concepts
Differential geometry, Ricci flows, etc
Morse-Smale Complex of a surface S = f−1(0):
Critical Points (max/min/saddle)
Integral Lines
OPEN: How to connect saddle to itsmaximas
Exactness Bottleneck: this “Continuous-to-Discrete” transformation
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 23 / 113
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Five Examples of Geometric Computation
(V) Discrete Morse Theory
Edelsbrunner, Harer, Zomorodian (2003)
Methodology: discrete analogues of continuous concepts
Differential geometry, Ricci flows, etc
Morse-Smale Complex of a surface S = f−1(0):
Critical Points (max/min/saddle)
Integral Lines
OPEN: How to connect saddle to itsmaximas
Exactness Bottleneck: this “Continuous-to-Discrete” transformation
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 23 / 113
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Five Examples of Geometric Computation
(V) Discrete Morse Theory
Edelsbrunner, Harer, Zomorodian (2003)
Methodology: discrete analogues of continuous concepts
Differential geometry, Ricci flows, etc
Morse-Smale Complex of a surface S = f−1(0):
Critical Points (max/min/saddle)
Integral Lines
OPEN: How to connect saddle to itsmaximas
Exactness Bottleneck: this “Continuous-to-Discrete” transformation
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 23 / 113
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Five Examples of Geometric Computation
(V) Discrete Morse Theory
Edelsbrunner, Harer, Zomorodian (2003)
Methodology: discrete analogues of continuous concepts
Differential geometry, Ricci flows, etc
Morse-Smale Complex of a surface S = f−1(0):
Critical Points (max/min/saddle)
Integral Lines
OPEN: How to connect saddle to itsmaximas
Exactness Bottleneck: this “Continuous-to-Discrete” transformation
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 23 / 113
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Five Examples of Geometric Computation
(V) Discrete Morse Theory
Edelsbrunner, Harer, Zomorodian (2003)
Methodology: discrete analogues of continuous concepts
Differential geometry, Ricci flows, etc
Morse-Smale Complex of a surface S = f−1(0):
Critical Points (max/min/saddle)
Integral Lines
OPEN: How to connect saddle to itsmaximas
Exactness Bottleneck: this “Continuous-to-Discrete” transformation
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 23 / 113
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Five Examples of Geometric Computation
(V) Discrete Morse Theory
Edelsbrunner, Harer, Zomorodian (2003)
Methodology: discrete analogues of continuous concepts
Differential geometry, Ricci flows, etc
Morse-Smale Complex of a surface S = f−1(0):
Critical Points (max/min/saddle)
Integral Lines
OPEN: How to connect saddle to itsmaximas
Exactness Bottleneck: this “Continuous-to-Discrete” transformation
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 23 / 113
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Mini Summary
We saw 5 Geometric Problems:I=classic, II=hard, III=very hard, IV=current, V=open
Up Next : Let us examine their underlying computationalmodels...
![Page 153: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/153.jpg)
Mini Summary
We saw 5 Geometric Problems:I=classic, II=hard, III=very hard, IV=current, V=open
Up Next : Let us examine their underlying computationalmodels...
![Page 154: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/154.jpg)
Mini Summary
We saw 5 Geometric Problems:I=classic, II=hard, III=very hard, IV=current, V=open
Up Next : Let us examine their underlying computationalmodels...
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Mini Summary
We saw 5 Geometric Problems:I=classic, II=hard, III=very hard, IV=current, V=open
Up Next : Let us examine their underlying computationalmodels...
![Page 156: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/156.jpg)
ENC
Coming Up Next
1 Introduction: What is Geometric Computation?
2 Five Examples of Geometric Computation
3 Exact Numeric Computation – A Synthesis
4 Exact Geometric Computation
5 Constructive Zero Bounds
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 25 / 113
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ENC
Two Worlds of Computing
(EX) Discrete, Combinatorial, Exact .
Theoretical Computer Science, Computer Algebra
(AP) Continuous, Numerical, Approximate. Computational Science & Engineering (CS&E) or Physics Problems too hard in exact framework (e.g., 3D Ising Model) Even when exact solution is possible,...
The 2 Worlds meet in Geometry Solving Linear Systems (Gaussian vs. Gauss-Seidel)
Linear Programming (Simplex vs. Interior-Point)
Solving Numerical PDE (Symbolic vs. Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 26 / 113
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ENC
Two Worlds of Computing
(EX) Discrete, Combinatorial, Exact .
Theoretical Computer Science, Computer Algebra
(AP) Continuous, Numerical, Approximate. Computational Science & Engineering (CS&E) or Physics Problems too hard in exact framework (e.g., 3D Ising Model) Even when exact solution is possible,...
The 2 Worlds meet in Geometry Solving Linear Systems (Gaussian vs. Gauss-Seidel)
Linear Programming (Simplex vs. Interior-Point)
Solving Numerical PDE (Symbolic vs. Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 26 / 113
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ENC
Two Worlds of Computing
(EX) Discrete, Combinatorial, Exact .
Theoretical Computer Science, Computer Algebra
(AP) Continuous, Numerical, Approximate. Computational Science & Engineering (CS&E) or Physics Problems too hard in exact framework (e.g., 3D Ising Model) Even when exact solution is possible,...
The 2 Worlds meet in Geometry Solving Linear Systems (Gaussian vs. Gauss-Seidel)
Linear Programming (Simplex vs. Interior-Point)
Solving Numerical PDE (Symbolic vs. Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 26 / 113
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ENC
Two Worlds of Computing
(EX) Discrete, Combinatorial, Exact .
Theoretical Computer Science, Computer Algebra
(AP) Continuous, Numerical, Approximate. Computational Science & Engineering (CS&E) or Physics Problems too hard in exact framework (e.g., 3D Ising Model) Even when exact solution is possible,...
The 2 Worlds meet in Geometry Solving Linear Systems (Gaussian vs. Gauss-Seidel)
Linear Programming (Simplex vs. Interior-Point)
Solving Numerical PDE (Symbolic vs. Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 26 / 113
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ENC
Two Worlds of Computing
(EX) Discrete, Combinatorial, Exact .
Theoretical Computer Science, Computer Algebra
(AP) Continuous, Numerical, Approximate. Computational Science & Engineering (CS&E) or Physics Problems too hard in exact framework (e.g., 3D Ising Model) Even when exact solution is possible,...
The 2 Worlds meet in Geometry Solving Linear Systems (Gaussian vs. Gauss-Seidel)
Linear Programming (Simplex vs. Interior-Point)
Solving Numerical PDE (Symbolic vs. Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 26 / 113
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ENC
Two Worlds of Computing
(EX) Discrete, Combinatorial, Exact .
Theoretical Computer Science, Computer Algebra
(AP) Continuous, Numerical, Approximate. Computational Science & Engineering (CS&E) or Physics Problems too hard in exact framework (e.g., 3D Ising Model) Even when exact solution is possible,...
The 2 Worlds meet in Geometry Solving Linear Systems (Gaussian vs. Gauss-Seidel)
Linear Programming (Simplex vs. Interior-Point)
Solving Numerical PDE (Symbolic vs. Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 26 / 113
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ENC
Two Worlds of Computing
(EX) Discrete, Combinatorial, Exact .
Theoretical Computer Science, Computer Algebra
(AP) Continuous, Numerical, Approximate. Computational Science & Engineering (CS&E) or Physics Problems too hard in exact framework (e.g., 3D Ising Model) Even when exact solution is possible,...
The 2 Worlds meet in Geometry Solving Linear Systems (Gaussian vs. Gauss-Seidel)
Linear Programming (Simplex vs. Interior-Point)
Solving Numerical PDE (Symbolic vs. Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 26 / 113
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ENC
Two Worlds of Computing
(EX) Discrete, Combinatorial, Exact .
Theoretical Computer Science, Computer Algebra
(AP) Continuous, Numerical, Approximate. Computational Science & Engineering (CS&E) or Physics Problems too hard in exact framework (e.g., 3D Ising Model) Even when exact solution is possible,...
The 2 Worlds meet in Geometry Solving Linear Systems (Gaussian vs. Gauss-Seidel)
Linear Programming (Simplex vs. Interior-Point)
Solving Numerical PDE (Symbolic vs. Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 26 / 113
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ENC
Two Worlds of Computing
(EX) Discrete, Combinatorial, Exact .
Theoretical Computer Science, Computer Algebra
(AP) Continuous, Numerical, Approximate. Computational Science & Engineering (CS&E) or Physics Problems too hard in exact framework (e.g., 3D Ising Model) Even when exact solution is possible,...
The 2 Worlds meet in Geometry Solving Linear Systems (Gaussian vs. Gauss-Seidel)
Linear Programming (Simplex vs. Interior-Point)
Solving Numerical PDE (Symbolic vs. Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 26 / 113
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ENC
Two Worlds of Computing
(EX) Discrete, Combinatorial, Exact .
Theoretical Computer Science, Computer Algebra
(AP) Continuous, Numerical, Approximate. Computational Science & Engineering (CS&E) or Physics Problems too hard in exact framework (e.g., 3D Ising Model) Even when exact solution is possible,...
The 2 Worlds meet in Geometry Solving Linear Systems (Gaussian vs. Gauss-Seidel)
Linear Programming (Simplex vs. Interior-Point)
Solving Numerical PDE (Symbolic vs. Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 26 / 113
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ENC
Two Worlds of Computing
(EX) Discrete, Combinatorial, Exact .
Theoretical Computer Science, Computer Algebra
(AP) Continuous, Numerical, Approximate. Computational Science & Engineering (CS&E) or Physics Problems too hard in exact framework (e.g., 3D Ising Model) Even when exact solution is possible,...
The 2 Worlds meet in Geometry Solving Linear Systems (Gaussian vs. Gauss-Seidel)
Linear Programming (Simplex vs. Interior-Point)
Solving Numerical PDE (Symbolic vs. Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 26 / 113
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ENC
Two Worlds of Computing
(EX) Discrete, Combinatorial, Exact .
Theoretical Computer Science, Computer Algebra
(AP) Continuous, Numerical, Approximate. Computational Science & Engineering (CS&E) or Physics Problems too hard in exact framework (e.g., 3D Ising Model) Even when exact solution is possible,...
The 2 Worlds meet in Geometry Solving Linear Systems (Gaussian vs. Gauss-Seidel)
Linear Programming (Simplex vs. Interior-Point)
Solving Numerical PDE (Symbolic vs. Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 26 / 113
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ENC
Again, What is Geometry?
Geometry is always about zeros
Problem (I): Is a Point on a Hyperplane?
Problems (II),(III): Are two path lengths are equal?
Problems (IV),(V): Continuous-to-discrete transformations,defined by zero sets
These zero decisions are captured by geometric predicates
View developed by CG’ers in robust geometric computation
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 27 / 113
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ENC
Again, What is Geometry?
Geometry is always about zeros
Problem (I): Is a Point on a Hyperplane?
Problems (II),(III): Are two path lengths are equal?
Problems (IV),(V): Continuous-to-discrete transformations,defined by zero sets
These zero decisions are captured by geometric predicates
View developed by CG’ers in robust geometric computation
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 27 / 113
![Page 171: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/171.jpg)
ENC
Again, What is Geometry?
Geometry is always about zeros
Problem (I): Is a Point on a Hyperplane?
Problems (II),(III): Are two path lengths are equal?
Problems (IV),(V): Continuous-to-discrete transformations,defined by zero sets
These zero decisions are captured by geometric predicates
View developed by CG’ers in robust geometric computation
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 27 / 113
![Page 172: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/172.jpg)
ENC
Again, What is Geometry?
Geometry is always about zeros
Problem (I): Is a Point on a Hyperplane?
Problems (II),(III): Are two path lengths are equal?
Problems (IV),(V): Continuous-to-discrete transformations,defined by zero sets
These zero decisions are captured by geometric predicates
View developed by CG’ers in robust geometric computation
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 27 / 113
![Page 173: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/173.jpg)
ENC
Again, What is Geometry?
Geometry is always about zeros
Problem (I): Is a Point on a Hyperplane?
Problems (II),(III): Are two path lengths are equal?
Problems (IV),(V): Continuous-to-discrete transformations,defined by zero sets
These zero decisions are captured by geometric predicates
View developed by CG’ers in robust geometric computation
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 27 / 113
![Page 174: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/174.jpg)
ENC
Again, What is Geometry?
Geometry is always about zeros
Problem (I): Is a Point on a Hyperplane?
Problems (II),(III): Are two path lengths are equal?
Problems (IV),(V): Continuous-to-discrete transformations,defined by zero sets
These zero decisions are captured by geometric predicates
View developed by CG’ers in robust geometric computation
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 27 / 113
![Page 175: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/175.jpg)
ENC
Again, What is Geometry?
Geometry is always about zeros
Problem (I): Is a Point on a Hyperplane?
Problems (II),(III): Are two path lengths are equal?
Problems (IV),(V): Continuous-to-discrete transformations,defined by zero sets
These zero decisions are captured by geometric predicates
View developed by CG’ers in robust geometric computation
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 27 / 113
![Page 176: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/176.jpg)
ENC
Four Computational Models for Geometry
How to compute in a Continuum (Rn)?
(EX) Algebraic Computational Model(e.g., Real RAM, Blum-Shub-Smale model, Disc Shortest Path)
PROBLEM: Zero is trivial
(EX’) Abstract Operational Models(e.g., CG, Traub, Orientation, Ray shooting, Giftwrap)
PROBLEM: Zero is hidden
(AP) Analytic Computational Model (e.g., Ko, Weihrauch)
PROBLEM: Zero is undecidable
(AP’) Numerical Analysis Model (e.g., x⊕ y = (x + y)(1+ ε))
PROBLEM: Zero is abolished
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 28 / 113
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ENC
Four Computational Models for Geometry
How to compute in a Continuum (Rn)?
(EX) Algebraic Computational Model(e.g., Real RAM, Blum-Shub-Smale model, Disc Shortest Path)
PROBLEM: Zero is trivial
(EX’) Abstract Operational Models(e.g., CG, Traub, Orientation, Ray shooting, Giftwrap)
PROBLEM: Zero is hidden
(AP) Analytic Computational Model (e.g., Ko, Weihrauch)
PROBLEM: Zero is undecidable
(AP’) Numerical Analysis Model (e.g., x⊕ y = (x + y)(1+ ε))
PROBLEM: Zero is abolished
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 28 / 113
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ENC
Four Computational Models for Geometry
How to compute in a Continuum (Rn)?
(EX) Algebraic Computational Model(e.g., Real RAM, Blum-Shub-Smale model, Disc Shortest Path)
PROBLEM: Zero is trivial
(EX’) Abstract Operational Models(e.g., CG, Traub, Orientation, Ray shooting, Giftwrap)
PROBLEM: Zero is hidden
(AP) Analytic Computational Model (e.g., Ko, Weihrauch)
PROBLEM: Zero is undecidable
(AP’) Numerical Analysis Model (e.g., x⊕ y = (x + y)(1+ ε))
PROBLEM: Zero is abolished
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 28 / 113
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ENC
Four Computational Models for Geometry
How to compute in a Continuum (Rn)?
(EX) Algebraic Computational Model(e.g., Real RAM, Blum-Shub-Smale model, Disc Shortest Path)
PROBLEM: Zero is trivial
(EX’) Abstract Operational Models(e.g., CG, Traub, Orientation, Ray shooting, Giftwrap)
PROBLEM: Zero is hidden
(AP) Analytic Computational Model (e.g., Ko, Weihrauch)
PROBLEM: Zero is undecidable
(AP’) Numerical Analysis Model (e.g., x⊕ y = (x + y)(1+ ε))
PROBLEM: Zero is abolished
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 28 / 113
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ENC
Four Computational Models for Geometry
How to compute in a Continuum (Rn)?
(EX) Algebraic Computational Model(e.g., Real RAM, Blum-Shub-Smale model, Disc Shortest Path)
PROBLEM: Zero is trivial
(EX’) Abstract Operational Models(e.g., CG, Traub, Orientation, Ray shooting, Giftwrap)
PROBLEM: Zero is hidden
(AP) Analytic Computational Model (e.g., Ko, Weihrauch)
PROBLEM: Zero is undecidable
(AP’) Numerical Analysis Model (e.g., x⊕ y = (x + y)(1+ ε))
PROBLEM: Zero is abolished
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 28 / 113
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ENC
Four Computational Models for Geometry
How to compute in a Continuum (Rn)?
(EX) Algebraic Computational Model(e.g., Real RAM, Blum-Shub-Smale model, Disc Shortest Path)
PROBLEM: Zero is trivial
(EX’) Abstract Operational Models(e.g., CG, Traub, Orientation, Ray shooting, Giftwrap)
PROBLEM: Zero is hidden
(AP) Analytic Computational Model (e.g., Ko, Weihrauch)
PROBLEM: Zero is undecidable
(AP’) Numerical Analysis Model (e.g., x⊕ y = (x + y)(1+ ε))
PROBLEM: Zero is abolished
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 28 / 113
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ENC
Four Computational Models for Geometry
How to compute in a Continuum (Rn)?
(EX) Algebraic Computational Model(e.g., Real RAM, Blum-Shub-Smale model, Disc Shortest Path)
PROBLEM: Zero is trivial
(EX’) Abstract Operational Models(e.g., CG, Traub, Orientation, Ray shooting, Giftwrap)
PROBLEM: Zero is hidden
(AP) Analytic Computational Model (e.g., Ko, Weihrauch)
PROBLEM: Zero is undecidable
(AP’) Numerical Analysis Model (e.g., x⊕ y = (x + y)(1+ ε))
PROBLEM: Zero is abolished
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 28 / 113
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ENC
Four Computational Models for Geometry
How to compute in a Continuum (Rn)?
(EX) Algebraic Computational Model(e.g., Real RAM, Blum-Shub-Smale model, Disc Shortest Path)
PROBLEM: Zero is trivial
(EX’) Abstract Operational Models(e.g., CG, Traub, Orientation, Ray shooting, Giftwrap)
PROBLEM: Zero is hidden
(AP) Analytic Computational Model (e.g., Ko, Weihrauch)
PROBLEM: Zero is undecidable
(AP’) Numerical Analysis Model (e.g., x⊕ y = (x + y)(1+ ε))
PROBLEM: Zero is abolished
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 28 / 113
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ENC
Four Computational Models for Geometry
How to compute in a Continuum (Rn)?
(EX) Algebraic Computational Model(e.g., Real RAM, Blum-Shub-Smale model, Disc Shortest Path)
PROBLEM: Zero is trivial
(EX’) Abstract Operational Models(e.g., CG, Traub, Orientation, Ray shooting, Giftwrap)
PROBLEM: Zero is hidden
(AP) Analytic Computational Model (e.g., Ko, Weihrauch)
PROBLEM: Zero is undecidable
(AP’) Numerical Analysis Model (e.g., x⊕ y = (x + y)(1+ ε))
PROBLEM: Zero is abolished
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 28 / 113
![Page 185: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/185.jpg)
ENC
Four Computational Models for Geometry
How to compute in a Continuum (Rn)?
(EX) Algebraic Computational Model(e.g., Real RAM, Blum-Shub-Smale model, Disc Shortest Path)
PROBLEM: Zero is trivial
(EX’) Abstract Operational Models(e.g., CG, Traub, Orientation, Ray shooting, Giftwrap)
PROBLEM: Zero is hidden
(AP) Analytic Computational Model (e.g., Ko, Weihrauch)
PROBLEM: Zero is undecidable
(AP’) Numerical Analysis Model (e.g., x⊕ y = (x + y)(1+ ε))
PROBLEM: Zero is abolished
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 28 / 113
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ENC
Other Issues
You cannot avoid the Zero Problem(EX) How do you implement R?
(EX’) We may abstract away too much cf. Problems (II) and (III)
(AP) Only continuous functions are computable Geometry is a discontinuous phenomenon
(AP’) Approximate geometry maybe harder than exact geometry Exercise: Program a geometric algorithm w/o equality test
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ENC
Other Issues
You cannot avoid the Zero Problem(EX) How do you implement R?
(EX’) We may abstract away too much cf. Problems (II) and (III)
(AP) Only continuous functions are computable Geometry is a discontinuous phenomenon
(AP’) Approximate geometry maybe harder than exact geometry Exercise: Program a geometric algorithm w/o equality test
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 29 / 113
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ENC
Other Issues
You cannot avoid the Zero Problem(EX) How do you implement R?
(EX’) We may abstract away too much cf. Problems (II) and (III)
(AP) Only continuous functions are computable Geometry is a discontinuous phenomenon
(AP’) Approximate geometry maybe harder than exact geometry Exercise: Program a geometric algorithm w/o equality test
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 29 / 113
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ENC
Other Issues
You cannot avoid the Zero Problem(EX) How do you implement R?
(EX’) We may abstract away too much cf. Problems (II) and (III)
(AP) Only continuous functions are computable Geometry is a discontinuous phenomenon
(AP’) Approximate geometry maybe harder than exact geometry Exercise: Program a geometric algorithm w/o equality test
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 29 / 113
![Page 190: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/190.jpg)
ENC
Other Issues
You cannot avoid the Zero Problem(EX) How do you implement R?
(EX’) We may abstract away too much cf. Problems (II) and (III)
(AP) Only continuous functions are computable Geometry is a discontinuous phenomenon
(AP’) Approximate geometry maybe harder than exact geometry Exercise: Program a geometric algorithm w/o equality test
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 29 / 113
![Page 191: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/191.jpg)
ENC
Other Issues
You cannot avoid the Zero Problem(EX) How do you implement R?
(EX’) We may abstract away too much cf. Problems (II) and (III)
(AP) Only continuous functions are computable Geometry is a discontinuous phenomenon
(AP’) Approximate geometry maybe harder than exact geometry Exercise: Program a geometric algorithm w/o equality test
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 29 / 113
![Page 192: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/192.jpg)
ENC
Other Issues
You cannot avoid the Zero Problem(EX) How do you implement R?
(EX’) We may abstract away too much cf. Problems (II) and (III)
(AP) Only continuous functions are computable Geometry is a discontinuous phenomenon
(AP’) Approximate geometry maybe harder than exact geometry Exercise: Program a geometric algorithm w/o equality test
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 29 / 113
![Page 193: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/193.jpg)
ENC
Other Issues
You cannot avoid the Zero Problem(EX) How do you implement R?
(EX’) We may abstract away too much cf. Problems (II) and (III)
(AP) Only continuous functions are computable Geometry is a discontinuous phenomenon
(AP’) Approximate geometry maybe harder than exact geometry Exercise: Program a geometric algorithm w/o equality test
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 29 / 113
![Page 194: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/194.jpg)
ENC
Other Issues
You cannot avoid the Zero Problem(EX) How do you implement R?
(EX’) We may abstract away too much cf. Problems (II) and (III)
(AP) Only continuous functions are computable Geometry is a discontinuous phenomenon
(AP’) Approximate geometry maybe harder than exact geometry Exercise: Program a geometric algorithm w/o equality test
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 29 / 113
![Page 195: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/195.jpg)
ENC
Duality in Numbers
Physics Analogy:
Discrete Continuous
Light particle waveR field metric spaceNumbers algebraic analytic
α =√
15−√
224 ≈ 0.0223√
15−√
224 is exact, but 0.0223 is more useful! WHY? Want the locus of α in the continuum JOKE: a physicist and an engineer were in a hot-air balloon...
How to capture this Duality? For exact computation, need algebraic representation. For analytic properties, need an approximation process What about deciding zero? (Algebraic or Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 30 / 113
![Page 196: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/196.jpg)
ENC
Duality in Numbers
Physics Analogy:
Discrete Continuous
Light particle waveR field metric spaceNumbers algebraic analytic
α =√
15−√
224 ≈ 0.0223√
15−√
224 is exact, but 0.0223 is more useful! WHY? Want the locus of α in the continuum JOKE: a physicist and an engineer were in a hot-air balloon...
How to capture this Duality? For exact computation, need algebraic representation. For analytic properties, need an approximation process What about deciding zero? (Algebraic or Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 30 / 113
![Page 197: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/197.jpg)
ENC
Duality in Numbers
Physics Analogy:
Discrete Continuous
Light particle waveR field metric spaceNumbers algebraic analytic
α =√
15−√
224 ≈ 0.0223√
15−√
224 is exact, but 0.0223 is more useful! WHY? Want the locus of α in the continuum JOKE: a physicist and an engineer were in a hot-air balloon...
How to capture this Duality? For exact computation, need algebraic representation. For analytic properties, need an approximation process What about deciding zero? (Algebraic or Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 30 / 113
![Page 198: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/198.jpg)
ENC
Duality in Numbers
Physics Analogy:
Discrete Continuous
Light particle waveR field metric spaceNumbers algebraic analytic
α =√
15−√
224 ≈ 0.0223√
15−√
224 is exact, but 0.0223 is more useful! WHY? Want the locus of α in the continuum JOKE: a physicist and an engineer were in a hot-air balloon...
How to capture this Duality? For exact computation, need algebraic representation. For analytic properties, need an approximation process What about deciding zero? (Algebraic or Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 30 / 113
![Page 199: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/199.jpg)
ENC
Duality in Numbers
Physics Analogy:
Discrete Continuous
Light particle waveR field metric spaceNumbers algebraic analytic
α =√
15−√
224 ≈ 0.0223√
15−√
224 is exact, but 0.0223 is more useful! WHY? Want the locus of α in the continuum JOKE: a physicist and an engineer were in a hot-air balloon...
How to capture this Duality? For exact computation, need algebraic representation. For analytic properties, need an approximation process What about deciding zero? (Algebraic or Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 30 / 113
![Page 200: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/200.jpg)
ENC
Duality in Numbers
Physics Analogy:
Discrete Continuous
Light particle waveR field metric spaceNumbers algebraic analytic
α =√
15−√
224 ≈ 0.0223√
15−√
224 is exact, but 0.0223 is more useful! WHY? Want the locus of α in the continuum JOKE: a physicist and an engineer were in a hot-air balloon...
How to capture this Duality? For exact computation, need algebraic representation. For analytic properties, need an approximation process What about deciding zero? (Algebraic or Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 30 / 113
![Page 201: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/201.jpg)
ENC
Duality in Numbers
Physics Analogy:
Discrete Continuous
Light particle waveR field metric spaceNumbers algebraic analytic
α =√
15−√
224 ≈ 0.0223√
15−√
224 is exact, but 0.0223 is more useful! WHY? Want the locus of α in the continuum JOKE: a physicist and an engineer were in a hot-air balloon...
How to capture this Duality? For exact computation, need algebraic representation. For analytic properties, need an approximation process What about deciding zero? (Algebraic or Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 30 / 113
![Page 202: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/202.jpg)
ENC
Duality in Numbers
Physics Analogy:
Discrete Continuous
Light particle waveR field metric spaceNumbers algebraic analytic
α =√
15−√
224 ≈ 0.0223√
15−√
224 is exact, but 0.0223 is more useful! WHY? Want the locus of α in the continuum JOKE: a physicist and an engineer were in a hot-air balloon...
How to capture this Duality? For exact computation, need algebraic representation. For analytic properties, need an approximation process What about deciding zero? (Algebraic or Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 30 / 113
![Page 203: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/203.jpg)
ENC
Duality in Numbers
Physics Analogy:
Discrete Continuous
Light particle waveR field metric spaceNumbers algebraic analytic
α =√
15−√
224 ≈ 0.0223√
15−√
224 is exact, but 0.0223 is more useful! WHY? Want the locus of α in the continuum JOKE: a physicist and an engineer were in a hot-air balloon...
How to capture this Duality? For exact computation, need algebraic representation. For analytic properties, need an approximation process What about deciding zero? (Algebraic or Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 30 / 113
![Page 204: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/204.jpg)
ENC
Duality in Numbers
Physics Analogy:
Discrete Continuous
Light particle waveR field metric spaceNumbers algebraic analytic
α =√
15−√
224 ≈ 0.0223√
15−√
224 is exact, but 0.0223 is more useful! WHY? Want the locus of α in the continuum JOKE: a physicist and an engineer were in a hot-air balloon...
How to capture this Duality? For exact computation, need algebraic representation. For analytic properties, need an approximation process What about deciding zero? (Algebraic or Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 30 / 113
![Page 205: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/205.jpg)
ENC
Duality in Numbers
Physics Analogy:
Discrete Continuous
Light particle waveR field metric spaceNumbers algebraic analytic
α =√
15−√
224 ≈ 0.0223√
15−√
224 is exact, but 0.0223 is more useful! WHY? Want the locus of α in the continuum JOKE: a physicist and an engineer were in a hot-air balloon...
How to capture this Duality? For exact computation, need algebraic representation. For analytic properties, need an approximation process What about deciding zero? (Algebraic or Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 30 / 113
![Page 206: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/206.jpg)
ENC
Duality in Numbers
Physics Analogy:
Discrete Continuous
Light particle waveR field metric spaceNumbers algebraic analytic
α =√
15−√
224 ≈ 0.0223√
15−√
224 is exact, but 0.0223 is more useful! WHY? Want the locus of α in the continuum JOKE: a physicist and an engineer were in a hot-air balloon...
How to capture this Duality? For exact computation, need algebraic representation. For analytic properties, need an approximation process What about deciding zero? (Algebraic or Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 30 / 113
![Page 207: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/207.jpg)
ENC
Duality in Numbers
Physics Analogy:
Discrete Continuous
Light particle waveR field metric spaceNumbers algebraic analytic
α =√
15−√
224 ≈ 0.0223√
15−√
224 is exact, but 0.0223 is more useful! WHY? Want the locus of α in the continuum JOKE: a physicist and an engineer were in a hot-air balloon...
How to capture this Duality? For exact computation, need algebraic representation. For analytic properties, need an approximation process What about deciding zero? (Algebraic or Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 30 / 113
![Page 208: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/208.jpg)
ENC
Duality in Numbers
Physics Analogy:
Discrete Continuous
Light particle waveR field metric spaceNumbers algebraic analytic
α =√
15−√
224 ≈ 0.0223√
15−√
224 is exact, but 0.0223 is more useful! WHY? Want the locus of α in the continuum JOKE: a physicist and an engineer were in a hot-air balloon...
How to capture this Duality? For exact computation, need algebraic representation. For analytic properties, need an approximation process What about deciding zero? (Algebraic or Numeric)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 30 / 113
![Page 209: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/209.jpg)
Mini Summary
Geometry is decided by Zeros
Zero is a special number
Numbers have a dual nature: need dual representation
Up Next : A General Solution
![Page 210: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/210.jpg)
Mini Summary
Geometry is decided by Zeros
Zero is a special number
Numbers have a dual nature: need dual representation
Up Next : A General Solution
![Page 211: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/211.jpg)
Mini Summary
Geometry is decided by Zeros
Zero is a special number
Numbers have a dual nature: need dual representation
Up Next : A General Solution
![Page 212: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/212.jpg)
Mini Summary
Geometry is decided by Zeros
Zero is a special number
Numbers have a dual nature: need dual representation
Up Next : A General Solution
![Page 213: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/213.jpg)
Mini Summary
Geometry is decided by Zeros
Zero is a special number
Numbers have a dual nature: need dual representation
Up Next : A General Solution
![Page 214: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/214.jpg)
Mini Summary
Geometry is decided by Zeros
Zero is a special number
Numbers have a dual nature: need dual representation
Up Next : A General Solution
![Page 215: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/215.jpg)
EGC
Coming Up Next
1 Introduction: What is Geometric Computation?
2 Five Examples of Geometric Computation
3 Exact Numeric Computation – A Synthesis
4 Exact Geometric Computation
5 Constructive Zero Bounds
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 32 / 113
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EGC
The Universal Solution (EGC)
Key Principle of Exact Geometric Computation (EGC)Algorithm = Sequence of Steps
Steps = Construction x := y +2; or Tests if x = 0 goto L
Geometric relations determined by Tests (Zero or Sign)
THUS: if Tests are error free , the Geometry is exact
Numerical robustness follows! Take-home message
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 33 / 113
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EGC
The Universal Solution (EGC)
Key Principle of Exact Geometric Computation (EGC)Algorithm = Sequence of Steps
Steps = Construction x := y +2; or Tests if x = 0 goto L
Geometric relations determined by Tests (Zero or Sign)
THUS: if Tests are error free , the Geometry is exact
Numerical robustness follows! Take-home message
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 33 / 113
![Page 218: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/218.jpg)
EGC
The Universal Solution (EGC)
Key Principle of Exact Geometric Computation (EGC)Algorithm = Sequence of Steps
Steps = Construction x := y +2; or Tests if x = 0 goto L
Geometric relations determined by Tests (Zero or Sign)
THUS: if Tests are error free , the Geometry is exact
Numerical robustness follows! Take-home message
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 33 / 113
![Page 219: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/219.jpg)
EGC
The Universal Solution (EGC)
Key Principle of Exact Geometric Computation (EGC)Algorithm = Sequence of Steps
Steps = Construction x := y +2; or Tests if x = 0 goto L
Geometric relations determined by Tests (Zero or Sign)
THUS: if Tests are error free , the Geometry is exact
Numerical robustness follows! Take-home message
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 33 / 113
![Page 220: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/220.jpg)
EGC
The Universal Solution (EGC)
Key Principle of Exact Geometric Computation (EGC)Algorithm = Sequence of Steps
Steps = Construction x := y +2; or Tests if x = 0 goto L
Geometric relations determined by Tests (Zero or Sign)
THUS: if Tests are error free , the Geometry is exact
Numerical robustness follows! Take-home message
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 33 / 113
![Page 221: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/221.jpg)
EGC
The Universal Solution (EGC)
Key Principle of Exact Geometric Computation (EGC)Algorithm = Sequence of Steps
Steps = Construction x := y +2; or Tests if x = 0 goto L
Geometric relations determined by Tests (Zero or Sign)
THUS: if Tests are error free , the Geometry is exact
Numerical robustness follows! Take-home message
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 33 / 113
![Page 222: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/222.jpg)
EGC
The Universal Solution (EGC)
Key Principle of Exact Geometric Computation (EGC)Algorithm = Sequence of Steps
Steps = Construction x := y +2; or Tests if x = 0 goto L
Geometric relations determined by Tests (Zero or Sign)
THUS: if Tests are error free , the Geometry is exact
Numerical robustness follows! Take-home message
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 33 / 113
![Page 223: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/223.jpg)
EGC
Implementing the Universal Solution (CoreLibrary)Any programmer can access this capability
#define Core Level 3
#include ”CORE.h”
.... Standard C++ Program ....
Numerical Accuracy API
Level 1: Machine Accuracy (int, long, float, double)
Level 2: Arbitrary Accuracy (BigInt, BigRat, BigFloat)
Level 3: Guaranteed Accuracy (Expr)
Program should compile at every Accuracy LevelYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 34 / 113
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EGC
Implementing the Universal Solution (CoreLibrary)Any programmer can access this capability
#define Core Level 3
#include ”CORE.h”
.... Standard C++ Program ....
Numerical Accuracy API
Level 1: Machine Accuracy (int, long, float, double)
Level 2: Arbitrary Accuracy (BigInt, BigRat, BigFloat)
Level 3: Guaranteed Accuracy (Expr)
Program should compile at every Accuracy LevelYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 34 / 113
![Page 225: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/225.jpg)
EGC
Implementing the Universal Solution (CoreLibrary)Any programmer can access this capability
#define Core Level 3
#include ”CORE.h”
.... Standard C++ Program ....
Numerical Accuracy API
Level 1: Machine Accuracy (int, long, float, double)
Level 2: Arbitrary Accuracy (BigInt, BigRat, BigFloat)
Level 3: Guaranteed Accuracy (Expr)
Program should compile at every Accuracy LevelYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 34 / 113
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EGC
Implementing the Universal Solution (CoreLibrary)Any programmer can access this capability
#define Core Level 3
#include ”CORE.h”
.... Standard C++ Program ....
Numerical Accuracy API
Level 1: Machine Accuracy (int, long, float, double)
Level 2: Arbitrary Accuracy (BigInt, BigRat, BigFloat)
Level 3: Guaranteed Accuracy (Expr)
Program should compile at every Accuracy LevelYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 34 / 113
![Page 227: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/227.jpg)
EGC
Implementing the Universal Solution (CoreLibrary)Any programmer can access this capability
#define Core Level 3
#include ”CORE.h”
.... Standard C++ Program ....
Numerical Accuracy API
Level 1: Machine Accuracy (int, long, float, double)
Level 2: Arbitrary Accuracy (BigInt, BigRat, BigFloat)
Level 3: Guaranteed Accuracy (Expr)
Program should compile at every Accuracy LevelYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 34 / 113
![Page 228: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/228.jpg)
EGC
Implementing the Universal Solution (CoreLibrary)Any programmer can access this capability
#define Core Level 3
#include ”CORE.h”
.... Standard C++ Program ....
Numerical Accuracy API
Level 1: Machine Accuracy (int, long, float, double)
Level 2: Arbitrary Accuracy (BigInt, BigRat, BigFloat)
Level 3: Guaranteed Accuracy (Expr)
Program should compile at every Accuracy LevelYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 34 / 113
![Page 229: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/229.jpg)
EGC
Implementing the Universal Solution (CoreLibrary)Any programmer can access this capability
#define Core Level 3
#include ”CORE.h”
.... Standard C++ Program ....
Numerical Accuracy API
Level 1: Machine Accuracy (int, long, float, double)
Level 2: Arbitrary Accuracy (BigInt, BigRat, BigFloat)
Level 3: Guaranteed Accuracy (Expr)
Program should compile at every Accuracy LevelYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 34 / 113
![Page 230: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/230.jpg)
EGC
Implementing the Universal Solution (CoreLibrary)Any programmer can access this capability
#define Core Level 3
#include ”CORE.h”
.... Standard C++ Program ....
Numerical Accuracy API
Level 1: Machine Accuracy (int, long, float, double)
Level 2: Arbitrary Accuracy (BigInt, BigRat, BigFloat)
Level 3: Guaranteed Accuracy (Expr)
Program should compile at every Accuracy LevelYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 34 / 113
![Page 231: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/231.jpg)
EGC
What is Achieved?
FeaturesRemoved numerical non-robustness from geometry (!)
Algorithm-independent solution to non-robustness
Standard (Euclidean) geometry (why important?)
Exactness in geometry ( can use approximate numbers !)
Implemented in LEDA , CGAL , Core Library
Other ImplicationsA new approach to do algebraic number computation
Euclidean Shortest Path need signs of expressions like∑100
i=1 ai√
bi .Standard algebraic approach is doomed
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 35 / 113
![Page 232: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/232.jpg)
EGC
What is Achieved?
FeaturesRemoved numerical non-robustness from geometry (!)
Algorithm-independent solution to non-robustness
Standard (Euclidean) geometry (why important?)
Exactness in geometry ( can use approximate numbers !)
Implemented in LEDA , CGAL , Core Library
Other ImplicationsA new approach to do algebraic number computation
Euclidean Shortest Path need signs of expressions like∑100
i=1 ai√
bi .Standard algebraic approach is doomed
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 35 / 113
![Page 233: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/233.jpg)
EGC
What is Achieved?
FeaturesRemoved numerical non-robustness from geometry (!)
Algorithm-independent solution to non-robustness
Standard (Euclidean) geometry (why important?)
Exactness in geometry ( can use approximate numbers !)
Implemented in LEDA , CGAL , Core Library
Other ImplicationsA new approach to do algebraic number computation
Euclidean Shortest Path need signs of expressions like∑100
i=1 ai√
bi .Standard algebraic approach is doomed
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 35 / 113
![Page 234: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/234.jpg)
EGC
What is Achieved?
FeaturesRemoved numerical non-robustness from geometry (!)
Algorithm-independent solution to non-robustness
Standard (Euclidean) geometry (why important?)
Exactness in geometry ( can use approximate numbers !)
Implemented in LEDA , CGAL , Core Library
Other ImplicationsA new approach to do algebraic number computation
Euclidean Shortest Path need signs of expressions like∑100
i=1 ai√
bi .Standard algebraic approach is doomed
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 35 / 113
![Page 235: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/235.jpg)
EGC
What is Achieved?
FeaturesRemoved numerical non-robustness from geometry (!)
Algorithm-independent solution to non-robustness
Standard (Euclidean) geometry (why important?)
Exactness in geometry ( can use approximate numbers !)
Implemented in LEDA , CGAL , Core Library
Other ImplicationsA new approach to do algebraic number computation
Euclidean Shortest Path need signs of expressions like∑100
i=1 ai√
bi .Standard algebraic approach is doomed
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 35 / 113
![Page 236: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/236.jpg)
EGC
What is Achieved?
FeaturesRemoved numerical non-robustness from geometry (!)
Algorithm-independent solution to non-robustness
Standard (Euclidean) geometry (why important?)
Exactness in geometry ( can use approximate numbers !)
Implemented in LEDA , CGAL , Core Library
Other ImplicationsA new approach to do algebraic number computation
Euclidean Shortest Path need signs of expressions like∑100
i=1 ai√
bi .Standard algebraic approach is doomed
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 35 / 113
![Page 237: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/237.jpg)
EGC
What is Achieved?
FeaturesRemoved numerical non-robustness from geometry (!)
Algorithm-independent solution to non-robustness
Standard (Euclidean) geometry (why important?)
Exactness in geometry ( can use approximate numbers !)
Implemented in LEDA , CGAL , Core Library
Other ImplicationsA new approach to do algebraic number computation
Euclidean Shortest Path need signs of expressions like∑100
i=1 ai√
bi .Standard algebraic approach is doomed
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 35 / 113
![Page 238: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/238.jpg)
EGC
What is Achieved?
FeaturesRemoved numerical non-robustness from geometry (!)
Algorithm-independent solution to non-robustness
Standard (Euclidean) geometry (why important?)
Exactness in geometry ( can use approximate numbers !)
Implemented in LEDA , CGAL , Core Library
Other ImplicationsA new approach to do algebraic number computation
Euclidean Shortest Path need signs of expressions like∑100
i=1 ai√
bi .Standard algebraic approach is doomed
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 35 / 113
![Page 239: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/239.jpg)
EGC
What is Achieved?
FeaturesRemoved numerical non-robustness from geometry (!)
Algorithm-independent solution to non-robustness
Standard (Euclidean) geometry (why important?)
Exactness in geometry ( can use approximate numbers !)
Implemented in LEDA , CGAL , Core Library
Other ImplicationsA new approach to do algebraic number computation
Euclidean Shortest Path need signs of expressions like∑100
i=1 ai√
bi .Standard algebraic approach is doomed
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 35 / 113
![Page 240: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/240.jpg)
Zero Bounds
Coming Up Next
1 Introduction: What is Geometric Computation?
2 Five Examples of Geometric Computation
3 Exact Numeric Computation – A Synthesis
4 Exact Geometric Computation
5 Constructive Zero Bounds
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Zero Bounds
Adaptive Zero Determination
Core of Core LibraryMust use numerical method based on Zero Bounds
Must NOT use algebraic methods!
Let Ω = +,−,×, . . .∪Z be a class of operators
ZERO(Ω) is the corresponding zero problem
Zero Bound for Ω is a function B : Expr(Ω)→R≥0
such that e ∈ Expr(Ω) is non-zero implies
|e|> B(e)
How to use zero bounds? Combine with approximation.
Zero Bound is the bottleneck only in case of zero.
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Zero Bounds
Adaptive Zero Determination
Core of Core LibraryMust use numerical method based on Zero Bounds
Must NOT use algebraic methods!
Let Ω = +,−,×, . . .∪Z be a class of operators
ZERO(Ω) is the corresponding zero problem
Zero Bound for Ω is a function B : Expr(Ω)→R≥0
such that e ∈ Expr(Ω) is non-zero implies
|e|> B(e)
How to use zero bounds? Combine with approximation.
Zero Bound is the bottleneck only in case of zero.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 37 / 113
![Page 243: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/243.jpg)
Zero Bounds
Adaptive Zero Determination
Core of Core LibraryMust use numerical method based on Zero Bounds
Must NOT use algebraic methods!
Let Ω = +,−,×, . . .∪Z be a class of operators
ZERO(Ω) is the corresponding zero problem
Zero Bound for Ω is a function B : Expr(Ω)→R≥0
such that e ∈ Expr(Ω) is non-zero implies
|e|> B(e)
How to use zero bounds? Combine with approximation.
Zero Bound is the bottleneck only in case of zero.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 37 / 113
![Page 244: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/244.jpg)
Zero Bounds
Adaptive Zero Determination
Core of Core LibraryMust use numerical method based on Zero Bounds
Must NOT use algebraic methods!
Let Ω = +,−,×, . . .∪Z be a class of operators
ZERO(Ω) is the corresponding zero problem
Zero Bound for Ω is a function B : Expr(Ω)→R≥0
such that e ∈ Expr(Ω) is non-zero implies
|e|> B(e)
How to use zero bounds? Combine with approximation.
Zero Bound is the bottleneck only in case of zero.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 37 / 113
![Page 245: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/245.jpg)
Zero Bounds
Adaptive Zero Determination
Core of Core LibraryMust use numerical method based on Zero Bounds
Must NOT use algebraic methods!
Let Ω = +,−,×, . . .∪Z be a class of operators
ZERO(Ω) is the corresponding zero problem
Zero Bound for Ω is a function B : Expr(Ω)→R≥0
such that e ∈ Expr(Ω) is non-zero implies
|e|> B(e)
How to use zero bounds? Combine with approximation.
Zero Bound is the bottleneck only in case of zero.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 37 / 113
![Page 246: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/246.jpg)
Zero Bounds
Adaptive Zero Determination
Core of Core LibraryMust use numerical method based on Zero Bounds
Must NOT use algebraic methods!
Let Ω = +,−,×, . . .∪Z be a class of operators
ZERO(Ω) is the corresponding zero problem
Zero Bound for Ω is a function B : Expr(Ω)→R≥0
such that e ∈ Expr(Ω) is non-zero implies
|e|> B(e)
How to use zero bounds? Combine with approximation.
Zero Bound is the bottleneck only in case of zero.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 37 / 113
![Page 247: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/247.jpg)
Zero Bounds
Adaptive Zero Determination
Core of Core LibraryMust use numerical method based on Zero Bounds
Must NOT use algebraic methods!
Let Ω = +,−,×, . . .∪Z be a class of operators
ZERO(Ω) is the corresponding zero problem
Zero Bound for Ω is a function B : Expr(Ω)→R≥0
such that e ∈ Expr(Ω) is non-zero implies
|e|> B(e)
How to use zero bounds? Combine with approximation.
Zero Bound is the bottleneck only in case of zero.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 37 / 113
![Page 248: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/248.jpg)
Zero Bounds
Some Constructive Bounds
Degree-Measure Bounds [Mignotte (1982)], [Sekigawa (1997)]
Degree-Height, Degree-Length [Yap-Dube (1994)]
BFMS Bound [Burnikel et al (1989)]
Eigenvalue Bounds [Scheinerman (2000)]
Conjugate Bounds [Li-Yap (2001)]
BFMSS Bound [Burnikel et al (2001)] One of the best bounds
k-ary Method [Pion-Yap (2002)] Idea: division is bad. k-ary numbers are good
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Zero Bounds
An Example
Consider the e =√
x +√
y−√
x + y +2√
xy .
Assume x = a/b and y = c/d where a,b,c,d are L-bit integers.
Then Li-Yap Bound is 28L+60 bits, BFMSS is 96L+30 andDegree-Measure is 80L+56.
Timing in seconds (Core 1.6):
L 50 100 500 5000BFMS 0.637 9.12 101.9 202.9
Measure 0.063 0.07 1.93 15.26BFMSS 0.073 0.61 1.95 15.41Li-Yap 0.013 0.07 1.88 1.89
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Zero Bounds
An Example
Consider the e =√
x +√
y−√
x + y +2√
xy .
Assume x = a/b and y = c/d where a,b,c,d are L-bit integers.
Then Li-Yap Bound is 28L+60 bits, BFMSS is 96L+30 andDegree-Measure is 80L+56.
Timing in seconds (Core 1.6):
L 50 100 500 5000BFMS 0.637 9.12 101.9 202.9
Measure 0.063 0.07 1.93 15.26BFMSS 0.073 0.61 1.95 15.41Li-Yap 0.013 0.07 1.88 1.89
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Zero Bounds
An Example
Consider the e =√
x +√
y−√
x + y +2√
xy .
Assume x = a/b and y = c/d where a,b,c,d are L-bit integers.
Then Li-Yap Bound is 28L+60 bits, BFMSS is 96L+30 andDegree-Measure is 80L+56.
Timing in seconds (Core 1.6):
L 50 100 500 5000BFMS 0.637 9.12 101.9 202.9
Measure 0.063 0.07 1.93 15.26BFMSS 0.073 0.61 1.95 15.41Li-Yap 0.013 0.07 1.88 1.89
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Zero Bounds
An Example
Consider the e =√
x +√
y−√
x + y +2√
xy .
Assume x = a/b and y = c/d where a,b,c,d are L-bit integers.
Then Li-Yap Bound is 28L+60 bits, BFMSS is 96L+30 andDegree-Measure is 80L+56.
Timing in seconds (Core 1.6):
L 50 100 500 5000BFMS 0.637 9.12 101.9 202.9
Measure 0.063 0.07 1.93 15.26BFMSS 0.073 0.61 1.95 15.41Li-Yap 0.013 0.07 1.88 1.89
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Zero Bounds
An Example
Consider the e =√
x +√
y−√
x + y +2√
xy .
Assume x = a/b and y = c/d where a,b,c,d are L-bit integers.
Then Li-Yap Bound is 28L+60 bits, BFMSS is 96L+30 andDegree-Measure is 80L+56.
Timing in seconds (Core 1.6):
L 50 100 500 5000BFMS 0.637 9.12 101.9 202.9
Measure 0.063 0.07 1.93 15.26BFMSS 0.073 0.61 1.95 15.41Li-Yap 0.013 0.07 1.88 1.89
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Zero Bounds
An Example
Consider the e =√
x +√
y−√
x + y +2√
xy .
Assume x = a/b and y = c/d where a,b,c,d are L-bit integers.
Then Li-Yap Bound is 28L+60 bits, BFMSS is 96L+30 andDegree-Measure is 80L+56.
Timing in seconds (Core 1.6):
L 50 100 500 5000BFMS 0.637 9.12 101.9 202.9
Measure 0.063 0.07 1.93 15.26BFMSS 0.073 0.61 1.95 15.41Li-Yap 0.013 0.07 1.88 1.89
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Mini Summary
There is a “Universal Solution” for synthesizing the Algebraic andthe Geometric viewpoints
Slogan: Algebraic computation without Algebra
(Use approximations & zero bounds)
PUZZLE 3: What was the answer to PUZZLE 2?
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Mini Summary
There is a “Universal Solution” for synthesizing the Algebraic andthe Geometric viewpoints
Slogan: Algebraic computation without Algebra
(Use approximations & zero bounds)
PUZZLE 3: What was the answer to PUZZLE 2?
![Page 257: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/257.jpg)
Mini Summary
There is a “Universal Solution” for synthesizing the Algebraic andthe Geometric viewpoints
Slogan: Algebraic computation without Algebra
(Use approximations & zero bounds)
PUZZLE 3: What was the answer to PUZZLE 2?
![Page 258: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/258.jpg)
Mini Summary
There is a “Universal Solution” for synthesizing the Algebraic andthe Geometric viewpoints
Slogan: Algebraic computation without Algebra
(Use approximations & zero bounds)
PUZZLE 3: What was the answer to PUZZLE 2?
![Page 259: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/259.jpg)
Mini Summary
There is a “Universal Solution” for synthesizing the Algebraic andthe Geometric viewpoints
Slogan: Algebraic computation without Algebra
(Use approximations & zero bounds)
PUZZLE 3: What was the answer to PUZZLE 2?
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Summary of Lecture 1
Nature of Geometric Computation: Discrete as well as Continuous Algebraic as well as Analytic
It is possible to provide a fairly general solution (ENC) thatcombines the dual nature of numbers
Up Next : Directly design ENC algorithms
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Summary of Lecture 1
Nature of Geometric Computation: Discrete as well as Continuous Algebraic as well as Analytic
It is possible to provide a fairly general solution (ENC) thatcombines the dual nature of numbers
Up Next : Directly design ENC algorithms
![Page 262: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/262.jpg)
Summary of Lecture 1
Nature of Geometric Computation: Discrete as well as Continuous Algebraic as well as Analytic
It is possible to provide a fairly general solution (ENC) thatcombines the dual nature of numbers
Up Next : Directly design ENC algorithms
![Page 263: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/263.jpg)
Summary of Lecture 1
Nature of Geometric Computation: Discrete as well as Continuous Algebraic as well as Analytic
It is possible to provide a fairly general solution (ENC) thatcombines the dual nature of numbers
Up Next : Directly design ENC algorithms
![Page 264: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/264.jpg)
Summary of Lecture 1
Nature of Geometric Computation: Discrete as well as Continuous Algebraic as well as Analytic
It is possible to provide a fairly general solution (ENC) thatcombines the dual nature of numbers
Up Next : Directly design ENC algorithms
![Page 265: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/265.jpg)
Summary of Lecture 1
Nature of Geometric Computation: Discrete as well as Continuous Algebraic as well as Analytic
It is possible to provide a fairly general solution (ENC) thatcombines the dual nature of numbers
Up Next : Directly design ENC algorithms
![Page 266: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/266.jpg)
Summary of Lecture 1
Nature of Geometric Computation: Discrete as well as Continuous Algebraic as well as Analytic
It is possible to provide a fairly general solution (ENC) thatcombines the dual nature of numbers
Up Next : Directly design ENC algorithms
![Page 267: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/267.jpg)
PART 2
Explicitization and Subdivision
“It can be of no practical use to know that π is irrational, but if we canknow, it surely would be intolerable not to know.”
— E.C. Titchmarsh
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 42 / 113
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Introduction
Coming Up Next
6 Introduction
7 Review of Subdivision Algorithms
8 Cxy Algorithm
9 Extensions of Cxy
10 How to treat Boundary
11 How to treat Singularity
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Introduction
Towards Exact Numerical Computation (ENC)
Beyond the Universal Solution
Design algorithms directly incorporating the principles of EGC
What do we need? What are its features? It must be numerical in nature It must be arbitrary precision It must respect zero It must be adaptive
⋆ actively control precision⋆ exploit filters
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 44 / 113
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Introduction
Towards Exact Numerical Computation (ENC)
Beyond the Universal Solution
Design algorithms directly incorporating the principles of EGC
What do we need? What are its features? It must be numerical in nature It must be arbitrary precision It must respect zero It must be adaptive
⋆ actively control precision⋆ exploit filters
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 44 / 113
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Introduction
Towards Exact Numerical Computation (ENC)
Beyond the Universal Solution
Design algorithms directly incorporating the principles of EGC
What do we need? What are its features? It must be numerical in nature It must be arbitrary precision It must respect zero It must be adaptive
⋆ actively control precision⋆ exploit filters
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 44 / 113
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Introduction
Towards Exact Numerical Computation (ENC)
Beyond the Universal Solution
Design algorithms directly incorporating the principles of EGC
What do we need? What are its features? It must be numerical in nature It must be arbitrary precision It must respect zero It must be adaptive
⋆ actively control precision⋆ exploit filters
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 44 / 113
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Introduction
Towards Exact Numerical Computation (ENC)
Beyond the Universal Solution
Design algorithms directly incorporating the principles of EGC
What do we need? What are its features? It must be numerical in nature It must be arbitrary precision It must respect zero It must be adaptive
⋆ actively control precision⋆ exploit filters
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 44 / 113
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Introduction
Towards Exact Numerical Computation (ENC)
Beyond the Universal Solution
Design algorithms directly incorporating the principles of EGC
What do we need? What are its features? It must be numerical in nature It must be arbitrary precision It must respect zero It must be adaptive
⋆ actively control precision⋆ exploit filters
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 44 / 113
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Introduction
Towards Exact Numerical Computation (ENC)
Beyond the Universal Solution
Design algorithms directly incorporating the principles of EGC
What do we need? What are its features? It must be numerical in nature It must be arbitrary precision It must respect zero It must be adaptive
⋆ actively control precision⋆ exploit filters
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 44 / 113
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Introduction
Towards Exact Numerical Computation (ENC)
Beyond the Universal Solution
Design algorithms directly incorporating the principles of EGC
What do we need? What are its features? It must be numerical in nature It must be arbitrary precision It must respect zero It must be adaptive
⋆ actively control precision⋆ exploit filters
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 44 / 113
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Introduction
Towards Exact Numerical Computation (ENC)
Beyond the Universal Solution
Design algorithms directly incorporating the principles of EGC
What do we need? What are its features? It must be numerical in nature It must be arbitrary precision It must respect zero It must be adaptive
⋆ actively control precision⋆ exploit filters
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 44 / 113
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Introduction
Computational Ring ApproachComputational Ring (D,0,1,+,−,×,÷2)
D is countable, dense subset of R
D is a ring extension of Z
Efficient representation ρ : 0,1∗ ≻D for implementing ringoperations, and exact comparison.
Examples of D
BigFloats or dyadic numbers:
F := m2n : m,n ∈Z= Z[1
2]
Rationals: Q (avoid, if possible)
Real Algebraic Numbers: A (AVOID!)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 45 / 113
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Introduction
Computational Ring ApproachComputational Ring (D,0,1,+,−,×,÷2)
D is countable, dense subset of R
D is a ring extension of Z
Efficient representation ρ : 0,1∗ ≻D for implementing ringoperations, and exact comparison.
Examples of D
BigFloats or dyadic numbers:
F := m2n : m,n ∈Z= Z[1
2]
Rationals: Q (avoid, if possible)
Real Algebraic Numbers: A (AVOID!)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 45 / 113
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Introduction
Computational Ring ApproachComputational Ring (D,0,1,+,−,×,÷2)
D is countable, dense subset of R
D is a ring extension of Z
Efficient representation ρ : 0,1∗ ≻D for implementing ringoperations, and exact comparison.
Examples of D
BigFloats or dyadic numbers:
F := m2n : m,n ∈Z= Z[1
2]
Rationals: Q (avoid, if possible)
Real Algebraic Numbers: A (AVOID!)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 45 / 113
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Introduction
Computational Ring ApproachComputational Ring (D,0,1,+,−,×,÷2)
D is countable, dense subset of R
D is a ring extension of Z
Efficient representation ρ : 0,1∗ ≻D for implementing ringoperations, and exact comparison.
Examples of D
BigFloats or dyadic numbers:
F := m2n : m,n ∈Z= Z[1
2]
Rationals: Q (avoid, if possible)
Real Algebraic Numbers: A (AVOID!)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 45 / 113
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Introduction
Computational Ring ApproachComputational Ring (D,0,1,+,−,×,÷2)
D is countable, dense subset of R
D is a ring extension of Z
Efficient representation ρ : 0,1∗ ≻D for implementing ringoperations, and exact comparison.
Examples of D
BigFloats or dyadic numbers:
F := m2n : m,n ∈Z= Z[1
2]
Rationals: Q (avoid, if possible)
Real Algebraic Numbers: A (AVOID!)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 45 / 113
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Introduction
Computational Ring ApproachComputational Ring (D,0,1,+,−,×,÷2)
D is countable, dense subset of R
D is a ring extension of Z
Efficient representation ρ : 0,1∗ ≻D for implementing ringoperations, and exact comparison.
Examples of D
BigFloats or dyadic numbers:
F := m2n : m,n ∈Z= Z[1
2]
Rationals: Q (avoid, if possible)
Real Algebraic Numbers: A (AVOID!)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 45 / 113
![Page 284: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/284.jpg)
Introduction
Computational Ring ApproachComputational Ring (D,0,1,+,−,×,÷2)
D is countable, dense subset of R
D is a ring extension of Z
Efficient representation ρ : 0,1∗ ≻D for implementing ringoperations, and exact comparison.
Examples of D
BigFloats or dyadic numbers:
F := m2n : m,n ∈Z= Z[1
2]
Rationals: Q (avoid, if possible)
Real Algebraic Numbers: A (AVOID!)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 45 / 113
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Introduction
Computational Ring ApproachComputational Ring (D,0,1,+,−,×,÷2)
D is countable, dense subset of R
D is a ring extension of Z
Efficient representation ρ : 0,1∗ ≻D for implementing ringoperations, and exact comparison.
Examples of D
BigFloats or dyadic numbers:
F := m2n : m,n ∈Z= Z[1
2]
Rationals: Q (avoid, if possible)
Real Algebraic Numbers: A (AVOID!)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 45 / 113
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Introduction
What else is needed in ENC Algorithms?
IntervalsD: set of dyadic intervals
Dn: set of n-boxes
Box FunctionsLet f : Dm→D.
Box function f : m(D)→ (D) (1) Inclusion: f (B)⊆ f (B). (2) Convergence: limi→∞ f (Bi) = f (limi→∞ Bi).
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Introduction
What else is needed in ENC Algorithms?
IntervalsD: set of dyadic intervals
Dn: set of n-boxes
Box FunctionsLet f : Dm→D.
Box function f : m(D)→ (D) (1) Inclusion: f (B)⊆ f (B). (2) Convergence: limi→∞ f (Bi) = f (limi→∞ Bi).
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 46 / 113
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Introduction
What else is needed in ENC Algorithms?
IntervalsD: set of dyadic intervals
Dn: set of n-boxes
Box FunctionsLet f : Dm→D.
Box function f : m(D)→ (D) (1) Inclusion: f (B)⊆ f (B). (2) Convergence: limi→∞ f (Bi) = f (limi→∞ Bi).
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 46 / 113
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Introduction
What else is needed in ENC Algorithms?
IntervalsD: set of dyadic intervals
Dn: set of n-boxes
Box FunctionsLet f : Dm→D.
Box function f : m(D)→ (D) (1) Inclusion: f (B)⊆ f (B). (2) Convergence: limi→∞ f (Bi) = f (limi→∞ Bi).
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 46 / 113
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Introduction
What else is needed in ENC Algorithms?
IntervalsD: set of dyadic intervals
Dn: set of n-boxes
Box FunctionsLet f : Dm→D.
Box function f : m(D)→ (D) (1) Inclusion: f (B)⊆ f (B). (2) Convergence: limi→∞ f (Bi) = f (limi→∞ Bi).
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 46 / 113
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Introduction
What else is needed in ENC Algorithms?
IntervalsD: set of dyadic intervals
Dn: set of n-boxes
Box FunctionsLet f : Dm→D.
Box function f : m(D)→ (D) (1) Inclusion: f (B)⊆ f (B). (2) Convergence: limi→∞ f (Bi) = f (limi→∞ Bi).
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 46 / 113
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Introduction
What else is needed in ENC Algorithms?
IntervalsD: set of dyadic intervals
Dn: set of n-boxes
Box FunctionsLet f : Dm→D.
Box function f : m(D)→ (D) (1) Inclusion: f (B)⊆ f (B). (2) Convergence: limi→∞ f (Bi) = f (limi→∞ Bi).
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 46 / 113
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Introduction
What else is needed in ENC Algorithms?
IntervalsD: set of dyadic intervals
Dn: set of n-boxes
Box FunctionsLet f : Dm→D.
Box function f : m(D)→ (D) (1) Inclusion: f (B)⊆ f (B). (2) Convergence: limi→∞ f (Bi) = f (limi→∞ Bi).
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 46 / 113
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Introduction
What else is needed in ENC Algorithms?
IntervalsD: set of dyadic intervals
Dn: set of n-boxes
Box FunctionsLet f : Dm→D.
Box function f : m(D)→ (D) (1) Inclusion: f (B)⊆ f (B). (2) Convergence: limi→∞ f (Bi) = f (limi→∞ Bi).
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 46 / 113
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Introduction
Our Target: Explicitization ProblemsFrom Implicit to Explicit Representation
Mesh generation [Problem (IV)]
Discrete Morse-Smale complex [Problem (V)]
Arrangement of hypersurfaces
Voronoi diagram of a collection of objects
Cell complex approximation of algebraic variety
Representation of Flow fields
From Parameter Space to Ambient SpaceWhy this class? Interface between Continuous and Discrete!
ENC Algorithms is ideal for this class
Interplay of Topological and Geometric requirements
Domain subdivision as the general algorithmic paradigmYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 47 / 113
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Introduction
Our Target: Explicitization ProblemsFrom Implicit to Explicit Representation
Mesh generation [Problem (IV)]
Discrete Morse-Smale complex [Problem (V)]
Arrangement of hypersurfaces
Voronoi diagram of a collection of objects
Cell complex approximation of algebraic variety
Representation of Flow fields
From Parameter Space to Ambient SpaceWhy this class? Interface between Continuous and Discrete!
ENC Algorithms is ideal for this class
Interplay of Topological and Geometric requirements
Domain subdivision as the general algorithmic paradigmYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 47 / 113
![Page 297: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/297.jpg)
Introduction
Our Target: Explicitization ProblemsFrom Implicit to Explicit Representation
Mesh generation [Problem (IV)]
Discrete Morse-Smale complex [Problem (V)]
Arrangement of hypersurfaces
Voronoi diagram of a collection of objects
Cell complex approximation of algebraic variety
Representation of Flow fields
From Parameter Space to Ambient SpaceWhy this class? Interface between Continuous and Discrete!
ENC Algorithms is ideal for this class
Interplay of Topological and Geometric requirements
Domain subdivision as the general algorithmic paradigmYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 47 / 113
![Page 298: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/298.jpg)
Introduction
Our Target: Explicitization ProblemsFrom Implicit to Explicit Representation
Mesh generation [Problem (IV)]
Discrete Morse-Smale complex [Problem (V)]
Arrangement of hypersurfaces
Voronoi diagram of a collection of objects
Cell complex approximation of algebraic variety
Representation of Flow fields
From Parameter Space to Ambient SpaceWhy this class? Interface between Continuous and Discrete!
ENC Algorithms is ideal for this class
Interplay of Topological and Geometric requirements
Domain subdivision as the general algorithmic paradigmYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 47 / 113
![Page 299: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/299.jpg)
Introduction
Our Target: Explicitization ProblemsFrom Implicit to Explicit Representation
Mesh generation [Problem (IV)]
Discrete Morse-Smale complex [Problem (V)]
Arrangement of hypersurfaces
Voronoi diagram of a collection of objects
Cell complex approximation of algebraic variety
Representation of Flow fields
From Parameter Space to Ambient SpaceWhy this class? Interface between Continuous and Discrete!
ENC Algorithms is ideal for this class
Interplay of Topological and Geometric requirements
Domain subdivision as the general algorithmic paradigmYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 47 / 113
![Page 300: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/300.jpg)
Introduction
Our Target: Explicitization ProblemsFrom Implicit to Explicit Representation
Mesh generation [Problem (IV)]
Discrete Morse-Smale complex [Problem (V)]
Arrangement of hypersurfaces
Voronoi diagram of a collection of objects
Cell complex approximation of algebraic variety
Representation of Flow fields
From Parameter Space to Ambient SpaceWhy this class? Interface between Continuous and Discrete!
ENC Algorithms is ideal for this class
Interplay of Topological and Geometric requirements
Domain subdivision as the general algorithmic paradigmYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 47 / 113
![Page 301: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/301.jpg)
Introduction
Our Target: Explicitization ProblemsFrom Implicit to Explicit Representation
Mesh generation [Problem (IV)]
Discrete Morse-Smale complex [Problem (V)]
Arrangement of hypersurfaces
Voronoi diagram of a collection of objects
Cell complex approximation of algebraic variety
Representation of Flow fields
From Parameter Space to Ambient SpaceWhy this class? Interface between Continuous and Discrete!
ENC Algorithms is ideal for this class
Interplay of Topological and Geometric requirements
Domain subdivision as the general algorithmic paradigmYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 47 / 113
![Page 302: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/302.jpg)
Introduction
Our Target: Explicitization ProblemsFrom Implicit to Explicit Representation
Mesh generation [Problem (IV)]
Discrete Morse-Smale complex [Problem (V)]
Arrangement of hypersurfaces
Voronoi diagram of a collection of objects
Cell complex approximation of algebraic variety
Representation of Flow fields
From Parameter Space to Ambient SpaceWhy this class? Interface between Continuous and Discrete!
ENC Algorithms is ideal for this class
Interplay of Topological and Geometric requirements
Domain subdivision as the general algorithmic paradigmYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 47 / 113
![Page 303: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/303.jpg)
Introduction
Our Target: Explicitization ProblemsFrom Implicit to Explicit Representation
Mesh generation [Problem (IV)]
Discrete Morse-Smale complex [Problem (V)]
Arrangement of hypersurfaces
Voronoi diagram of a collection of objects
Cell complex approximation of algebraic variety
Representation of Flow fields
From Parameter Space to Ambient SpaceWhy this class? Interface between Continuous and Discrete!
ENC Algorithms is ideal for this class
Interplay of Topological and Geometric requirements
Domain subdivision as the general algorithmic paradigmYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 47 / 113
![Page 304: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/304.jpg)
Introduction
Our Target: Explicitization ProblemsFrom Implicit to Explicit Representation
Mesh generation [Problem (IV)]
Discrete Morse-Smale complex [Problem (V)]
Arrangement of hypersurfaces
Voronoi diagram of a collection of objects
Cell complex approximation of algebraic variety
Representation of Flow fields
From Parameter Space to Ambient SpaceWhy this class? Interface between Continuous and Discrete!
ENC Algorithms is ideal for this class
Interplay of Topological and Geometric requirements
Domain subdivision as the general algorithmic paradigmYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 47 / 113
![Page 305: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/305.jpg)
Introduction
Our Target: Explicitization ProblemsFrom Implicit to Explicit Representation
Mesh generation [Problem (IV)]
Discrete Morse-Smale complex [Problem (V)]
Arrangement of hypersurfaces
Voronoi diagram of a collection of objects
Cell complex approximation of algebraic variety
Representation of Flow fields
From Parameter Space to Ambient SpaceWhy this class? Interface between Continuous and Discrete!
ENC Algorithms is ideal for this class
Interplay of Topological and Geometric requirements
Domain subdivision as the general algorithmic paradigmYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 47 / 113
![Page 306: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/306.jpg)
Introduction
Our Target: Explicitization ProblemsFrom Implicit to Explicit Representation
Mesh generation [Problem (IV)]
Discrete Morse-Smale complex [Problem (V)]
Arrangement of hypersurfaces
Voronoi diagram of a collection of objects
Cell complex approximation of algebraic variety
Representation of Flow fields
From Parameter Space to Ambient SpaceWhy this class? Interface between Continuous and Discrete!
ENC Algorithms is ideal for this class
Interplay of Topological and Geometric requirements
Domain subdivision as the general algorithmic paradigmYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 47 / 113
![Page 307: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/307.jpg)
Introduction
Our Target: Explicitization ProblemsFrom Implicit to Explicit Representation
Mesh generation [Problem (IV)]
Discrete Morse-Smale complex [Problem (V)]
Arrangement of hypersurfaces
Voronoi diagram of a collection of objects
Cell complex approximation of algebraic variety
Representation of Flow fields
From Parameter Space to Ambient SpaceWhy this class? Interface between Continuous and Discrete!
ENC Algorithms is ideal for this class
Interplay of Topological and Geometric requirements
Domain subdivision as the general algorithmic paradigmYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 47 / 113
![Page 308: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/308.jpg)
Introduction
Three Approaches to Meshing, I:
1. Algebraic Approach
Projection Based (Refinements of CAD)E.g., [Mourrain and Tecourt (2005); Cheng, Gao, and Li (2005) ]
Algebraic Subdivision SchemesE.g., [Wolpert and Seidel (2005)]
Properties Exact; complete (usually); slow (in general); hard to
implement
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Introduction
Three Approaches to Meshing, I:
1. Algebraic Approach
Projection Based (Refinements of CAD)E.g., [Mourrain and Tecourt (2005); Cheng, Gao, and Li (2005) ]
Algebraic Subdivision SchemesE.g., [Wolpert and Seidel (2005)]
Properties Exact; complete (usually); slow (in general); hard to
implement
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Introduction
Three Approaches to Meshing, I:
1. Algebraic Approach
Projection Based (Refinements of CAD)E.g., [Mourrain and Tecourt (2005); Cheng, Gao, and Li (2005) ]
Algebraic Subdivision SchemesE.g., [Wolpert and Seidel (2005)]
Properties Exact; complete (usually); slow (in general); hard to
implement
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Introduction
Three Approaches to Meshing, I:
1. Algebraic Approach
Projection Based (Refinements of CAD)E.g., [Mourrain and Tecourt (2005); Cheng, Gao, and Li (2005) ]
Algebraic Subdivision SchemesE.g., [Wolpert and Seidel (2005)]
Properties Exact; complete (usually); slow (in general); hard to
implement
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Introduction
Three Approaches to Meshing, I:
1. Algebraic Approach
Projection Based (Refinements of CAD)E.g., [Mourrain and Tecourt (2005); Cheng, Gao, and Li (2005) ]
Algebraic Subdivision SchemesE.g., [Wolpert and Seidel (2005)]
Properties Exact; complete (usually); slow (in general); hard to
implement
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Introduction
Three Approaches to Meshing, II:
2. Geometric Approach
Sampling Approach (Ray Shooting)E.g., [Boissonnat & Oudot (2005); Cheng, Dey, Ramos and Ray (2004)]
Morse theoryE.g., [Stander & Hart (1997); Boissonnat, Cohen-Steiner & Vegter (2004)]
Properties Implementation gaps; requires “niceness conditions”
(Morseness, non-singularity, etc)
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Introduction
Three Approaches to Meshing, II:
2. Geometric Approach
Sampling Approach (Ray Shooting)E.g., [Boissonnat & Oudot (2005); Cheng, Dey, Ramos and Ray (2004)]
Morse theoryE.g., [Stander & Hart (1997); Boissonnat, Cohen-Steiner & Vegter (2004)]
Properties Implementation gaps; requires “niceness conditions”
(Morseness, non-singularity, etc)
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Introduction
Three Approaches to Meshing, II:
2. Geometric Approach
Sampling Approach (Ray Shooting)E.g., [Boissonnat & Oudot (2005); Cheng, Dey, Ramos and Ray (2004)]
Morse theoryE.g., [Stander & Hart (1997); Boissonnat, Cohen-Steiner & Vegter (2004)]
Properties Implementation gaps; requires “niceness conditions”
(Morseness, non-singularity, etc)
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Introduction
Three Approaches to Meshing, II:
2. Geometric Approach
Sampling Approach (Ray Shooting)E.g., [Boissonnat & Oudot (2005); Cheng, Dey, Ramos and Ray (2004)]
Morse theoryE.g., [Stander & Hart (1997); Boissonnat, Cohen-Steiner & Vegter (2004)]
Properties Implementation gaps; requires “niceness conditions”
(Morseness, non-singularity, etc)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 49 / 113
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Introduction
Three Approaches to Meshing, II:
2. Geometric Approach
Sampling Approach (Ray Shooting)E.g., [Boissonnat & Oudot (2005); Cheng, Dey, Ramos and Ray (2004)]
Morse theoryE.g., [Stander & Hart (1997); Boissonnat, Cohen-Steiner & Vegter (2004)]
Properties Implementation gaps; requires “niceness conditions”
(Morseness, non-singularity, etc)
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Introduction
Three Approaches to Meshing, III:
3. Numeric Approach
Curve Tracing Literature[Ratschek & Rokne (2005)]
Subdivision Approach[Marching Cube (1987); Snyder (1992); Plantinga & Vegter (2004)]
Properties Practical; easy to implement; adaptive; incomplete (until
recently)
This is our focus
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Introduction
Three Approaches to Meshing, III:
3. Numeric Approach
Curve Tracing Literature[Ratschek & Rokne (2005)]
Subdivision Approach[Marching Cube (1987); Snyder (1992); Plantinga & Vegter (2004)]
Properties Practical; easy to implement; adaptive; incomplete (until
recently)
This is our focus
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 50 / 113
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Introduction
Three Approaches to Meshing, III:
3. Numeric Approach
Curve Tracing Literature[Ratschek & Rokne (2005)]
Subdivision Approach[Marching Cube (1987); Snyder (1992); Plantinga & Vegter (2004)]
Properties Practical; easy to implement; adaptive; incomplete (until
recently)
This is our focus
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 50 / 113
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Introduction
Three Approaches to Meshing, III:
3. Numeric Approach
Curve Tracing Literature[Ratschek & Rokne (2005)]
Subdivision Approach[Marching Cube (1987); Snyder (1992); Plantinga & Vegter (2004)]
Properties Practical; easy to implement; adaptive; incomplete (until
recently)
This is our focus
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 50 / 113
![Page 322: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/322.jpg)
Introduction
Three Approaches to Meshing, III:
3. Numeric Approach
Curve Tracing Literature[Ratschek & Rokne (2005)]
Subdivision Approach[Marching Cube (1987); Snyder (1992); Plantinga & Vegter (2004)]
Properties Practical; easy to implement; adaptive; incomplete (until
recently)
This is our focus
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 50 / 113
![Page 323: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/323.jpg)
Introduction
Three Approaches to Meshing, III:
3. Numeric Approach
Curve Tracing Literature[Ratschek & Rokne (2005)]
Subdivision Approach[Marching Cube (1987); Snyder (1992); Plantinga & Vegter (2004)]
Properties Practical; easy to implement; adaptive; incomplete (until
recently)
This is our focus
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 50 / 113
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Introduction
Two Criteria of Meshing
I. Topological Correctness
The approximation S is isotopic to the S.
S1 S2 S1 and S2 arehomeomorphic , but notisotopic
Ambient space property!
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Introduction
(contd.) Two Criteria of Meshing
II. Geometrical Accuracy (ε-closeness)
For any given ε > 0, the Hausdorff distance d(S, S) should notexceed ε .
Set ε = ∞ to focus on isotopy.
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Mini Summary
Want ENC algorithms for Explicitization Problems
Focus on (purely) Numerical Subdivision methods
Algorithms for Meshing Curves (and Surfaces)
What will be New?Numerical methods that are exact and can handle singularities
![Page 327: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/327.jpg)
Mini Summary
Want ENC algorithms for Explicitization Problems
Focus on (purely) Numerical Subdivision methods
Algorithms for Meshing Curves (and Surfaces)
What will be New?Numerical methods that are exact and can handle singularities
![Page 328: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/328.jpg)
Mini Summary
Want ENC algorithms for Explicitization Problems
Focus on (purely) Numerical Subdivision methods
Algorithms for Meshing Curves (and Surfaces)
What will be New?Numerical methods that are exact and can handle singularities
![Page 329: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/329.jpg)
Mini Summary
Want ENC algorithms for Explicitization Problems
Focus on (purely) Numerical Subdivision methods
Algorithms for Meshing Curves (and Surfaces)
What will be New?Numerical methods that are exact and can handle singularities
![Page 330: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/330.jpg)
Mini Summary
Want ENC algorithms for Explicitization Problems
Focus on (purely) Numerical Subdivision methods
Algorithms for Meshing Curves (and Surfaces)
What will be New?Numerical methods that are exact and can handle singularities
![Page 331: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/331.jpg)
Mini Summary
Want ENC algorithms for Explicitization Problems
Focus on (purely) Numerical Subdivision methods
Algorithms for Meshing Curves (and Surfaces)
What will be New?Numerical methods that are exact and can handle singularities
![Page 332: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/332.jpg)
Review of Subdivision Algorithms
Coming Up Next
6 Introduction
7 Review of Subdivision Algorithms
8 Cxy Algorithm
9 Extensions of Cxy
10 How to treat Boundary
11 How to treat Singularity
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Review of Subdivision Algorithms
Subdivision Algorithms
Viewed as generalized binary search, organized as a quadtree.
Here is a typical output:
Figure: Approximation of the curve f (X ,Y ) = Y 2−X2 +X3 +0.02 = 0
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Review of Subdivision Algorithms
The Generic Subdivision Algorithm
INPUT: Curve S = f−1(0), box B0 ⊆R2, and ε > 0
OUTPUT: Graph G = (V ,E),representing an isotopic ε-approximation of S∩B0.
1 Let Qin← B0 be a queue of boxes2 SUBDIVISION PHASE: Qout ← SUBDIVIDE(Qin)3 REFINEMENT PHASE: Qref ← REFINE(Qout)4 CONSTRUCTION PHASE: G← CONSTRUCT (Qref )
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Review of Subdivision Algorithms
The Generic Subdivision Algorithm
INPUT: Curve S = f−1(0), box B0 ⊆R2, and ε > 0
OUTPUT: Graph G = (V ,E),representing an isotopic ε-approximation of S∩B0.
1 Let Qin← B0 be a queue of boxes2 SUBDIVISION PHASE: Qout ← SUBDIVIDE(Qin)3 REFINEMENT PHASE: Qref ← REFINE(Qout)4 CONSTRUCTION PHASE: G← CONSTRUCT (Qref )
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Review of Subdivision Algorithms
The Generic Subdivision Algorithm
INPUT: Curve S = f−1(0), box B0 ⊆R2, and ε > 0
OUTPUT: Graph G = (V ,E),representing an isotopic ε-approximation of S∩B0.
1 Let Qin← B0 be a queue of boxes2 SUBDIVISION PHASE: Qout ← SUBDIVIDE(Qin)3 REFINEMENT PHASE: Qref ← REFINE(Qout)4 CONSTRUCTION PHASE: G← CONSTRUCT (Qref )
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Review of Subdivision Algorithms
The Generic Subdivision Algorithm
INPUT: Curve S = f−1(0), box B0 ⊆R2, and ε > 0
OUTPUT: Graph G = (V ,E),representing an isotopic ε-approximation of S∩B0.
1 Let Qin← B0 be a queue of boxes2 SUBDIVISION PHASE: Qout ← SUBDIVIDE(Qin)3 REFINEMENT PHASE: Qref ← REFINE(Qout)4 CONSTRUCTION PHASE: G← CONSTRUCT (Qref )
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Review of Subdivision Algorithms
The Generic Subdivision Algorithm
INPUT: Curve S = f−1(0), box B0 ⊆R2, and ε > 0
OUTPUT: Graph G = (V ,E),representing an isotopic ε-approximation of S∩B0.
1 Let Qin← B0 be a queue of boxes2 SUBDIVISION PHASE: Qout ← SUBDIVIDE(Qin)3 REFINEMENT PHASE: Qref ← REFINE(Qout)4 CONSTRUCTION PHASE: G← CONSTRUCT (Qref )
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Review of Subdivision Algorithms
The Generic Subdivision Algorithm
INPUT: Curve S = f−1(0), box B0 ⊆R2, and ε > 0
OUTPUT: Graph G = (V ,E),representing an isotopic ε-approximation of S∩B0.
1 Let Qin← B0 be a queue of boxes2 SUBDIVISION PHASE: Qout ← SUBDIVIDE(Qin)3 REFINEMENT PHASE: Qref ← REFINE(Qout)4 CONSTRUCTION PHASE: G← CONSTRUCT (Qref )
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Review of Subdivision Algorithms
The Generic Subdivision Algorithm
INPUT: Curve S = f−1(0), box B0 ⊆R2, and ε > 0
OUTPUT: Graph G = (V ,E),representing an isotopic ε-approximation of S∩B0.
1 Let Qin← B0 be a queue of boxes2 SUBDIVISION PHASE: Qout ← SUBDIVIDE(Qin)3 REFINEMENT PHASE: Qref ← REFINE(Qout)4 CONSTRUCTION PHASE: G← CONSTRUCT (Qref )
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Review of Subdivision Algorithms
The Generic Subdivision Algorithm
INPUT: Curve S = f−1(0), box B0 ⊆R2, and ε > 0
OUTPUT: Graph G = (V ,E),representing an isotopic ε-approximation of S∩B0.
1 Let Qin← B0 be a queue of boxes2 SUBDIVISION PHASE: Qout ← SUBDIVIDE(Qin)3 REFINEMENT PHASE: Qref ← REFINE(Qout)4 CONSTRUCTION PHASE: G← CONSTRUCT (Qref )
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Review of Subdivision Algorithms
E.g., Marching Cube
Subdivision PhaseSubdivide until size of each box ≤ ε .
Construction Phase(1) Evaluate sign of f at grid points, (2) insert vertices, and (3) connectthem in each box:
+
+
−
−
(b)
+
+
−
−
(d)
+
+−
−
(c)
+
+
(a)
+
−
Cannot guarantee the topological correctness
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Review of Subdivision Algorithms
Parametrizability and Normal Variation
Parametrizable in X -direction
(d)(c)(b)(a)
(a) Parametrizable in X -direction
(b) Non-parametrizable in X - or Y -direction
(c) Small normal variation
(d) Big normal variation
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Review of Subdivision Algorithms
Box Predicates
Three Conditions (Predicates)
C0 0 /∈ f (B) Exclusion
Cxy 0 /∈ fx(B) or 0 /∈ fy (B) Parametrizability
C1 0 /∈ fx(B)2 + fy(B)2 Small Normal Variation
Implementation: e.g., f (x,y) = x2−2xy +3yInterval Arithmetic (Box):
f (I,J) = I2−2IJ +3J
Interval Taylor (Disc): f (x ,y , r) = [f (x ,y)± r(|2(x − y)|+ |−2x +3|+3r2)]
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Review of Subdivision Algorithms
Box Predicates
Three Conditions (Predicates)
C0 0 /∈ f (B) Exclusion
Cxy 0 /∈ fx(B) or 0 /∈ fy (B) Parametrizability
C1 0 /∈ fx(B)2 + fy(B)2 Small Normal Variation
Implementation: e.g., f (x,y) = x2−2xy +3yInterval Arithmetic (Box):
f (I,J) = I2−2IJ +3J
Interval Taylor (Disc): f (x ,y , r) = [f (x ,y)± r(|2(x − y)|+ |−2x +3|+3r2)]
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Review of Subdivision Algorithms
Box Predicates
Three Conditions (Predicates)
C0 0 /∈ f (B) Exclusion
Cxy 0 /∈ fx(B) or 0 /∈ fy (B) Parametrizability
C1 0 /∈ fx(B)2 + fy(B)2 Small Normal Variation
Implementation: e.g., f (x,y) = x2−2xy +3yInterval Arithmetic (Box):
f (I,J) = I2−2IJ +3J
Interval Taylor (Disc): f (x ,y , r) = [f (x ,y)± r(|2(x − y)|+ |−2x +3|+3r2)]
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Review of Subdivision Algorithms
Box Predicates
Three Conditions (Predicates)
C0 0 /∈ f (B) Exclusion
Cxy 0 /∈ fx(B) or 0 /∈ fy (B) Parametrizability
C1 0 /∈ fx(B)2 + fy(B)2 Small Normal Variation
Implementation: e.g., f (x,y) = x2−2xy +3yInterval Arithmetic (Box):
f (I,J) = I2−2IJ +3J
Interval Taylor (Disc): f (x ,y , r) = [f (x ,y)± r(|2(x − y)|+ |−2x +3|+3r2)]
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Review of Subdivision Algorithms
Box Predicates
Three Conditions (Predicates)
C0 0 /∈ f (B) Exclusion
Cxy 0 /∈ fx(B) or 0 /∈ fy (B) Parametrizability
C1 0 /∈ fx(B)2 + fy(B)2 Small Normal Variation
Implementation: e.g., f (x,y) = x2−2xy +3yInterval Arithmetic (Box):
f (I,J) = I2−2IJ +3J
Interval Taylor (Disc): f (x ,y , r) = [f (x ,y)± r(|2(x − y)|+ |−2x +3|+3r2)]
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Review of Subdivision Algorithms
Box Predicates
Three Conditions (Predicates)
C0 0 /∈ f (B) Exclusion
Cxy 0 /∈ fx(B) or 0 /∈ fy (B) Parametrizability
C1 0 /∈ fx(B)2 + fy(B)2 Small Normal Variation
Implementation: e.g., f (x,y) = x2−2xy +3yInterval Arithmetic (Box):
f (I,J) = I2−2IJ +3J
Interval Taylor (Disc): f (x ,y , r) = [f (x ,y)± r(|2(x − y)|+ |−2x +3|+3r2)]
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Review of Subdivision Algorithms
Box Predicates
Three Conditions (Predicates)
C0 0 /∈ f (B) Exclusion
Cxy 0 /∈ fx(B) or 0 /∈ fy (B) Parametrizability
C1 0 /∈ fx(B)2 + fy(B)2 Small Normal Variation
Implementation: e.g., f (x,y) = x2−2xy +3yInterval Arithmetic (Box):
f (I,J) = I2−2IJ +3J
Interval Taylor (Disc): f (x ,y , r) = [f (x ,y)± r(|2(x − y)|+ |−2x +3|+3r2)]
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Review of Subdivision Algorithms
Box Predicates
Three Conditions (Predicates)
C0 0 /∈ f (B) Exclusion
Cxy 0 /∈ fx(B) or 0 /∈ fy (B) Parametrizability
C1 0 /∈ fx(B)2 + fy(B)2 Small Normal Variation
Implementation: e.g., f (x,y) = x2−2xy +3yInterval Arithmetic (Box):
f (I,J) = I2−2IJ +3J
Interval Taylor (Disc): f (x ,y , r) = [f (x ,y)± r(|2(x − y)|+ |−2x +3|+3r2)]
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Review of Subdivision Algorithms
Box Predicates
Three Conditions (Predicates)
C0 0 /∈ f (B) Exclusion
Cxy 0 /∈ fx(B) or 0 /∈ fy (B) Parametrizability
C1 0 /∈ fx(B)2 + fy(B)2 Small Normal Variation
Implementation: e.g., f (x,y) = x2−2xy +3yInterval Arithmetic (Box):
f (I,J) = I2−2IJ +3J
Interval Taylor (Disc): f (x ,y , r) = [f (x ,y)± r(|2(x − y)|+ |−2x +3|+3r2)]
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Review of Subdivision Algorithms
Box Predicates
Three Conditions (Predicates)
C0 0 /∈ f (B) Exclusion
Cxy 0 /∈ fx(B) or 0 /∈ fy (B) Parametrizability
C1 0 /∈ fx(B)2 + fy(B)2 Small Normal Variation
Implementation: e.g., f (x,y) = x2−2xy +3yInterval Arithmetic (Box):
f (I,J) = I2−2IJ +3J
Interval Taylor (Disc): f (x ,y , r) = [f (x ,y)± r(|2(x − y)|+ |−2x +3|+3r2)]
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Review of Subdivision Algorithms
Box Predicates
Three Conditions (Predicates)
C0 0 /∈ f (B) Exclusion
Cxy 0 /∈ fx(B) or 0 /∈ fy (B) Parametrizability
C1 0 /∈ fx(B)2 + fy(B)2 Small Normal Variation
Implementation: e.g., f (x,y) = x2−2xy +3yInterval Arithmetic (Box):
f (I,J) = I2−2IJ +3J
Interval Taylor (Disc): f (x ,y , r) = [f (x ,y)± r(|2(x − y)|+ |−2x +3|+3r2)]
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Review of Subdivision Algorithms
Snyder’s Algorithm
Subdivision PhaseFor each box B:
C0(B)⇒ discard
¬Cxy(B)⇒ subdivide B
Construction PhaseDetermine intersections on boundary
Connect the intersections
(Non-trivial, unbounded complexity)
Boundary Analysis is not good (may not even terminate).
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Review of Subdivision Algorithms
Snyder’s Algorithm
Subdivision PhaseFor each box B:
C0(B)⇒ discard
¬Cxy(B)⇒ subdivide B
Construction PhaseDetermine intersections on boundary
Connect the intersections
(Non-trivial, unbounded complexity)
Boundary Analysis is not good (may not even terminate).
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Review of Subdivision Algorithms
Idea of Plantinga and VegterIntroduce a strong predicate C1 predicate
Allow local NON-isotopyLocal incursion and excursions
B
B′
B
B′
Locally, graph is not isotopic
Simple box geometry(simpler than Snyder, less simple than Marching Cube)
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Review of Subdivision Algorithms
Plantinga and Vegter’s Algorithm
Exploit the global isotopy
Subdivision Phase: For each box B: C0(B)⇒ discard ¬C1(B)⇒ subdivide B
Refinement Phase: Balance!
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Review of Subdivision Algorithms
(contd.) Plantinga and Vegter’s Algorithm
Global, not local, isotopy
Construction Phase:
(b)
+
+
+
+
−
+
+
−
+
+
−
+
−
(e)
++
+ −
−
−+
(f)(d)
+−
(c)
+
(a)
+
− +
+
−
−
+
Figure: Extended Rules
Local isotopy is NOT good !
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Cxy Algorithm
Coming Up Next
6 Introduction
7 Review of Subdivision Algorithms
8 Cxy Algorithm
9 Extensions of Cxy
10 How to treat Boundary
11 How to treat Singularity
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Cxy Algorithm
Idea of Cxy Algorithm
Replace C1 by CxyC1(B) implies Cxy(B)
This would produce fewer boxes.
Exploit local non-isotopyLocal isotopy is an artifact!
This also avoid boundary analysis.
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Cxy Algorithm
Obstructions to Cxy Idea
Replace C1 by CxyJust run PV Algorithm but using Cxy instead:
What can go wrong?
x
y
(b)
(a)
positive corner
vertexB2B1
negative corner
KEY:
B′2 B′1
(−5,−1)
(5,1)
Figure: Elongated hyperbola
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Cxy Algorithm
Cxy AlgorithmSubdivision and Refinement Phases: As beforeConstruction Phase:
(full-split)
− +
+
(a)
+/−
+
+
+ +
(c’)
+ +
+
+ − +
+ +
(c”)
+ +−
+
−
+
+ − +
+ −(b’)
+ +
+
+ − +
+ −
+ +
(b)
+
+ − +
+ +
(c)
+ +
−
+
−
+
+ − +
Figure: Resolution of Ambiguity
Nontrivial proof of correctnessYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 67 / 113
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Mini Summary
What has Cxy Algorithm done? Exploit Parametrizability (like Snyder) Rejected local isotopy (like PV)
Up Next: More improvements
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Mini Summary
What has Cxy Algorithm done? Exploit Parametrizability (like Snyder) Rejected local isotopy (like PV)
Up Next: More improvements
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Mini Summary
What has Cxy Algorithm done? Exploit Parametrizability (like Snyder) Rejected local isotopy (like PV)
Up Next: More improvements
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Mini Summary
What has Cxy Algorithm done? Exploit Parametrizability (like Snyder) Rejected local isotopy (like PV)
Up Next: More improvements
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Mini Summary
What has Cxy Algorithm done? Exploit Parametrizability (like Snyder) Rejected local isotopy (like PV)
Up Next: More improvements
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Mini Summary
What has Cxy Algorithm done? Exploit Parametrizability (like Snyder) Rejected local isotopy (like PV)
Up Next: More improvements
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Extensions of Cxy
Coming Up Next
6 Introduction
7 Review of Subdivision Algorithms
8 Cxy Algorithm
9 Extensions of Cxy
10 How to treat Boundary
11 How to treat Singularity
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Extensions of Cxy
Idea of Rectangular Cxy Algorithm
Exploit Anisotropy
(b) PV (c) Snyder
(d) Balanced Cxy (e) Rectangular Cxy
(a) Original Curve
“Heel Curve”X2Y 2−X +Y −1 = 0 inboxB = [(−2,−10),(10,2)]
Comparing PV, Snyder,Cxy, Rect Cxy
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Extensions of Cxy
Partial Splits for Rectangles
Splits
Full-splits:B→ (B1,B2,B3,B4)
1
3
B14
4
B12
B342
B23
HorizontalHalf-split:
B→ (B12,B34)
Vertical Half-split:
B→ (B14,B23)
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Extensions of Cxy
Rectangular Cxy Algorithm
What is neededAspect Ratio Bound: r > 1 arbitrary but fixed.
Splitting Procedure: do full-split if none of these hold
L0 : C0(B),Cxy (B) TerminateLout : C0(B12),C0(B34),C0(B14),C0(B23) Half-splitLin : Cxy (B12),Cxy (B34),Cxy (B14),Cxy (B23) Half-split
Axis-dependent balancing: each node has a X -depth andY -depth.
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Extensions of Cxy
Rectangular Cxy Algorithm
What is neededAspect Ratio Bound: r > 1 arbitrary but fixed.
Splitting Procedure: do full-split if none of these hold
L0 : C0(B),Cxy (B) TerminateLout : C0(B12),C0(B34),C0(B14),C0(B23) Half-splitLin : Cxy (B12),Cxy (B34),Cxy (B14),Cxy (B23) Half-split
Axis-dependent balancing: each node has a X -depth andY -depth.
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Extensions of Cxy
Rectangular Cxy Algorithm
What is neededAspect Ratio Bound: r > 1 arbitrary but fixed.
Splitting Procedure: do full-split if none of these hold
L0 : C0(B),Cxy (B) TerminateLout : C0(B12),C0(B34),C0(B14),C0(B23) Half-splitLin : Cxy (B12),Cxy (B34),Cxy (B14),Cxy (B23) Half-split
Axis-dependent balancing: each node has a X -depth andY -depth.
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Extensions of Cxy
Rectangular Cxy Algorithm
What is neededAspect Ratio Bound: r > 1 arbitrary but fixed.
Splitting Procedure: do full-split if none of these hold
L0 : C0(B),Cxy (B) TerminateLout : C0(B12),C0(B34),C0(B14),C0(B23) Half-splitLin : Cxy (B12),Cxy (B34),Cxy (B14),Cxy (B23) Half-split
Axis-dependent balancing: each node has a X -depth andY -depth.
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Extensions of Cxy
Rectangular Cxy Algorithm
What is neededAspect Ratio Bound: r > 1 arbitrary but fixed.
Splitting Procedure: do full-split if none of these hold
L0 : C0(B),Cxy (B) TerminateLout : C0(B12),C0(B34),C0(B14),C0(B23) Half-splitLin : Cxy (B12),Cxy (B34),Cxy (B14),Cxy (B23) Half-split
Axis-dependent balancing: each node has a X -depth andY -depth.
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Extensions of Cxy
Rectangular Cxy Algorithm
What is neededAspect Ratio Bound: r > 1 arbitrary but fixed.
Splitting Procedure: do full-split if none of these hold
L0 : C0(B),Cxy (B) TerminateLout : C0(B12),C0(B34),C0(B14),C0(B23) Half-splitLin : Cxy (B12),Cxy (B34),Cxy (B14),Cxy (B23) Half-split
Axis-dependent balancing: each node has a X -depth andY -depth.
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Extensions of Cxy
Rectangular Cxy Algorithm
What is neededAspect Ratio Bound: r > 1 arbitrary but fixed.
Splitting Procedure: do full-split if none of these hold
L0 : C0(B),Cxy (B) TerminateLout : C0(B12),C0(B34),C0(B14),C0(B23) Half-splitLin : Cxy (B12),Cxy (B34),Cxy (B14),Cxy (B23) Half-split
Axis-dependent balancing: each node has a X -depth andY -depth.
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Extensions of Cxy
Rectangular Cxy Algorithm
What is neededAspect Ratio Bound: r > 1 arbitrary but fixed.
Splitting Procedure: do full-split if none of these hold
L0 : C0(B),Cxy (B) TerminateLout : C0(B12),C0(B34),C0(B14),C0(B23) Half-splitLin : Cxy (B12),Cxy (B34),Cxy (B14),Cxy (B23) Half-split
Axis-dependent balancing: each node has a X -depth andY -depth.
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Extensions of Cxy
Ensuring Geometric Accuracy
Buffer Property of C1 predicateAspect Ratio ≤ 2:
a b w
S
va′
b′
q
u
p
B
e
Half-circle argument
Generalize C1(B) to C∗1(B). for any box B
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Extensions of Cxy
Comparisons
Compare Rect Cxy to PV (note: Snyder has degeneracy). Curve X(XY −1) = 0, box Bs := [(−s,−s),(s,s)], Aspect ratio
bound r = 5: (JSO=Java stack overflow)
Increasing r can increase the performance of Rect Cxy. r = 80,s = 100⇒ Boxes/Time(ms) = 751/78
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Extensions of Cxy
Comparisons (2)
Compare to Snyder’s Algorithm. Curve X(XY −1) = 0, box
Bn := [(−14×10n,−14×10n),(15×10n,15×10n)]. Maximumaspect ratio r = 257.
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Extensions of Cxy
Comparisons (2)
Compare to Snyder’s Algorithm. Curve X(XY −1) = 0, box
Bn := [(−14×10n,−14×10n),(15×10n,15×10n)]. Maximumaspect ratio r = 257.
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Extensions of Cxy
Comparisons (2)
Compare to Snyder’s Algorithm. Curve X(XY −1) = 0, box
Bn := [(−14×10n,−14×10n),(15×10n,15×10n)]. Maximumaspect ratio r = 257.
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Extensions of Cxy
Summary of Experimental Results
Cxy combines the advantages of Snyder & PV Algorithms.
Can be significantly faster than PV & Snyder’s algorithm.
Rectangular Cxy Algorithm can be significantly faster thanBalanced Cxy algorithm.
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How to treat Boundary
Coming Up Next
6 Introduction
7 Review of Subdivision Algorithms
8 Cxy Algorithm
9 Extensions of Cxy
10 How to treat Boundary
11 How to treat Singularity
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How to treat Boundary
Boundary (Summary)
An Obvious Way and a Better Way Exact Way : Recursively solve the problem on ∂B0
Better Way : Exploit isotopy
Price for Better Way: Weaker Correctness StatementFor some B0 ⊆ B+
0 ⊆ B0⊕B(ε),G is isotopic to S∩B+
0 .
APPLICATIONS: Singularity (below) Input region B0 to have “any” geometry, even holes, provided it
contains no singularities.
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How to treat Boundary
Boundary (Summary)
An Obvious Way and a Better Way Exact Way : Recursively solve the problem on ∂B0
Better Way : Exploit isotopy
Price for Better Way: Weaker Correctness StatementFor some B0 ⊆ B+
0 ⊆ B0⊕B(ε),G is isotopic to S∩B+
0 .
APPLICATIONS: Singularity (below) Input region B0 to have “any” geometry, even holes, provided it
contains no singularities.
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How to treat Boundary
Boundary (Summary)
An Obvious Way and a Better Way Exact Way : Recursively solve the problem on ∂B0
Better Way : Exploit isotopy
Price for Better Way: Weaker Correctness StatementFor some B0 ⊆ B+
0 ⊆ B0⊕B(ε),G is isotopic to S∩B+
0 .
APPLICATIONS: Singularity (below) Input region B0 to have “any” geometry, even holes, provided it
contains no singularities.
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How to treat Boundary
Boundary (Summary)
An Obvious Way and a Better Way Exact Way : Recursively solve the problem on ∂B0
Better Way : Exploit isotopy
Price for Better Way: Weaker Correctness StatementFor some B0 ⊆ B+
0 ⊆ B0⊕B(ε),G is isotopic to S∩B+
0 .
APPLICATIONS: Singularity (below) Input region B0 to have “any” geometry, even holes, provided it
contains no singularities.
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How to treat Boundary
Boundary (Summary)
An Obvious Way and a Better Way Exact Way : Recursively solve the problem on ∂B0
Better Way : Exploit isotopy
Price for Better Way: Weaker Correctness StatementFor some B0 ⊆ B+
0 ⊆ B0⊕B(ε),G is isotopic to S∩B+
0 .
APPLICATIONS: Singularity (below) Input region B0 to have “any” geometry, even holes, provided it
contains no singularities.
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How to treat Boundary
Boundary (Summary)
An Obvious Way and a Better Way Exact Way : Recursively solve the problem on ∂B0
Better Way : Exploit isotopy
Price for Better Way: Weaker Correctness StatementFor some B0 ⊆ B+
0 ⊆ B0⊕B(ε),G is isotopic to S∩B+
0 .
APPLICATIONS: Singularity (below) Input region B0 to have “any” geometry, even holes, provided it
contains no singularities.
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How to treat Boundary
Boundary (Summary)
An Obvious Way and a Better Way Exact Way : Recursively solve the problem on ∂B0
Better Way : Exploit isotopy
Price for Better Way: Weaker Correctness StatementFor some B0 ⊆ B+
0 ⊆ B0⊕B(ε),G is isotopic to S∩B+
0 .
APPLICATIONS: Singularity (below) Input region B0 to have “any” geometry, even holes, provided it
contains no singularities.
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How to treat Boundary
Boundary (Summary)
An Obvious Way and a Better Way Exact Way : Recursively solve the problem on ∂B0
Better Way : Exploit isotopy
Price for Better Way: Weaker Correctness StatementFor some B0 ⊆ B+
0 ⊆ B0⊕B(ε),G is isotopic to S∩B+
0 .
APPLICATIONS: Singularity (below) Input region B0 to have “any” geometry, even holes, provided it
contains no singularities.
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How to treat Boundary
Boundary (Summary)
An Obvious Way and a Better Way Exact Way : Recursively solve the problem on ∂B0
Better Way : Exploit isotopy
Price for Better Way: Weaker Correctness StatementFor some B0 ⊆ B+
0 ⊆ B0⊕B(ε),G is isotopic to S∩B+
0 .
APPLICATIONS: Singularity (below) Input region B0 to have “any” geometry, even holes, provided it
contains no singularities.
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How to treat Boundary
Boundary (Summary)
An Obvious Way and a Better Way Exact Way : Recursively solve the problem on ∂B0
Better Way : Exploit isotopy
Price for Better Way: Weaker Correctness StatementFor some B0 ⊆ B+
0 ⊆ B0⊕B(ε),G is isotopic to S∩B+
0 .
APPLICATIONS: Singularity (below) Input region B0 to have “any” geometry, even holes, provided it
contains no singularities.
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How to treat Singularity
Coming Up Next
6 Introduction
7 Review of Subdivision Algorithms
8 Cxy Algorithm
9 Extensions of Cxy
10 How to treat Boundary
11 How to treat Singularity
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How to treat Singularity
Singularity : Algebraic Preliminary
Square-free part of f (X1, . . . ,Xn) ∈ Z[X1, . . . ,Xn]:f
GCD(f ,∂1f ,...,∂nf )= f
GCD(f ,∇(f ))
For n = 1: square-free implies no singularities
Generally:Singular set sing(f ) := Zero(f ,∇(f )) has co-dimension ≥ 2.
For Curves:we now assume f (X ,Y ) ∈Z[X ,Y ] has isolated singularities.
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How to treat Singularity
Singularity : Algebraic Preliminary
Square-free part of f (X1, . . . ,Xn) ∈ Z[X1, . . . ,Xn]:f
GCD(f ,∂1f ,...,∂nf )= f
GCD(f ,∇(f ))
For n = 1: square-free implies no singularities
Generally:Singular set sing(f ) := Zero(f ,∇(f )) has co-dimension ≥ 2.
For Curves:we now assume f (X ,Y ) ∈Z[X ,Y ] has isolated singularities.
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How to treat Singularity
Singularity : Algebraic Preliminary
Square-free part of f (X1, . . . ,Xn) ∈ Z[X1, . . . ,Xn]:f
GCD(f ,∂1f ,...,∂nf )= f
GCD(f ,∇(f ))
For n = 1: square-free implies no singularities
Generally:Singular set sing(f ) := Zero(f ,∇(f )) has co-dimension ≥ 2.
For Curves:we now assume f (X ,Y ) ∈Z[X ,Y ] has isolated singularities.
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How to treat Singularity
Singularity : Algebraic Preliminary
Square-free part of f (X1, . . . ,Xn) ∈ Z[X1, . . . ,Xn]:f
GCD(f ,∂1f ,...,∂nf )= f
GCD(f ,∇(f ))
For n = 1: square-free implies no singularities
Generally:Singular set sing(f ) := Zero(f ,∇(f )) has co-dimension ≥ 2.
For Curves:we now assume f (X ,Y ) ∈Z[X ,Y ] has isolated singularities.
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How to treat Singularity
Singularity : Algebraic Preliminary
Square-free part of f (X1, . . . ,Xn) ∈ Z[X1, . . . ,Xn]:f
GCD(f ,∂1f ,...,∂nf )= f
GCD(f ,∇(f ))
For n = 1: square-free implies no singularities
Generally:Singular set sing(f ) := Zero(f ,∇(f )) has co-dimension ≥ 2.
For Curves:we now assume f (X ,Y ) ∈Z[X ,Y ] has isolated singularities.
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How to treat Singularity
Singularity : Algebraic Preliminary
Square-free part of f (X1, . . . ,Xn) ∈ Z[X1, . . . ,Xn]:f
GCD(f ,∂1f ,...,∂nf )= f
GCD(f ,∇(f ))
For n = 1: square-free implies no singularities
Generally:Singular set sing(f ) := Zero(f ,∇(f )) has co-dimension ≥ 2.
For Curves:we now assume f (X ,Y ) ∈Z[X ,Y ] has isolated singularities.
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How to treat Singularity
Some Zero BoundsEvaluation Bound LemmaIf f (X ,Y ) has degree d and height L then
− logEV (f ) = O(d2(L+d logd))
where EV (f ) := min|f (α)| : ∇(α) = 0, f (α) 6= 0
Singularity Separation Bound [Y. (2006)]
Any two singularities of f = 0 are separated by
δ3 ≥ (16d+2256L812dd5)−d
Closest Approach BoundThe “locally closest” approach of a curve f = 0 to its own singularities is
δ4 ≥ (62e7)−30D(44 ·5 ·2L)−5D4
where D = max2,deg fYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 81 / 113
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How to treat Singularity
Some Zero BoundsEvaluation Bound LemmaIf f (X ,Y ) has degree d and height L then
− logEV (f ) = O(d2(L+d logd))
where EV (f ) := min|f (α)| : ∇(α) = 0, f (α) 6= 0
Singularity Separation Bound [Y. (2006)]
Any two singularities of f = 0 are separated by
δ3 ≥ (16d+2256L812dd5)−d
Closest Approach BoundThe “locally closest” approach of a curve f = 0 to its own singularities is
δ4 ≥ (62e7)−30D(44 ·5 ·2L)−5D4
where D = max2,deg fYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 81 / 113
![Page 407: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/407.jpg)
How to treat Singularity
Some Zero BoundsEvaluation Bound LemmaIf f (X ,Y ) has degree d and height L then
− logEV (f ) = O(d2(L+d logd))
where EV (f ) := min|f (α)| : ∇(α) = 0, f (α) 6= 0
Singularity Separation Bound [Y. (2006)]
Any two singularities of f = 0 are separated by
δ3 ≥ (16d+2256L812dd5)−d
Closest Approach BoundThe “locally closest” approach of a curve f = 0 to its own singularities is
δ4 ≥ (62e7)−30D(44 ·5 ·2L)−5D4
where D = max2,deg fYap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 81 / 113
![Page 408: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/408.jpg)
How to treat Singularity
Isolating Singularities
Mountain Pass TheoremConsider F := f 2 + f 2
X + f 2Y .
Any 2 singularities in B0 are connected by paths γ : [0,1]→R2
satisfyingminγ(F([0,1])) ≥ ε0
where
ε0 := minEV (f ),min F(∂B0)
Can provide a subdivision algorithm using F ,ε0 to isolate regionscontaining singularities.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 82 / 113
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How to treat Singularity
Isolating Singularities
Mountain Pass TheoremConsider F := f 2 + f 2
X + f 2Y .
Any 2 singularities in B0 are connected by paths γ : [0,1]→R2
satisfyingminγ(F([0,1])) ≥ ε0
where
ε0 := minEV (f ),min F(∂B0)
Can provide a subdivision algorithm using F ,ε0 to isolate regionscontaining singularities.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 82 / 113
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How to treat Singularity
Isolating Singularities
Mountain Pass TheoremConsider F := f 2 + f 2
X + f 2Y .
Any 2 singularities in B0 are connected by paths γ : [0,1]→R2
satisfyingminγ(F([0,1])) ≥ ε0
where
ε0 := minEV (f ),min F(∂B0)
Can provide a subdivision algorithm using F ,ε0 to isolate regionscontaining singularities.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 82 / 113
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How to treat Singularity
Degree of Singularites
Degree of singularity := number of half-branches
Use two concentric boxes B2 ⊆ B1:inner box has singularity, outer radius less than δ3,δ4
p
(3)
(1)
(2)
(2)
(1)
B1
B2
(b)(a)
(a) Singularity p with3 types of components
(b) Concentric boxes(B1,B2)
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Mini Summary
We have seen how to combine Snyder and PV, and make severalpractical improvements
Future Work: Extend 3D (and beyond?)
Improve efficiency of refinement
Improve efficiency of singularity Nonsingular case is fast Singular case is (currently) not fast
![Page 413: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/413.jpg)
Mini Summary
We have seen how to combine Snyder and PV, and make severalpractical improvements
Future Work: Extend 3D (and beyond?)
Improve efficiency of refinement
Improve efficiency of singularity Nonsingular case is fast Singular case is (currently) not fast
![Page 414: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/414.jpg)
Mini Summary
We have seen how to combine Snyder and PV, and make severalpractical improvements
Future Work: Extend 3D (and beyond?)
Improve efficiency of refinement
Improve efficiency of singularity Nonsingular case is fast Singular case is (currently) not fast
![Page 415: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/415.jpg)
Mini Summary
We have seen how to combine Snyder and PV, and make severalpractical improvements
Future Work: Extend 3D (and beyond?)
Improve efficiency of refinement
Improve efficiency of singularity Nonsingular case is fast Singular case is (currently) not fast
![Page 416: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/416.jpg)
Mini Summary
We have seen how to combine Snyder and PV, and make severalpractical improvements
Future Work: Extend 3D (and beyond?)
Improve efficiency of refinement
Improve efficiency of singularity Nonsingular case is fast Singular case is (currently) not fast
![Page 417: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/417.jpg)
Mini Summary
We have seen how to combine Snyder and PV, and make severalpractical improvements
Future Work: Extend 3D (and beyond?)
Improve efficiency of refinement
Improve efficiency of singularity Nonsingular case is fast Singular case is (currently) not fast
![Page 418: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/418.jpg)
Mini Summary
We have seen how to combine Snyder and PV, and make severalpractical improvements
Future Work: Extend 3D (and beyond?)
Improve efficiency of refinement
Improve efficiency of singularity Nonsingular case is fast Singular case is (currently) not fast
![Page 419: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/419.jpg)
Mini Summary
We have seen how to combine Snyder and PV, and make severalpractical improvements
Future Work: Extend 3D (and beyond?)
Improve efficiency of refinement
Improve efficiency of singularity Nonsingular case is fast Singular case is (currently) not fast
![Page 420: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/420.jpg)
Mini Summary
We have seen how to combine Snyder and PV, and make severalpractical improvements
Future Work: Extend 3D (and beyond?)
Improve efficiency of refinement
Improve efficiency of singularity Nonsingular case is fast Singular case is (currently) not fast
![Page 421: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/421.jpg)
Summary of Lecture 2
Problems at the interface of continuous and discrete:Explicitization Problems
ENC Algorithms for them are novel
Numerical Treatment of Singularity and Degeneracy Possible in theory, but severe practical challenge
![Page 422: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/422.jpg)
Summary of Lecture 2
Problems at the interface of continuous and discrete:Explicitization Problems
ENC Algorithms for them are novel
Numerical Treatment of Singularity and Degeneracy Possible in theory, but severe practical challenge
![Page 423: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/423.jpg)
Summary of Lecture 2
Problems at the interface of continuous and discrete:Explicitization Problems
ENC Algorithms for them are novel
Numerical Treatment of Singularity and Degeneracy Possible in theory, but severe practical challenge
![Page 424: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/424.jpg)
Summary of Lecture 2
Problems at the interface of continuous and discrete:Explicitization Problems
ENC Algorithms for them are novel
Numerical Treatment of Singularity and Degeneracy Possible in theory, but severe practical challenge
![Page 425: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/425.jpg)
Summary of Lecture 2
Problems at the interface of continuous and discrete:Explicitization Problems
ENC Algorithms for them are novel
Numerical Treatment of Singularity and Degeneracy Possible in theory, but severe practical challenge
![Page 426: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/426.jpg)
Summary of Lecture 2
Problems at the interface of continuous and discrete:Explicitization Problems
ENC Algorithms for them are novel
Numerical Treatment of Singularity and Degeneracy Possible in theory, but severe practical challenge
![Page 427: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/427.jpg)
PART 3
Complexity Analysis of Adaptivity
“A rapacious monster lurks within every computer,and it dines exclusively on accurate digits.”
— B.D. McCullough (2000)
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Analysis of Adaptive Complexity
Coming Up Next
12 Analysis of Adaptive Complexity
13 Analysis of Descartes Method
14 Integral Bounds and Framework of Stopping Functions
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Analysis of Adaptive Complexity
Towards Analysis of Adaptive Algorithms
Major Challenge in Theoretical Computer Science Analysis of discrete algorithms is highly developed What about continuous , adaptive algorithms?
Previous such analysis requires probabilistic assumptions. Basically in Linear Programming: [Smale, Borgwardt,
Teng-Spielman]
We focus on the recursion tree size Return to 1-D !
Adaptive algorithms may have some deep paths, but overall sizeis only polynomial in depth.
Previous (trivial) result – size is exponential in depth [Kearfott(1987)]
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 88 / 113
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Analysis of Adaptive Complexity
Towards Analysis of Adaptive Algorithms
Major Challenge in Theoretical Computer Science Analysis of discrete algorithms is highly developed What about continuous , adaptive algorithms?
Previous such analysis requires probabilistic assumptions. Basically in Linear Programming: [Smale, Borgwardt,
Teng-Spielman]
We focus on the recursion tree size Return to 1-D !
Adaptive algorithms may have some deep paths, but overall sizeis only polynomial in depth.
Previous (trivial) result – size is exponential in depth [Kearfott(1987)]
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 88 / 113
![Page 431: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/431.jpg)
Analysis of Adaptive Complexity
Towards Analysis of Adaptive Algorithms
Major Challenge in Theoretical Computer Science Analysis of discrete algorithms is highly developed What about continuous , adaptive algorithms?
Previous such analysis requires probabilistic assumptions. Basically in Linear Programming: [Smale, Borgwardt,
Teng-Spielman]
We focus on the recursion tree size Return to 1-D !
Adaptive algorithms may have some deep paths, but overall sizeis only polynomial in depth.
Previous (trivial) result – size is exponential in depth [Kearfott(1987)]
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 88 / 113
![Page 432: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/432.jpg)
Analysis of Adaptive Complexity
Towards Analysis of Adaptive Algorithms
Major Challenge in Theoretical Computer Science Analysis of discrete algorithms is highly developed What about continuous , adaptive algorithms?
Previous such analysis requires probabilistic assumptions. Basically in Linear Programming: [Smale, Borgwardt,
Teng-Spielman]
We focus on the recursion tree size Return to 1-D !
Adaptive algorithms may have some deep paths, but overall sizeis only polynomial in depth.
Previous (trivial) result – size is exponential in depth [Kearfott(1987)]
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 88 / 113
![Page 433: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/433.jpg)
Analysis of Adaptive Complexity
Towards Analysis of Adaptive Algorithms
Major Challenge in Theoretical Computer Science Analysis of discrete algorithms is highly developed What about continuous , adaptive algorithms?
Previous such analysis requires probabilistic assumptions. Basically in Linear Programming: [Smale, Borgwardt,
Teng-Spielman]
We focus on the recursion tree size Return to 1-D !
Adaptive algorithms may have some deep paths, but overall sizeis only polynomial in depth.
Previous (trivial) result – size is exponential in depth [Kearfott(1987)]
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 88 / 113
![Page 434: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/434.jpg)
Analysis of Adaptive Complexity
Towards Analysis of Adaptive Algorithms
Major Challenge in Theoretical Computer Science Analysis of discrete algorithms is highly developed What about continuous , adaptive algorithms?
Previous such analysis requires probabilistic assumptions. Basically in Linear Programming: [Smale, Borgwardt,
Teng-Spielman]
We focus on the recursion tree size Return to 1-D !
Adaptive algorithms may have some deep paths, but overall sizeis only polynomial in depth.
Previous (trivial) result – size is exponential in depth [Kearfott(1987)]
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 88 / 113
![Page 435: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/435.jpg)
Analysis of Adaptive Complexity
Towards Analysis of Adaptive Algorithms
Major Challenge in Theoretical Computer Science Analysis of discrete algorithms is highly developed What about continuous , adaptive algorithms?
Previous such analysis requires probabilistic assumptions. Basically in Linear Programming: [Smale, Borgwardt,
Teng-Spielman]
We focus on the recursion tree size Return to 1-D !
Adaptive algorithms may have some deep paths, but overall sizeis only polynomial in depth.
Previous (trivial) result – size is exponential in depth [Kearfott(1987)]
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 88 / 113
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Analysis of Adaptive Complexity
Towards Analysis of Adaptive Algorithms
Major Challenge in Theoretical Computer Science Analysis of discrete algorithms is highly developed What about continuous , adaptive algorithms?
Previous such analysis requires probabilistic assumptions. Basically in Linear Programming: [Smale, Borgwardt,
Teng-Spielman]
We focus on the recursion tree size Return to 1-D !
Adaptive algorithms may have some deep paths, but overall sizeis only polynomial in depth.
Previous (trivial) result – size is exponential in depth [Kearfott(1987)]
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 88 / 113
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Analysis of Adaptive Complexity
Towards Analysis of Adaptive Algorithms
Major Challenge in Theoretical Computer Science Analysis of discrete algorithms is highly developed What about continuous , adaptive algorithms?
Previous such analysis requires probabilistic assumptions. Basically in Linear Programming: [Smale, Borgwardt,
Teng-Spielman]
We focus on the recursion tree size Return to 1-D !
Adaptive algorithms may have some deep paths, but overall sizeis only polynomial in depth.
Previous (trivial) result – size is exponential in depth [Kearfott(1987)]
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 88 / 113
![Page 438: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/438.jpg)
Analysis of Adaptive Complexity
Towards Analysis of Adaptive Algorithms
Major Challenge in Theoretical Computer Science Analysis of discrete algorithms is highly developed What about continuous , adaptive algorithms?
Previous such analysis requires probabilistic assumptions. Basically in Linear Programming: [Smale, Borgwardt,
Teng-Spielman]
We focus on the recursion tree size Return to 1-D !
Adaptive algorithms may have some deep paths, but overall sizeis only polynomial in depth.
Previous (trivial) result – size is exponential in depth [Kearfott(1987)]
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 88 / 113
![Page 439: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/439.jpg)
Analysis of Adaptive Complexity
Towards Analysis of Adaptive Algorithms
Major Challenge in Theoretical Computer Science Analysis of discrete algorithms is highly developed What about continuous , adaptive algorithms?
Previous such analysis requires probabilistic assumptions. Basically in Linear Programming: [Smale, Borgwardt,
Teng-Spielman]
We focus on the recursion tree size Return to 1-D !
Adaptive algorithms may have some deep paths, but overall sizeis only polynomial in depth.
Previous (trivial) result – size is exponential in depth [Kearfott(1987)]
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 88 / 113
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Analysis of Adaptive Complexity
Analytic Approach to Root Isolation
Suppose you want to isolate real roots of f (x) in I = [a,b]
Midpoint m(I) := (a+b)/2, Width w(I) := b−a
Exclusion Predicate: C0(I) : |f (m)|> ∑i≥1|f (i)(m)|
i!
(w(I)
2
)i
Inclusion Predicate: C1(I) : |f ′(m)|> ∑i≥1|f (i+1)(m)|
i!
(w(I)
2
)i
Confirmation (Bolzano) Test: f (a)f (b) < 0
Simple analytic method for root isolation!
Simpler than algebraic subdivision methods:
STURM > DESCARTES > BOLZANO
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Analysis of Adaptive Complexity
Analytic Approach to Root Isolation
Suppose you want to isolate real roots of f (x) in I = [a,b]
Midpoint m(I) := (a+b)/2, Width w(I) := b−a
Exclusion Predicate: C0(I) : |f (m)|> ∑i≥1|f (i)(m)|
i!
(w(I)
2
)i
Inclusion Predicate: C1(I) : |f ′(m)|> ∑i≥1|f (i+1)(m)|
i!
(w(I)
2
)i
Confirmation (Bolzano) Test: f (a)f (b) < 0
Simple analytic method for root isolation!
Simpler than algebraic subdivision methods:
STURM > DESCARTES > BOLZANO
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 89 / 113
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Analysis of Adaptive Complexity
Analytic Approach to Root Isolation
Suppose you want to isolate real roots of f (x) in I = [a,b]
Midpoint m(I) := (a+b)/2, Width w(I) := b−a
Exclusion Predicate: C0(I) : |f (m)|> ∑i≥1|f (i)(m)|
i!
(w(I)
2
)i
Inclusion Predicate: C1(I) : |f ′(m)|> ∑i≥1|f (i+1)(m)|
i!
(w(I)
2
)i
Confirmation (Bolzano) Test: f (a)f (b) < 0
Simple analytic method for root isolation!
Simpler than algebraic subdivision methods:
STURM > DESCARTES > BOLZANO
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 89 / 113
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Analysis of Adaptive Complexity
Analytic Approach to Root Isolation
Suppose you want to isolate real roots of f (x) in I = [a,b]
Midpoint m(I) := (a+b)/2, Width w(I) := b−a
Exclusion Predicate: C0(I) : |f (m)|> ∑i≥1|f (i)(m)|
i!
(w(I)
2
)i
Inclusion Predicate: C1(I) : |f ′(m)|> ∑i≥1|f (i+1)(m)|
i!
(w(I)
2
)i
Confirmation (Bolzano) Test: f (a)f (b) < 0
Simple analytic method for root isolation!
Simpler than algebraic subdivision methods:
STURM > DESCARTES > BOLZANO
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 89 / 113
![Page 444: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/444.jpg)
Analysis of Adaptive Complexity
Analytic Approach to Root Isolation
Suppose you want to isolate real roots of f (x) in I = [a,b]
Midpoint m(I) := (a+b)/2, Width w(I) := b−a
Exclusion Predicate: C0(I) : |f (m)|> ∑i≥1|f (i)(m)|
i!
(w(I)
2
)i
Inclusion Predicate: C1(I) : |f ′(m)|> ∑i≥1|f (i+1)(m)|
i!
(w(I)
2
)i
Confirmation (Bolzano) Test: f (a)f (b) < 0
Simple analytic method for root isolation!
Simpler than algebraic subdivision methods:
STURM > DESCARTES > BOLZANO
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 89 / 113
![Page 445: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/445.jpg)
Analysis of Adaptive Complexity
Analytic Approach to Root Isolation
Suppose you want to isolate real roots of f (x) in I = [a,b]
Midpoint m(I) := (a+b)/2, Width w(I) := b−a
Exclusion Predicate: C0(I) : |f (m)|> ∑i≥1|f (i)(m)|
i!
(w(I)
2
)i
Inclusion Predicate: C1(I) : |f ′(m)|> ∑i≥1|f (i+1)(m)|
i!
(w(I)
2
)i
Confirmation (Bolzano) Test: f (a)f (b) < 0
Simple analytic method for root isolation!
Simpler than algebraic subdivision methods:
STURM > DESCARTES > BOLZANO
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 89 / 113
![Page 446: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/446.jpg)
Analysis of Adaptive Complexity
Analytic Approach to Root Isolation
Suppose you want to isolate real roots of f (x) in I = [a,b]
Midpoint m(I) := (a+b)/2, Width w(I) := b−a
Exclusion Predicate: C0(I) : |f (m)|> ∑i≥1|f (i)(m)|
i!
(w(I)
2
)i
Inclusion Predicate: C1(I) : |f ′(m)|> ∑i≥1|f (i+1)(m)|
i!
(w(I)
2
)i
Confirmation (Bolzano) Test: f (a)f (b) < 0
Simple analytic method for root isolation!
Simpler than algebraic subdivision methods:
STURM > DESCARTES > BOLZANO
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 89 / 113
![Page 447: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/447.jpg)
Analysis of Adaptive Complexity
Analytic Approach to Root Isolation
Suppose you want to isolate real roots of f (x) in I = [a,b]
Midpoint m(I) := (a+b)/2, Width w(I) := b−a
Exclusion Predicate: C0(I) : |f (m)|> ∑i≥1|f (i)(m)|
i!
(w(I)
2
)i
Inclusion Predicate: C1(I) : |f ′(m)|> ∑i≥1|f (i+1)(m)|
i!
(w(I)
2
)i
Confirmation (Bolzano) Test: f (a)f (b) < 0
Simple analytic method for root isolation!
Simpler than algebraic subdivision methods:
STURM > DESCARTES > BOLZANO
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 89 / 113
![Page 448: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/448.jpg)
Analysis of Adaptive Complexity
Analytic Approach to Root Isolation
Suppose you want to isolate real roots of f (x) in I = [a,b]
Midpoint m(I) := (a+b)/2, Width w(I) := b−a
Exclusion Predicate: C0(I) : |f (m)|> ∑i≥1|f (i)(m)|
i!
(w(I)
2
)i
Inclusion Predicate: C1(I) : |f ′(m)|> ∑i≥1|f (i+1)(m)|
i!
(w(I)
2
)i
Confirmation (Bolzano) Test: f (a)f (b) < 0
Simple analytic method for root isolation!
Simpler than algebraic subdivision methods:
STURM > DESCARTES > BOLZANO
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 89 / 113
![Page 449: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/449.jpg)
Analysis of Adaptive Complexity
EVAL Algorithm
EVALINPUT: Function f and interval I0 = [a,b]
OUTPUT: Isolation intervals of roots of f in I01 Let Qin← I0 be a queue2 WHILE (Q 6= /0) ⊳ Subdivision Phase3 I← Q.remove()4 IF (C0(I) holds), discard I5 ELIF (C1(I) holds), output I6 ELSE7 IF (f (m(I)) = 0), output [m(I),m(I)]8 Split I into two and insert in Q9 PROCESS output list ⊳ Consruction Phase
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 90 / 113
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Analysis of Adaptive Complexity
EVAL Algorithm
EVALINPUT: Function f and interval I0 = [a,b]
OUTPUT: Isolation intervals of roots of f in I01 Let Qin← I0 be a queue2 WHILE (Q 6= /0) ⊳ Subdivision Phase3 I← Q.remove()4 IF (C0(I) holds), discard I5 ELIF (C1(I) holds), output I6 ELSE7 IF (f (m(I)) = 0), output [m(I),m(I)]8 Split I into two and insert in Q9 PROCESS output list ⊳ Consruction Phase
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 90 / 113
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Analysis of Adaptive Complexity
EVAL Algorithm
EVALINPUT: Function f and interval I0 = [a,b]
OUTPUT: Isolation intervals of roots of f in I01 Let Qin← I0 be a queue2 WHILE (Q 6= /0) ⊳ Subdivision Phase3 I← Q.remove()4 IF (C0(I) holds), discard I5 ELIF (C1(I) holds), output I6 ELSE7 IF (f (m(I)) = 0), output [m(I),m(I)]8 Split I into two and insert in Q9 PROCESS output list ⊳ Consruction Phase
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 90 / 113
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Analysis of Adaptive Complexity
EVAL Algorithm
EVALINPUT: Function f and interval I0 = [a,b]
OUTPUT: Isolation intervals of roots of f in I01 Let Qin← I0 be a queue2 WHILE (Q 6= /0) ⊳ Subdivision Phase3 I← Q.remove()4 IF (C0(I) holds), discard I5 ELIF (C1(I) holds), output I6 ELSE7 IF (f (m(I)) = 0), output [m(I),m(I)]8 Split I into two and insert in Q9 PROCESS output list ⊳ Consruction Phase
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 90 / 113
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Analysis of Adaptive Complexity
EVAL Algorithm
EVALINPUT: Function f and interval I0 = [a,b]
OUTPUT: Isolation intervals of roots of f in I01 Let Qin← I0 be a queue2 WHILE (Q 6= /0) ⊳ Subdivision Phase3 I← Q.remove()4 IF (C0(I) holds), discard I5 ELIF (C1(I) holds), output I6 ELSE7 IF (f (m(I)) = 0), output [m(I),m(I)]8 Split I into two and insert in Q9 PROCESS output list ⊳ Consruction Phase
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 90 / 113
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Analysis of Adaptive Complexity
EVAL Algorithm
EVALINPUT: Function f and interval I0 = [a,b]
OUTPUT: Isolation intervals of roots of f in I01 Let Qin← I0 be a queue2 WHILE (Q 6= /0) ⊳ Subdivision Phase3 I← Q.remove()4 IF (C0(I) holds), discard I5 ELIF (C1(I) holds), output I6 ELSE7 IF (f (m(I)) = 0), output [m(I),m(I)]8 Split I into two and insert in Q9 PROCESS output list ⊳ Consruction Phase
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 90 / 113
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Analysis of Adaptive Complexity
EVAL Algorithm
EVALINPUT: Function f and interval I0 = [a,b]
OUTPUT: Isolation intervals of roots of f in I01 Let Qin← I0 be a queue2 WHILE (Q 6= /0) ⊳ Subdivision Phase3 I← Q.remove()4 IF (C0(I) holds), discard I5 ELIF (C1(I) holds), output I6 ELSE7 IF (f (m(I)) = 0), output [m(I),m(I)]8 Split I into two and insert in Q9 PROCESS output list ⊳ Consruction Phase
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 90 / 113
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Analysis of Adaptive Complexity
EVAL Algorithm
EVALINPUT: Function f and interval I0 = [a,b]
OUTPUT: Isolation intervals of roots of f in I01 Let Qin← I0 be a queue2 WHILE (Q 6= /0) ⊳ Subdivision Phase3 I← Q.remove()4 IF (C0(I) holds), discard I5 ELIF (C1(I) holds), output I6 ELSE7 IF (f (m(I)) = 0), output [m(I),m(I)]8 Split I into two and insert in Q9 PROCESS output list ⊳ Consruction Phase
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 90 / 113
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Analysis of Adaptive Complexity
EVAL Algorithm
EVALINPUT: Function f and interval I0 = [a,b]
OUTPUT: Isolation intervals of roots of f in I01 Let Qin← I0 be a queue2 WHILE (Q 6= /0) ⊳ Subdivision Phase3 I← Q.remove()4 IF (C0(I) holds), discard I5 ELIF (C1(I) holds), output I6 ELSE7 IF (f (m(I)) = 0), output [m(I),m(I)]8 Split I into two and insert in Q9 PROCESS output list ⊳ Consruction Phase
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 90 / 113
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Analysis of Adaptive Complexity
EVAL Algorithm
EVALINPUT: Function f and interval I0 = [a,b]
OUTPUT: Isolation intervals of roots of f in I01 Let Qin← I0 be a queue2 WHILE (Q 6= /0) ⊳ Subdivision Phase3 I← Q.remove()4 IF (C0(I) holds), discard I5 ELIF (C1(I) holds), output I6 ELSE7 IF (f (m(I)) = 0), output [m(I),m(I)]8 Split I into two and insert in Q9 PROCESS output list ⊳ Consruction Phase
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 90 / 113
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Analysis of Adaptive Complexity
EVAL Algorithm
EVALINPUT: Function f and interval I0 = [a,b]
OUTPUT: Isolation intervals of roots of f in I01 Let Qin← I0 be a queue2 WHILE (Q 6= /0) ⊳ Subdivision Phase3 I← Q.remove()4 IF (C0(I) holds), discard I5 ELIF (C1(I) holds), output I6 ELSE7 IF (f (m(I)) = 0), output [m(I),m(I)]8 Split I into two and insert in Q9 PROCESS output list ⊳ Consruction Phase
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 90 / 113
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Analysis of Adaptive Complexity
EVAL Algorithm
EVALINPUT: Function f and interval I0 = [a,b]
OUTPUT: Isolation intervals of roots of f in I01 Let Qin← I0 be a queue2 WHILE (Q 6= /0) ⊳ Subdivision Phase3 I← Q.remove()4 IF (C0(I) holds), discard I5 ELIF (C1(I) holds), output I6 ELSE7 IF (f (m(I)) = 0), output [m(I),m(I)]8 Split I into two and insert in Q9 PROCESS output list ⊳ Consruction Phase
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 90 / 113
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Analysis of Adaptive Complexity
EVAL Algorithm
EVALINPUT: Function f and interval I0 = [a,b]
OUTPUT: Isolation intervals of roots of f in I01 Let Qin← I0 be a queue2 WHILE (Q 6= /0) ⊳ Subdivision Phase3 I← Q.remove()4 IF (C0(I) holds), discard I5 ELIF (C1(I) holds), output I6 ELSE7 IF (f (m(I)) = 0), output [m(I),m(I)]8 Split I into two and insert in Q9 PROCESS output list ⊳ Consruction Phase
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 90 / 113
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Analysis of Adaptive Complexity
Main Complexity Goal – Benchmark Problem
Benchmark Problem in Root IsolationProblem: isolate ALL (real) roots of square-free f (X) ∈Z[X ] ofdegree ≤ d and height < 2L.
Highly classical problem: Bit complexity is O(d3L) [Schohage 1982].
⋆ Improvement: O(d2L) arithmetic complexity [Pan]
Sturm tree size is O(d(L+ logd)) [Davenport, 1985] Descartes tree size is Θ(d(L+ logd)) [Eigenwillig-Sharma-Y,
2006]
MAIN RESULT: Bolzano tree size is O(d2(L+ logd)) Sketch in this lecture. See [Burr-Krahmer-Y-Sagraloff, 2008-9]
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 91 / 113
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Analysis of Adaptive Complexity
Main Complexity Goal – Benchmark Problem
Benchmark Problem in Root IsolationProblem: isolate ALL (real) roots of square-free f (X) ∈Z[X ] ofdegree ≤ d and height < 2L.
Highly classical problem: Bit complexity is O(d3L) [Schohage 1982].
⋆ Improvement: O(d2L) arithmetic complexity [Pan]
Sturm tree size is O(d(L+ logd)) [Davenport, 1985] Descartes tree size is Θ(d(L+ logd)) [Eigenwillig-Sharma-Y,
2006]
MAIN RESULT: Bolzano tree size is O(d2(L+ logd)) Sketch in this lecture. See [Burr-Krahmer-Y-Sagraloff, 2008-9]
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 91 / 113
![Page 464: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/464.jpg)
Analysis of Adaptive Complexity
Main Complexity Goal – Benchmark Problem
Benchmark Problem in Root IsolationProblem: isolate ALL (real) roots of square-free f (X) ∈Z[X ] ofdegree ≤ d and height < 2L.
Highly classical problem: Bit complexity is O(d3L) [Schohage 1982].
⋆ Improvement: O(d2L) arithmetic complexity [Pan]
Sturm tree size is O(d(L+ logd)) [Davenport, 1985] Descartes tree size is Θ(d(L+ logd)) [Eigenwillig-Sharma-Y,
2006]
MAIN RESULT: Bolzano tree size is O(d2(L+ logd)) Sketch in this lecture. See [Burr-Krahmer-Y-Sagraloff, 2008-9]
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 91 / 113
![Page 465: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/465.jpg)
Analysis of Adaptive Complexity
Main Complexity Goal – Benchmark Problem
Benchmark Problem in Root IsolationProblem: isolate ALL (real) roots of square-free f (X) ∈Z[X ] ofdegree ≤ d and height < 2L.
Highly classical problem: Bit complexity is O(d3L) [Schohage 1982].
⋆ Improvement: O(d2L) arithmetic complexity [Pan]
Sturm tree size is O(d(L+ logd)) [Davenport, 1985] Descartes tree size is Θ(d(L+ logd)) [Eigenwillig-Sharma-Y,
2006]
MAIN RESULT: Bolzano tree size is O(d2(L+ logd)) Sketch in this lecture. See [Burr-Krahmer-Y-Sagraloff, 2008-9]
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 91 / 113
![Page 466: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/466.jpg)
Analysis of Adaptive Complexity
Main Complexity Goal – Benchmark Problem
Benchmark Problem in Root IsolationProblem: isolate ALL (real) roots of square-free f (X) ∈Z[X ] ofdegree ≤ d and height < 2L.
Highly classical problem: Bit complexity is O(d3L) [Schohage 1982].
⋆ Improvement: O(d2L) arithmetic complexity [Pan]
Sturm tree size is O(d(L+ logd)) [Davenport, 1985] Descartes tree size is Θ(d(L+ logd)) [Eigenwillig-Sharma-Y,
2006]
MAIN RESULT: Bolzano tree size is O(d2(L+ logd)) Sketch in this lecture. See [Burr-Krahmer-Y-Sagraloff, 2008-9]
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 91 / 113
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Analysis of Adaptive Complexity
Main Complexity Goal – Benchmark Problem
Benchmark Problem in Root IsolationProblem: isolate ALL (real) roots of square-free f (X) ∈Z[X ] ofdegree ≤ d and height < 2L.
Highly classical problem: Bit complexity is O(d3L) [Schohage 1982].
⋆ Improvement: O(d2L) arithmetic complexity [Pan]
Sturm tree size is O(d(L+ logd)) [Davenport, 1985] Descartes tree size is Θ(d(L+ logd)) [Eigenwillig-Sharma-Y,
2006]
MAIN RESULT: Bolzano tree size is O(d2(L+ logd)) Sketch in this lecture. See [Burr-Krahmer-Y-Sagraloff, 2008-9]
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 91 / 113
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Analysis of Adaptive Complexity
Main Complexity Goal – Benchmark Problem
Benchmark Problem in Root IsolationProblem: isolate ALL (real) roots of square-free f (X) ∈Z[X ] ofdegree ≤ d and height < 2L.
Highly classical problem: Bit complexity is O(d3L) [Schohage 1982].
⋆ Improvement: O(d2L) arithmetic complexity [Pan]
Sturm tree size is O(d(L+ logd)) [Davenport, 1985] Descartes tree size is Θ(d(L+ logd)) [Eigenwillig-Sharma-Y,
2006]
MAIN RESULT: Bolzano tree size is O(d2(L+ logd)) Sketch in this lecture. See [Burr-Krahmer-Y-Sagraloff, 2008-9]
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 91 / 113
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Analysis of Adaptive Complexity
Main Complexity Goal – Benchmark Problem
Benchmark Problem in Root IsolationProblem: isolate ALL (real) roots of square-free f (X) ∈Z[X ] ofdegree ≤ d and height < 2L.
Highly classical problem: Bit complexity is O(d3L) [Schohage 1982].
⋆ Improvement: O(d2L) arithmetic complexity [Pan]
Sturm tree size is O(d(L+ logd)) [Davenport, 1985] Descartes tree size is Θ(d(L+ logd)) [Eigenwillig-Sharma-Y,
2006]
MAIN RESULT: Bolzano tree size is O(d2(L+ logd)) Sketch in this lecture. See [Burr-Krahmer-Y-Sagraloff, 2008-9]
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 91 / 113
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Analysis of Adaptive Complexity
Main Complexity Goal – Benchmark Problem
Benchmark Problem in Root IsolationProblem: isolate ALL (real) roots of square-free f (X) ∈Z[X ] ofdegree ≤ d and height < 2L.
Highly classical problem: Bit complexity is O(d3L) [Schohage 1982].
⋆ Improvement: O(d2L) arithmetic complexity [Pan]
Sturm tree size is O(d(L+ logd)) [Davenport, 1985] Descartes tree size is Θ(d(L+ logd)) [Eigenwillig-Sharma-Y,
2006]
MAIN RESULT: Bolzano tree size is O(d2(L+ logd)) Sketch in this lecture. See [Burr-Krahmer-Y-Sagraloff, 2008-9]
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 91 / 113
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Analysis of Adaptive Complexity
Main Complexity Goal – Benchmark Problem
Benchmark Problem in Root IsolationProblem: isolate ALL (real) roots of square-free f (X) ∈Z[X ] ofdegree ≤ d and height < 2L.
Highly classical problem: Bit complexity is O(d3L) [Schohage 1982].
⋆ Improvement: O(d2L) arithmetic complexity [Pan]
Sturm tree size is O(d(L+ logd)) [Davenport, 1985] Descartes tree size is Θ(d(L+ logd)) [Eigenwillig-Sharma-Y,
2006]
MAIN RESULT: Bolzano tree size is O(d2(L+ logd)) Sketch in this lecture. See [Burr-Krahmer-Y-Sagraloff, 2008-9]
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Analysis of Adaptive Complexity
Warm Up Technique: Algebraic Amortization
Idea of Amortization [Davenport (1985), Du/Sharma/Y. (2005)]
Let A(X) ∈Z[X ] have degree n and L-bit coefficients.
Root separation bound: − log |α−β |= O(n(L+ logn))
Amortized bound: −∏(α,β )∈E |β −α|= O(n(L+ logn))
What are restrictions on set E?
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Analysis of Adaptive Complexity
Warm Up Technique: Algebraic Amortization
Idea of Amortization [Davenport (1985), Du/Sharma/Y. (2005)]
Let A(X) ∈Z[X ] have degree n and L-bit coefficients.
Root separation bound: − log |α−β |= O(n(L+ logn))
Amortized bound: −∏(α,β )∈E |β −α|= O(n(L+ logn))
What are restrictions on set E?
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Analysis of Adaptive Complexity
Warm Up Technique: Algebraic Amortization
Idea of Amortization [Davenport (1985), Du/Sharma/Y. (2005)]
Let A(X) ∈Z[X ] have degree n and L-bit coefficients.
Root separation bound: − log |α−β |= O(n(L+ logn))
Amortized bound: −∏(α,β )∈E |β −α|= O(n(L+ logn))
What are restrictions on set E?
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Analysis of Adaptive Complexity
Warm Up Technique: Algebraic Amortization
Idea of Amortization [Davenport (1985), Du/Sharma/Y. (2005)]
Let A(X) ∈Z[X ] have degree n and L-bit coefficients.
Root separation bound: − log |α−β |= O(n(L+ logn))
Amortized bound: −∏(α,β )∈E |β −α|= O(n(L+ logn))
What are restrictions on set E?
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Analysis of Adaptive Complexity
Warm Up Technique: Algebraic Amortization
Idea of Amortization [Davenport (1985), Du/Sharma/Y. (2005)]
Let A(X) ∈Z[X ] have degree n and L-bit coefficients.
Root separation bound: − log |α−β |= O(n(L+ logn))
Amortized bound: −∏(α,β )∈E |β −α|= O(n(L+ logn))
What are restrictions on set E?
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Analysis of Adaptive Complexity
Warm Up Technique: Algebraic Amortization
Idea of Amortization [Davenport (1985), Du/Sharma/Y. (2005)]
Let A(X) ∈Z[X ] have degree n and L-bit coefficients.
Root separation bound: − log |α−β |= O(n(L+ logn))
Amortized bound: −∏(α,β )∈E |β −α|= O(n(L+ logn))
What are restrictions on set E?
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Analysis of Adaptive Complexity
The Davenport–Mahler Bound
Theorem ([Davenport (1985), Johnson (1991/98), Du/Sharma/Y. (2005)])
Consider a polynomial A(X) ∈ C[X ] of degree n. Let G = (V ,E) be adigraph whose node set V consists of the roots ϑ1, . . . ,ϑn of A(X). If
(i) (α,β ) ∈ E =⇒ |α| ≤ |β |,(ii) β ∈ V =⇒ indeg(β )≤ 1, and
(iii) G is acyclic,
then∏
(α,β )∈E
|β −α| ≥√|discr(A)|
M(A)n−1 ·2−O(n logn),
where
discr(A) := a2n−2n ∏
i>j
(ϑi −ϑj)2 and M(A) := |an|∏
i
max1, |ϑi |.
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Mini Summary
Adaptive analysis is important but virgin territory
Subdivision of Analytic Algorithms in 1-D is current challenge
Standard target is Benchmark Problem for root isolation
Warm-Up Exercise: Use Mahler-Davenport bound for DescartesMethod
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Mini Summary
Adaptive analysis is important but virgin territory
Subdivision of Analytic Algorithms in 1-D is current challenge
Standard target is Benchmark Problem for root isolation
Warm-Up Exercise: Use Mahler-Davenport bound for DescartesMethod
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Mini Summary
Adaptive analysis is important but virgin territory
Subdivision of Analytic Algorithms in 1-D is current challenge
Standard target is Benchmark Problem for root isolation
Warm-Up Exercise: Use Mahler-Davenport bound for DescartesMethod
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Mini Summary
Adaptive analysis is important but virgin territory
Subdivision of Analytic Algorithms in 1-D is current challenge
Standard target is Benchmark Problem for root isolation
Warm-Up Exercise: Use Mahler-Davenport bound for DescartesMethod
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Mini Summary
Adaptive analysis is important but virgin territory
Subdivision of Analytic Algorithms in 1-D is current challenge
Standard target is Benchmark Problem for root isolation
Warm-Up Exercise: Use Mahler-Davenport bound for DescartesMethod
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Mini Summary
Adaptive analysis is important but virgin territory
Subdivision of Analytic Algorithms in 1-D is current challenge
Standard target is Benchmark Problem for root isolation
Warm-Up Exercise: Use Mahler-Davenport bound for DescartesMethod
![Page 485: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/485.jpg)
Analysis of Descartes Method
Coming Up Next
12 Analysis of Adaptive Complexity
13 Analysis of Descartes Method
14 Integral Bounds and Framework of Stopping Functions
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Analysis of Descartes Method
What is the Descartes Method?
Same framework as EVAL or SturmTo isolate roots of square-free A(X) in interval I
Routine DescartesTest (A(X), I) gives an upper estimate on thenumber of real roots in I.
If DescartesTest (A(X), I) ∈ 0,1 then estimate is exact.
We keep splitting intervals until we get an exact estimate.
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Analysis of Descartes Method
What is the Descartes Method?
Same framework as EVAL or SturmTo isolate roots of square-free A(X) in interval I
Routine DescartesTest (A(X), I) gives an upper estimate on thenumber of real roots in I.
If DescartesTest (A(X), I) ∈ 0,1 then estimate is exact.
We keep splitting intervals until we get an exact estimate.
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Analysis of Descartes Method
What is the Descartes Method?
Same framework as EVAL or SturmTo isolate roots of square-free A(X) in interval I
Routine DescartesTest (A(X), I) gives an upper estimate on thenumber of real roots in I.
If DescartesTest (A(X), I) ∈ 0,1 then estimate is exact.
We keep splitting intervals until we get an exact estimate.
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Analysis of Descartes Method
What is the Descartes Method?
Same framework as EVAL or SturmTo isolate roots of square-free A(X) in interval I
Routine DescartesTest (A(X), I) gives an upper estimate on thenumber of real roots in I.
If DescartesTest (A(X), I) ∈ 0,1 then estimate is exact.
We keep splitting intervals until we get an exact estimate.
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Analysis of Descartes Method
What is the Descartes Method?
Same framework as EVAL or SturmTo isolate roots of square-free A(X) in interval I
Routine DescartesTest (A(X), I) gives an upper estimate on thenumber of real roots in I.
If DescartesTest (A(X), I) ∈ 0,1 then estimate is exact.
We keep splitting intervals until we get an exact estimate.
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Analysis of Descartes Method
What is the Descartes Method?
Same framework as EVAL or SturmTo isolate roots of square-free A(X) in interval I
Routine DescartesTest (A(X), I) gives an upper estimate on thenumber of real roots in I.
If DescartesTest (A(X), I) ∈ 0,1 then estimate is exact.
We keep splitting intervals until we get an exact estimate.
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Analysis of Descartes Method
Analysis of Descartes Method
Two-circle Theorem[Ostrowski (1950), Krandick/Mehlhorn (2006)]
If DescartesTest (A(X), [c,d ])≥ 2, then thetwo-circles figure in C around interval [c,d ]contains two roots α,β of A(X).
CorollaryCan choose α,β to be complex conjugate or adjacent real roots.Moreover, |β −α|<
√3(d−c); i.e., (d−c) > |β −α|/
√3.
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Analysis of Descartes Method
Analysis of Descartes Method
Two-circle Theorem[Ostrowski (1950), Krandick/Mehlhorn (2006)]
If DescartesTest (A(X), [c,d ])≥ 2, then thetwo-circles figure in C around interval [c,d ]contains two roots α,β of A(X).
CorollaryCan choose α,β to be complex conjugate or adjacent real roots.Moreover, |β −α|<
√3(d−c); i.e., (d−c) > |β −α|/
√3.
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Analysis of Descartes Method
Analysis of Descartes Method
Two-circle Theorem[Ostrowski (1950), Krandick/Mehlhorn (2006)]
If DescartesTest (A(X), [c,d ])≥ 2, then thetwo-circles figure in C around interval [c,d ]contains two roots α,β of A(X).
CorollaryCan choose α,β to be complex conjugate or adjacent real roots.Moreover, |β −α|<
√3(d−c); i.e., (d−c) > |β −α|/
√3.
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Analysis of Descartes Method
Analysis of Descartes Method
Two-circle Theorem[Ostrowski (1950), Krandick/Mehlhorn (2006)]
If DescartesTest (A(X), [c,d ])≥ 2, then thetwo-circles figure in C around interval [c,d ]contains two roots α,β of A(X).
CorollaryCan choose α,β to be complex conjugate or adjacent real roots.Moreover, |β −α|<
√3(d−c); i.e., (d−c) > |β −α|/
√3.
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Analysis of Descartes Method
Analysis of Descartes Method
Two-circle Theorem[Ostrowski (1950), Krandick/Mehlhorn (2006)]
If DescartesTest (A(X), [c,d ])≥ 2, then thetwo-circles figure in C around interval [c,d ]contains two roots α,β of A(X).
CorollaryCan choose α,β to be complex conjugate or adjacent real roots.Moreover, |β −α|<
√3(d−c); i.e., (d−c) > |β −α|/
√3.
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Analysis of Descartes Method
Analysis of Descartes Method
β
α Two-circle Theorem[Ostrowski (1950), Krandick/Mehlhorn (2006)]
If DescartesTest (A(X), [c,d ])≥ 2, then thetwo-circles figure in C around interval [c,d ]contains two roots α,β of A(X).
CorollaryCan choose α,β to be complex conjugate or adjacent real roots.Moreover, |β −α|<
√3(d−c); i.e., (d−c) > |β −α|/
√3.
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Analysis of Descartes Method
Analysis of Descartes Method
βα
Two-circle Theorem[Ostrowski (1950), Krandick/Mehlhorn (2006)]
If DescartesTest (A(X), [c,d ])≥ 2, then thetwo-circles figure in C around interval [c,d ]contains two roots α,β of A(X).
CorollaryCan choose α,β to be complex conjugate or adjacent real roots.Moreover, |β −α|<
√3(d−c); i.e., (d−c) > |β −α|/
√3.
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Analysis of Descartes Method
Analysis of Descartes Method
Two-circle Theorem[Ostrowski (1950), Krandick/Mehlhorn (2006)]
If DescartesTest (A(X), [c,d ])≥ 2, then thetwo-circles figure in C around interval [c,d ]contains two roots α,β of A(X).
CorollaryCan choose α,β to be complex conjugate or adjacent real roots.Moreover, |β −α|<
√3(d−c); i.e., (d−c) > |β −α|/
√3.
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Analysis of Descartes Method
Tree Bound in terms of Roots (1)
J =3d
0IA bound on path length
1 Consider any path in the recursion treefrom I0 to a parent J of two leaves.
2 At depth d , interval width is 2−d |I0|.Hence depth of J is d = log |I0|/|J|.
3 The path consists of d +1 internalnodes.
4 There is a pair of roots (αJ ,βJ)such that |J|> |βJ−αJ |/
√3; hence
d +1 < log |I0|− log |βJ−αJ |+2.
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Analysis of Descartes Method
Tree Bound in terms of Roots (1)
J =3d
0IA bound on path length
1 Consider any path in the recursion treefrom I0 to a parent J of two leaves.
2 At depth d , interval width is 2−d |I0|.Hence depth of J is d = log |I0|/|J|.
3 The path consists of d +1 internalnodes.
4 There is a pair of roots (αJ ,βJ)such that |J|> |βJ−αJ |/
√3; hence
d +1 < log |I0|− log |βJ−αJ |+2.
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Analysis of Descartes Method
Tree Bound in terms of Roots (1)
J =3d
0IA bound on path length
1 Consider any path in the recursion treefrom I0 to a parent J of two leaves.
2 At depth d , interval width is 2−d |I0|.Hence depth of J is d = log |I0|/|J|.
3 The path consists of d +1 internalnodes.
4 There is a pair of roots (αJ ,βJ)such that |J|> |βJ−αJ |/
√3; hence
d +1 < log |I0|− log |βJ−αJ |+2.
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Analysis of Descartes Method
Tree Bound in terms of Roots (1)
J =3d
0IA bound on path length
1 Consider any path in the recursion treefrom I0 to a parent J of two leaves.
2 At depth d , interval width is 2−d |I0|.Hence depth of J is d = log |I0|/|J|.
3 The path consists of d +1 internalnodes.
4 There is a pair of roots (αJ ,βJ)such that |J|> |βJ−αJ |/
√3; hence
d +1 < log |I0|− log |βJ−αJ |+2.
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Analysis of Descartes Method
Tree Bound in terms of Roots (2)
J
0I
#(internal nodes on path) < log |I0|− log |βJ−αJ |+2#(internal nodes in tree) < ∑J (log |I0|− log |βJ−αJ |+2)#(all nodes in tree) < 1+2 ·∑J (log |I0|− log |βJ−αJ |+2)
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Analysis of Descartes Method
Tree Bound in terms of Roots (2)
JJ
J
J
0I
#(internal nodes on path) < log |I0|− log |βJ−αJ |+2#(internal nodes in tree) < ∑J (log |I0|− log |βJ−αJ |+2)#(all nodes in tree) < 1+2 ·∑J (log |I0|− log |βJ−αJ |+2)
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Analysis of Descartes Method
Tree Bound in terms of Roots (2)
JJ
J
J
0I
#(internal nodes on path) < log |I0|− log |βJ−αJ |+2#(internal nodes in tree) < ∑J (log |I0|− log |βJ−αJ |+2)#(all nodes in tree) < 1+2 ·∑J (log |I0|− log |βJ−αJ |+2)
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Analysis of Descartes Method
Tree Bound in terms of Roots (2)
JJ
J
J
0I
#(internal nodes on path) < log |I0|− log |βJ−αJ |+2#(internal nodes in tree) < ∑J (log |I0|− log |βJ−αJ |+2)#(all nodes in tree) < 1+2 ·∑J (log |I0|− log |βJ−αJ |+2)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 99 / 113
PropositionThe size of the recursion tree is bounded by
−2 log ∏J
|βJ−αJ |+n log |I0|+2n +1
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Analysis of Descartes Method
Tree Bound in terms of Roots (2)
JJ
J
J
0I
#(internal nodes on path) < log |I0|− log |βJ−αJ |+2#(internal nodes in tree) < ∑J (log |I0|− log |βJ−αJ |+2)#(all nodes in tree) < 1+2 ·∑J (log |I0|− log |βJ−αJ |+2)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 99 / 113
PropositionThe size of the recursion tree is bounded by
−2 log ∏J
|βJ−αJ |+n log |I0|+2n +1
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Analysis of Descartes Method
Turning our Product into an Admissible Graph
We want to rewrite
∏J
|βJ−αJ | as ∏(α,β )∈E
|β −α|.
How often |βJ−αJ | appears?
adjacent real: ≤ 1
complex conjugate ≤ 2
We need two graphs. (Paper: just 1)
Conditions on G = (V ,E)
(i) (α,β ) ∈ E =⇒ |α| ≤ |β |(ii) β ∈ V =⇒ indeg(β )≤ 1 (iii) G is acyclic
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Analysis of Descartes Method
Turning our Product into an Admissible Graph
We want to rewrite
∏J
|βJ−αJ | as ∏(α,β )∈E
|β −α|.
How often |βJ−αJ | appears?
adjacent real: ≤ 1
complex conjugate ≤ 2
We need two graphs. (Paper: just 1)
Conditions on G = (V ,E)
(i) (α,β ) ∈ E =⇒ |α| ≤ |β |(ii) β ∈ V =⇒ indeg(β )≤ 1 (iii) G is acyclic
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 100 / 113
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Analysis of Descartes Method
Turning our Product into an Admissible Graph
We want to rewrite
∏J
|βJ−αJ | as ∏(α,β )∈E
|β −α|.
How often |βJ−αJ | appears?
adjacent real: ≤ 1
complex conjugate ≤ 2
We need two graphs. (Paper: just 1)
Conditions on G = (V ,E)
(i) (α,β ) ∈ E =⇒ |α| ≤ |β |(ii) β ∈ V =⇒ indeg(β )≤ 1 (iii) G is acyclic
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 100 / 113
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Analysis of Descartes Method
Turning our Product into an Admissible Graph
We want to rewrite
∏J
|βJ−αJ | as ∏(α,β )∈E
|β −α|.
How often |βJ−αJ | appears?
adjacent real: ≤ 1
complex conjugate ≤ 2
We need two graphs. (Paper: just 1)
Conditions on G = (V ,E)
(i) (α,β ) ∈ E =⇒ |α| ≤ |β |(ii) β ∈ V =⇒ indeg(β )≤ 1 (iii) G is acyclic
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 100 / 113
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Analysis of Descartes Method
Turning our Product into an Admissible Graph
We want to rewrite
∏J
|βJ−αJ | as ∏(α,β )∈E
|β −α|.
How often |βJ−αJ | appears?
adjacent real: ≤ 1
complex conjugate ≤ 2
We need two graphs. (Paper: just 1)
Conditions on G = (V ,E)
(i) (α,β ) ∈ E =⇒ |α| ≤ |β |(ii) β ∈ V =⇒ indeg(β )≤ 1 (iii) G is acyclic
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 100 / 113
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Analysis of Descartes Method
Turning our Product into an Admissible Graph
We want to rewrite
∏J
|βJ−αJ | as ∏(α,β )∈E
|β −α|.
How often |βJ−αJ | appears?
adjacent real: ≤ 1
complex conjugate ≤ 2
We need two graphs. (Paper: just 1)
Conditions on G = (V ,E)
(i) (α,β ) ∈ E =⇒ |α| ≤ |β |(ii) β ∈ V =⇒ indeg(β )≤ 1 (iii) G is acyclic
0
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 100 / 113
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Analysis of Descartes Method
Turning our Product into an Admissible Graph
We want to rewrite
∏J
|βJ−αJ | as ∏(α,β )∈E
|β −α|.
How often |βJ−αJ | appears?
adjacent real: ≤ 1
complex conjugate ≤ 2
We need two graphs. (Paper: just 1)
Conditions on G = (V ,E)
(i) (α,β ) ∈ E =⇒ |α| ≤ |β |(ii) β ∈ V =⇒ indeg(β )≤ 1 (iii) G is acyclic
0
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 100 / 113
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Analysis of Descartes Method
Turning our Product into an Admissible Graph
We want to rewrite
∏J
|βJ−αJ | as ∏(α,β )∈E
|β −α|.
How often |βJ−αJ | appears?
adjacent real: ≤ 1
complex conjugate ≤ 2
We need two graphs. (Paper: just 1)
Conditions on G = (V ,E)
(i) (α,β ) ∈ E =⇒ |α| ≤ |β |(ii) β ∈ V =⇒ indeg(β )≤ 1 (iii) G is acyclic
0
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 100 / 113
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Analysis of Descartes Method
Main Result on Descartes Analysis
Theorem (Eigenwillig/Sharma/Y. (2006))On the Benchmark Problem, we obtain
|T |= O(n(L+ logn)).
For L≥ logn, this is optimal.
Argument of [Krandick/Mehlhorn, 2006]: |T |= O(n logn (L+ logn)).
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Mini Summary
Almost Tight Bound on Descartes Method based on AlgebraicAmortization
Benchmark complexity of Sturm and Descartes are the same “theory caught up with practice”
What about EVAL? New ideas needed – one is Amortized Evaluation Bounds
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Mini Summary
Almost Tight Bound on Descartes Method based on AlgebraicAmortization
Benchmark complexity of Sturm and Descartes are the same “theory caught up with practice”
What about EVAL? New ideas needed – one is Amortized Evaluation Bounds
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Mini Summary
Almost Tight Bound on Descartes Method based on AlgebraicAmortization
Benchmark complexity of Sturm and Descartes are the same “theory caught up with practice”
What about EVAL? New ideas needed – one is Amortized Evaluation Bounds
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Mini Summary
Almost Tight Bound on Descartes Method based on AlgebraicAmortization
Benchmark complexity of Sturm and Descartes are the same “theory caught up with practice”
What about EVAL? New ideas needed – one is Amortized Evaluation Bounds
![Page 522: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/522.jpg)
Mini Summary
Almost Tight Bound on Descartes Method based on AlgebraicAmortization
Benchmark complexity of Sturm and Descartes are the same “theory caught up with practice”
What about EVAL? New ideas needed – one is Amortized Evaluation Bounds
![Page 523: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/523.jpg)
Mini Summary
Almost Tight Bound on Descartes Method based on AlgebraicAmortization
Benchmark complexity of Sturm and Descartes are the same “theory caught up with practice”
What about EVAL? New ideas needed – one is Amortized Evaluation Bounds
![Page 524: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/524.jpg)
Mini Summary
Almost Tight Bound on Descartes Method based on AlgebraicAmortization
Benchmark complexity of Sturm and Descartes are the same “theory caught up with practice”
What about EVAL? New ideas needed – one is Amortized Evaluation Bounds
![Page 525: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/525.jpg)
Integral Bounds and Framework of Stopping Functions
Coming Up Next
12 Analysis of Adaptive Complexity
13 Analysis of Descartes Method
14 Integral Bounds and Framework of Stopping Functions
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Integral Bounds and Framework of Stopping Functions
Subdivision Phase
Subdivision based on a Predicate C(I)Initialize a queue Q←I0
1 WHILE (Q 6= /0)2 I← Q.remove()3 IF (C(I) holds), output I4 ELSE5 Split I and insert children into Q
Goal – Bound the size of recursion tree T (I0)
NOTE: C(I)≡ C0(I)∨C1(I) in EVAL
The leaves of T (I0) induces a partition P(I) of I0Suffices to upper bound #P(I0)
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Integral Bounds and Framework of Stopping Functions
Subdivision Phase
Subdivision based on a Predicate C(I)Initialize a queue Q←I0
1 WHILE (Q 6= /0)2 I← Q.remove()3 IF (C(I) holds), output I4 ELSE5 Split I and insert children into Q
Goal – Bound the size of recursion tree T (I0)
NOTE: C(I)≡ C0(I)∨C1(I) in EVAL
The leaves of T (I0) induces a partition P(I) of I0Suffices to upper bound #P(I0)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 104 / 113
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Integral Bounds and Framework of Stopping Functions
Subdivision Phase
Subdivision based on a Predicate C(I)Initialize a queue Q←I0
1 WHILE (Q 6= /0)2 I← Q.remove()3 IF (C(I) holds), output I4 ELSE5 Split I and insert children into Q
Goal – Bound the size of recursion tree T (I0)
NOTE: C(I)≡ C0(I)∨C1(I) in EVAL
The leaves of T (I0) induces a partition P(I) of I0Suffices to upper bound #P(I0)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 104 / 113
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Integral Bounds and Framework of Stopping Functions
Subdivision Phase
Subdivision based on a Predicate C(I)Initialize a queue Q←I0
1 WHILE (Q 6= /0)2 I← Q.remove()3 IF (C(I) holds), output I4 ELSE5 Split I and insert children into Q
Goal – Bound the size of recursion tree T (I0)
NOTE: C(I)≡ C0(I)∨C1(I) in EVAL
The leaves of T (I0) induces a partition P(I) of I0Suffices to upper bound #P(I0)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 104 / 113
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Integral Bounds and Framework of Stopping Functions
Subdivision Phase
Subdivision based on a Predicate C(I)Initialize a queue Q←I0
1 WHILE (Q 6= /0)2 I← Q.remove()3 IF (C(I) holds), output I4 ELSE5 Split I and insert children into Q
Goal – Bound the size of recursion tree T (I0)
NOTE: C(I)≡ C0(I)∨C1(I) in EVAL
The leaves of T (I0) induces a partition P(I) of I0Suffices to upper bound #P(I0)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 104 / 113
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Integral Bounds and Framework of Stopping Functions
Subdivision Phase
Subdivision based on a Predicate C(I)Initialize a queue Q←I0
1 WHILE (Q 6= /0)2 I← Q.remove()3 IF (C(I) holds), output I4 ELSE5 Split I and insert children into Q
Goal – Bound the size of recursion tree T (I0)
NOTE: C(I)≡ C0(I)∨C1(I) in EVAL
The leaves of T (I0) induces a partition P(I) of I0Suffices to upper bound #P(I0)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 104 / 113
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Integral Bounds and Framework of Stopping Functions
Subdivision Phase
Subdivision based on a Predicate C(I)Initialize a queue Q←I0
1 WHILE (Q 6= /0)2 I← Q.remove()3 IF (C(I) holds), output I4 ELSE5 Split I and insert children into Q
Goal – Bound the size of recursion tree T (I0)
NOTE: C(I)≡ C0(I)∨C1(I) in EVAL
The leaves of T (I0) induces a partition P(I) of I0Suffices to upper bound #P(I0)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 104 / 113
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Integral Bounds and Framework of Stopping Functions
Subdivision Phase
Subdivision based on a Predicate C(I)Initialize a queue Q←I0
1 WHILE (Q 6= /0)2 I← Q.remove()3 IF (C(I) holds), output I4 ELSE5 Split I and insert children into Q
Goal – Bound the size of recursion tree T (I0)
NOTE: C(I)≡ C0(I)∨C1(I) in EVAL
The leaves of T (I0) induces a partition P(I) of I0Suffices to upper bound #P(I0)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 104 / 113
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Integral Bounds and Framework of Stopping Functions
Subdivision Phase
Subdivision based on a Predicate C(I)Initialize a queue Q←I0
1 WHILE (Q 6= /0)2 I← Q.remove()3 IF (C(I) holds), output I4 ELSE5 Split I and insert children into Q
Goal – Bound the size of recursion tree T (I0)
NOTE: C(I)≡ C0(I)∨C1(I) in EVAL
The leaves of T (I0) induces a partition P(I) of I0Suffices to upper bound #P(I0)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 104 / 113
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Integral Bounds and Framework of Stopping Functions
Subdivision Phase
Subdivision based on a Predicate C(I)Initialize a queue Q←I0
1 WHILE (Q 6= /0)2 I← Q.remove()3 IF (C(I) holds), output I4 ELSE5 Split I and insert children into Q
Goal – Bound the size of recursion tree T (I0)
NOTE: C(I)≡ C0(I)∨C1(I) in EVAL
The leaves of T (I0) induces a partition P(I) of I0Suffices to upper bound #P(I0)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 104 / 113
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Integral Bounds and Framework of Stopping Functions
Subdivision Phase
Subdivision based on a Predicate C(I)Initialize a queue Q←I0
1 WHILE (Q 6= /0)2 I← Q.remove()3 IF (C(I) holds), output I4 ELSE5 Split I and insert children into Q
Goal – Bound the size of recursion tree T (I0)
NOTE: C(I)≡ C0(I)∨C1(I) in EVAL
The leaves of T (I0) induces a partition P(I) of I0Suffices to upper bound #P(I0)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 104 / 113
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Integral Bounds and Framework of Stopping Functions
Framework of Stopping Functions
Stopping Function for C(I) is F : R→R≥0
For all interval I:If (∃b ∈ I)[w(I) < F(b)],
then C(I) holds.
How to use F? The Penultimate Property
Similar to Descartes proof
If J ∈ P(I0), its parent ( “penultimate leaf” ) has width 2w(J).
Conclude from definition of stopping function:(∀c ∈ J) [2w(J) ≥ F(c)].
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Integral Bounds and Framework of Stopping Functions
Framework of Stopping Functions
Stopping Function for C(I) is F : R→R≥0
For all interval I:If (∃b ∈ I)[w(I) < F(b)],
then C(I) holds.
How to use F? The Penultimate Property
Similar to Descartes proof
If J ∈ P(I0), its parent ( “penultimate leaf” ) has width 2w(J).
Conclude from definition of stopping function:(∀c ∈ J) [2w(J) ≥ F(c)].
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 105 / 113
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Integral Bounds and Framework of Stopping Functions
Framework of Stopping Functions
Stopping Function for C(I) is F : R→R≥0
For all interval I:If (∃b ∈ I)[w(I) < F(b)],
then C(I) holds.
How to use F? The Penultimate Property
Similar to Descartes proof
If J ∈ P(I0), its parent ( “penultimate leaf” ) has width 2w(J).
Conclude from definition of stopping function:(∀c ∈ J) [2w(J) ≥ F(c)].
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 105 / 113
![Page 540: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/540.jpg)
Integral Bounds and Framework of Stopping Functions
Framework of Stopping Functions
Stopping Function for C(I) is F : R→R≥0
For all interval I:If (∃b ∈ I)[w(I) < F(b)],
then C(I) holds.
How to use F? The Penultimate Property
Similar to Descartes proof
If J ∈ P(I0), its parent ( “penultimate leaf” ) has width 2w(J).
Conclude from definition of stopping function:(∀c ∈ J) [2w(J) ≥ F(c)].
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 105 / 113
![Page 541: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/541.jpg)
Integral Bounds and Framework of Stopping Functions
Framework of Stopping Functions
Stopping Function for C(I) is F : R→R≥0
For all interval I:If (∃b ∈ I)[w(I) < F(b)],
then C(I) holds.
How to use F? The Penultimate Property
Similar to Descartes proof
If J ∈ P(I0), its parent ( “penultimate leaf” ) has width 2w(J).
Conclude from definition of stopping function:(∀c ∈ J) [2w(J) ≥ F(c)].
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 105 / 113
![Page 542: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/542.jpg)
Integral Bounds and Framework of Stopping Functions
An Integral Bound
Theorem (Integral Bound[Burr/Krahmer/Y.] )
#P(I0)≤max
1,
∫
I0
2dx
F(x)
Proof.
1 If #P(I0) = 1, result is true.
2 Else pick any J ∈ P(I0): it hasthe penultimate property.
3 Choosing c∗ ∈ J such that F(c∗)is maximum
Pf (contd)
∫
J
2dx
F(x)≥
∫
J
2dx
F(c∗)
≥ 2
F(c∗)
∫
Jdx
=2w(J)
F(c∗)≥ 1 [PenultimateProp.]
∫
I0
2dx
F(x)= ∑
J∈P(I0)
∫
J
2dx
F(c∗)
≥ ∑J∈P(I0)
1 = #P(I0)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 106 / 113
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Integral Bounds and Framework of Stopping Functions
An Integral Bound
Theorem (Integral Bound[Burr/Krahmer/Y.] )
#P(I0)≤max
1,
∫
I0
2dx
F(x)
Proof.
1 If #P(I0) = 1, result is true.
2 Else pick any J ∈ P(I0): it hasthe penultimate property.
3 Choosing c∗ ∈ J such that F(c∗)is maximum
Pf (contd)
∫
J
2dx
F(x)≥
∫
J
2dx
F(c∗)
≥ 2
F(c∗)
∫
Jdx
=2w(J)
F(c∗)≥ 1 [PenultimateProp.]
∫
I0
2dx
F(x)= ∑
J∈P(I0)
∫
J
2dx
F(c∗)
≥ ∑J∈P(I0)
1 = #P(I0)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 106 / 113
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Integral Bounds and Framework of Stopping Functions
An Integral Bound
Theorem (Integral Bound[Burr/Krahmer/Y.] )
#P(I0)≤max
1,
∫
I0
2dx
F(x)
Proof.
1 If #P(I0) = 1, result is true.
2 Else pick any J ∈ P(I0): it hasthe penultimate property.
3 Choosing c∗ ∈ J such that F(c∗)is maximum
Pf (contd)
∫
J
2dx
F(x)≥
∫
J
2dx
F(c∗)
≥ 2
F(c∗)
∫
Jdx
=2w(J)
F(c∗)≥ 1 [PenultimateProp.]
∫
I0
2dx
F(x)= ∑
J∈P(I0)
∫
J
2dx
F(c∗)
≥ ∑J∈P(I0)
1 = #P(I0)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 106 / 113
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Integral Bounds and Framework of Stopping Functions
An Integral Bound
Theorem (Integral Bound[Burr/Krahmer/Y.] )
#P(I0)≤max
1,
∫
I0
2dx
F(x)
Proof.
1 If #P(I0) = 1, result is true.
2 Else pick any J ∈ P(I0): it hasthe penultimate property.
3 Choosing c∗ ∈ J such that F(c∗)is maximum
Pf (contd)
∫
J
2dx
F(x)≥
∫
J
2dx
F(c∗)
≥ 2
F(c∗)
∫
Jdx
=2w(J)
F(c∗)≥ 1 [PenultimateProp.]
∫
I0
2dx
F(x)= ∑
J∈P(I0)
∫
J
2dx
F(c∗)
≥ ∑J∈P(I0)
1 = #P(I0)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 106 / 113
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Integral Bounds and Framework of Stopping Functions
An Integral Bound
Theorem (Integral Bound[Burr/Krahmer/Y.] )
#P(I0)≤max
1,
∫
I0
2dx
F(x)
Proof.
1 If #P(I0) = 1, result is true.
2 Else pick any J ∈ P(I0): it hasthe penultimate property.
3 Choosing c∗ ∈ J such that F(c∗)is maximum
Pf (contd)
∫
J
2dx
F(x)≥
∫
J
2dx
F(c∗)
≥ 2
F(c∗)
∫
Jdx
=2w(J)
F(c∗)≥ 1 [PenultimateProp.]
∫
I0
2dx
F(x)= ∑
J∈P(I0)
∫
J
2dx
F(c∗)
≥ ∑J∈P(I0)
1 = #P(I0)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 106 / 113
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Integral Bounds and Framework of Stopping Functions
An Integral Bound
Theorem (Integral Bound[Burr/Krahmer/Y.] )
#P(I0)≤max
1,
∫
I0
2dx
F(x)
Proof.
1 If #P(I0) = 1, result is true.
2 Else pick any J ∈ P(I0): it hasthe penultimate property.
3 Choosing c∗ ∈ J such that F(c∗)is maximum
Pf (contd)
∫
J
2dx
F(x)≥
∫
J
2dx
F(c∗)
≥ 2
F(c∗)
∫
Jdx
=2w(J)
F(c∗)≥ 1 [PenultimateProp.]
∫
I0
2dx
F(x)= ∑
J∈P(I0)
∫
J
2dx
F(c∗)
≥ ∑J∈P(I0)
1 = #P(I0)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 106 / 113
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Integral Bounds and Framework of Stopping Functions
An Integral Bound
Theorem (Integral Bound[Burr/Krahmer/Y.] )
#P(I0)≤max
1,
∫
I0
2dx
F(x)
Proof.
1 If #P(I0) = 1, result is true.
2 Else pick any J ∈ P(I0): it hasthe penultimate property.
3 Choosing c∗ ∈ J such that F(c∗)is maximum
Pf (contd)
∫
J
2dx
F(x)≥
∫
J
2dx
F(c∗)
≥ 2
F(c∗)
∫
Jdx
=2w(J)
F(c∗)≥ 1 [PenultimateProp.]
∫
I0
2dx
F(x)= ∑
J∈P(I0)
∫
J
2dx
F(c∗)
≥ ∑J∈P(I0)
1 = #P(I0)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 106 / 113
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Integral Bounds and Framework of Stopping Functions
An Integral Bound
Theorem (Integral Bound[Burr/Krahmer/Y.] )
#P(I0)≤max
1,
∫
I0
2dx
F(x)
Proof.
1 If #P(I0) = 1, result is true.
2 Else pick any J ∈ P(I0): it hasthe penultimate property.
3 Choosing c∗ ∈ J such that F(c∗)is maximum
Pf (contd)
∫
J
2dx
F(x)≥
∫
J
2dx
F(c∗)
≥ 2
F(c∗)
∫
Jdx
=2w(J)
F(c∗)≥ 1 [PenultimateProp.]
∫
I0
2dx
F(x)= ∑
J∈P(I0)
∫
J
2dx
F(c∗)
≥ ∑J∈P(I0)
1 = #P(I0)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 106 / 113
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Integral Bounds and Framework of Stopping Functions
An Integral Bound
Theorem (Integral Bound[Burr/Krahmer/Y.] )
#P(I0)≤max
1,
∫
I0
2dx
F(x)
Proof.
1 If #P(I0) = 1, result is true.
2 Else pick any J ∈ P(I0): it hasthe penultimate property.
3 Choosing c∗ ∈ J such that F(c∗)is maximum
Pf (contd)
∫
J
2dx
F(x)≥
∫
J
2dx
F(c∗)
≥ 2
F(c∗)
∫
Jdx
=2w(J)
F(c∗)≥ 1 [PenultimateProp.]
∫
I0
2dx
F(x)= ∑
J∈P(I0)
∫
J
2dx
F(c∗)
≥ ∑J∈P(I0)
1 = #P(I0)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 106 / 113
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Integral Bounds and Framework of Stopping Functions
An Integral Bound
Theorem (Integral Bound[Burr/Krahmer/Y.] )
#P(I0)≤max
1,
∫
I0
2dx
F(x)
Proof.
1 If #P(I0) = 1, result is true.
2 Else pick any J ∈ P(I0): it hasthe penultimate property.
3 Choosing c∗ ∈ J such that F(c∗)is maximum
Pf (contd)
∫
J
2dx
F(x)≥
∫
J
2dx
F(c∗)
≥ 2
F(c∗)
∫
Jdx
=2w(J)
F(c∗)≥ 1 [PenultimateProp.]
∫
I0
2dx
F(x)= ∑
J∈P(I0)
∫
J
2dx
F(c∗)
≥ ∑J∈P(I0)
1 = #P(I0)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 106 / 113
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Integral Bounds and Framework of Stopping Functions
An Integral Bound
Theorem (Integral Bound[Burr/Krahmer/Y.] )
#P(I0)≤max
1,
∫
I0
2dx
F(x)
Proof.
1 If #P(I0) = 1, result is true.
2 Else pick any J ∈ P(I0): it hasthe penultimate property.
3 Choosing c∗ ∈ J such that F(c∗)is maximum
Pf (contd)
∫
J
2dx
F(x)≥
∫
J
2dx
F(c∗)
≥ 2
F(c∗)
∫
Jdx
=2w(J)
F(c∗)≥ 1 [PenultimateProp.]
∫
I0
2dx
F(x)= ∑
J∈P(I0)
∫
J
2dx
F(c∗)
≥ ∑J∈P(I0)
1 = #P(I0)
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 106 / 113
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Integral Bounds and Framework of Stopping Functions
Remarks on Integral Bound
Too hard to directly bound the integral implied by C0(I)∨C1(I). So we devise stopping functions F(x) that can be analyzed.
Technique of bounding∫
I φ(x)dx is Continuous Amortizationwhere φ(x) is charge function.
In discrete “amortization arguments”, we bound ∑ni=1 φ(i) where
φ(i) is “charge” for the ith operation.
Ruppert (1995) introduced a similar integral for triangulation. Unlike us, he does not evaluate his integral.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 107 / 113
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Integral Bounds and Framework of Stopping Functions
Remarks on Integral Bound
Too hard to directly bound the integral implied by C0(I)∨C1(I). So we devise stopping functions F(x) that can be analyzed.
Technique of bounding∫
I φ(x)dx is Continuous Amortizationwhere φ(x) is charge function.
In discrete “amortization arguments”, we bound ∑ni=1 φ(i) where
φ(i) is “charge” for the ith operation.
Ruppert (1995) introduced a similar integral for triangulation. Unlike us, he does not evaluate his integral.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 107 / 113
![Page 555: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/555.jpg)
Integral Bounds and Framework of Stopping Functions
Remarks on Integral Bound
Too hard to directly bound the integral implied by C0(I)∨C1(I). So we devise stopping functions F(x) that can be analyzed.
Technique of bounding∫
I φ(x)dx is Continuous Amortizationwhere φ(x) is charge function.
In discrete “amortization arguments”, we bound ∑ni=1 φ(i) where
φ(i) is “charge” for the ith operation.
Ruppert (1995) introduced a similar integral for triangulation. Unlike us, he does not evaluate his integral.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 107 / 113
![Page 556: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/556.jpg)
Integral Bounds and Framework of Stopping Functions
Remarks on Integral Bound
Too hard to directly bound the integral implied by C0(I)∨C1(I). So we devise stopping functions F(x) that can be analyzed.
Technique of bounding∫
I φ(x)dx is Continuous Amortizationwhere φ(x) is charge function.
In discrete “amortization arguments”, we bound ∑ni=1 φ(i) where
φ(i) is “charge” for the ith operation.
Ruppert (1995) introduced a similar integral for triangulation. Unlike us, he does not evaluate his integral.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 107 / 113
![Page 557: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/557.jpg)
Integral Bounds and Framework of Stopping Functions
Remarks on Integral Bound
Too hard to directly bound the integral implied by C0(I)∨C1(I). So we devise stopping functions F(x) that can be analyzed.
Technique of bounding∫
I φ(x)dx is Continuous Amortizationwhere φ(x) is charge function.
In discrete “amortization arguments”, we bound ∑ni=1 φ(i) where
φ(i) is “charge” for the ith operation.
Ruppert (1995) introduced a similar integral for triangulation. Unlike us, he does not evaluate his integral.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 107 / 113
![Page 558: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/558.jpg)
Integral Bounds and Framework of Stopping Functions
Remarks on Integral Bound
Too hard to directly bound the integral implied by C0(I)∨C1(I). So we devise stopping functions F(x) that can be analyzed.
Technique of bounding∫
I φ(x)dx is Continuous Amortizationwhere φ(x) is charge function.
In discrete “amortization arguments”, we bound ∑ni=1 φ(i) where
φ(i) is “charge” for the ith operation.
Ruppert (1995) introduced a similar integral for triangulation. Unlike us, he does not evaluate his integral.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 107 / 113
![Page 559: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/559.jpg)
Integral Bounds and Framework of Stopping Functions
An Amortized Evaluation BoundThe Idea
Want lower bounds on |f (α)|Multivariate version used in [Cheng/Gao/Y. ISSAC’2007]
Amortization: give lower bounds on ∏i∈J |f (αi)|.
Theorem
Let F ,H ∈Z[X ] be relatively prime such that F = φφ , H = ηη whereφ , φ ,η, η ∈ C[X ] have degrees m,m,n, n, respectively. If β1, . . . ,βnare all the zeros of η(X), then
n
∏i=1
|φ(βi)| ≥1
lc(η)m ((m +1)‖φ‖)n M(η)m((m +1)‖φ‖
)n+nM(H)m
.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 108 / 113
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Integral Bounds and Framework of Stopping Functions
An Amortized Evaluation BoundThe Idea
Want lower bounds on |f (α)|Multivariate version used in [Cheng/Gao/Y. ISSAC’2007]
Amortization: give lower bounds on ∏i∈J |f (αi)|.
Theorem
Let F ,H ∈Z[X ] be relatively prime such that F = φφ , H = ηη whereφ , φ ,η, η ∈ C[X ] have degrees m,m,n, n, respectively. If β1, . . . ,βnare all the zeros of η(X), then
n
∏i=1
|φ(βi)| ≥1
lc(η)m ((m +1)‖φ‖)n M(η)m((m +1)‖φ‖
)n+nM(H)m
.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 108 / 113
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Integral Bounds and Framework of Stopping Functions
An Amortized Evaluation BoundThe Idea
Want lower bounds on |f (α)|Multivariate version used in [Cheng/Gao/Y. ISSAC’2007]
Amortization: give lower bounds on ∏i∈J |f (αi)|.
Theorem
Let F ,H ∈Z[X ] be relatively prime such that F = φφ , H = ηη whereφ , φ ,η, η ∈ C[X ] have degrees m,m,n, n, respectively. If β1, . . . ,βnare all the zeros of η(X), then
n
∏i=1
|φ(βi)| ≥1
lc(η)m ((m +1)‖φ‖)n M(η)m((m +1)‖φ‖
)n+nM(H)m
.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 108 / 113
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Integral Bounds and Framework of Stopping Functions
An Amortized Evaluation BoundThe Idea
Want lower bounds on |f (α)|Multivariate version used in [Cheng/Gao/Y. ISSAC’2007]
Amortization: give lower bounds on ∏i∈J |f (αi)|.
Theorem
Let F ,H ∈Z[X ] be relatively prime such that F = φφ , H = ηη whereφ , φ ,η, η ∈ C[X ] have degrees m,m,n, n, respectively. If β1, . . . ,βnare all the zeros of η(X), then
n
∏i=1
|φ(βi)| ≥1
lc(η)m ((m +1)‖φ‖)n M(η)m((m +1)‖φ‖
)n+nM(H)m
.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 108 / 113
![Page 563: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/563.jpg)
Integral Bounds and Framework of Stopping Functions
An Amortized Evaluation BoundThe Idea
Want lower bounds on |f (α)|Multivariate version used in [Cheng/Gao/Y. ISSAC’2007]
Amortization: give lower bounds on ∏i∈J |f (αi)|.
Theorem
Let F ,H ∈Z[X ] be relatively prime such that F = φφ , H = ηη whereφ , φ ,η, η ∈ C[X ] have degrees m,m,n, n, respectively. If β1, . . . ,βnare all the zeros of η(X), then
n
∏i=1
|φ(βi)| ≥1
lc(η)m ((m +1)‖φ‖)n M(η)m((m +1)‖φ‖
)n+nM(H)m
.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 108 / 113
![Page 564: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/564.jpg)
Integral Bounds and Framework of Stopping Functions
Complex Roots: Lesson from Meshing Curves
How to isolate complex roots?Previous subdivision methods:
Pan-Weyl Algorithm (Turan Test) Root isolation on boundary of boxes (topological degree)
Hints from Curve Meshing (Snyder/PV/Cxy) – not good idea
New Result (with Sagraloff)There is an exact analog CEVAL for complex roots that is simple andeasy to implement exactly.It achieves the same bit complexity bound as in the real case.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 109 / 113
![Page 565: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/565.jpg)
Integral Bounds and Framework of Stopping Functions
Complex Roots: Lesson from Meshing Curves
How to isolate complex roots?Previous subdivision methods:
Pan-Weyl Algorithm (Turan Test) Root isolation on boundary of boxes (topological degree)
Hints from Curve Meshing (Snyder/PV/Cxy) – not good idea
New Result (with Sagraloff)There is an exact analog CEVAL for complex roots that is simple andeasy to implement exactly.It achieves the same bit complexity bound as in the real case.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 109 / 113
![Page 566: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/566.jpg)
Integral Bounds and Framework of Stopping Functions
Complex Roots: Lesson from Meshing Curves
How to isolate complex roots?Previous subdivision methods:
Pan-Weyl Algorithm (Turan Test) Root isolation on boundary of boxes (topological degree)
Hints from Curve Meshing (Snyder/PV/Cxy) – not good idea
New Result (with Sagraloff)There is an exact analog CEVAL for complex roots that is simple andeasy to implement exactly.It achieves the same bit complexity bound as in the real case.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 109 / 113
![Page 567: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/567.jpg)
Integral Bounds and Framework of Stopping Functions
Complex Roots: Lesson from Meshing Curves
How to isolate complex roots?Previous subdivision methods:
Pan-Weyl Algorithm (Turan Test) Root isolation on boundary of boxes (topological degree)
Hints from Curve Meshing (Snyder/PV/Cxy) – not good idea
New Result (with Sagraloff)There is an exact analog CEVAL for complex roots that is simple andeasy to implement exactly.It achieves the same bit complexity bound as in the real case.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 109 / 113
![Page 568: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/568.jpg)
Mini Summary
The Bolzano approach to Root Isolation is an Exact and Analyticapproach to root isolation
It seems to have complexity that matches Sturm and Descartes
It is much easier to implement than either
![Page 569: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/569.jpg)
Summary of Lecture 3
Complexity Analysis of Adaptivity at infancy
Analysis Techniques we have seen so far:
Continuous amortization via integral bounds Amortized root separation bounds Amortized evaluation bounds Cluster analysis
Major Open Problems How to characterize local complexity? How to extend to higher dimensions
![Page 570: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/570.jpg)
Summary of Lecture 3
Complexity Analysis of Adaptivity at infancy
Analysis Techniques we have seen so far:
Continuous amortization via integral bounds Amortized root separation bounds Amortized evaluation bounds Cluster analysis
Major Open Problems How to characterize local complexity? How to extend to higher dimensions
![Page 571: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/571.jpg)
Summary of Lecture 3
Complexity Analysis of Adaptivity at infancy
Analysis Techniques we have seen so far:
Continuous amortization via integral bounds Amortized root separation bounds Amortized evaluation bounds Cluster analysis
Major Open Problems How to characterize local complexity? How to extend to higher dimensions
![Page 572: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/572.jpg)
Summary of Lecture 3
Complexity Analysis of Adaptivity at infancy
Analysis Techniques we have seen so far:
Continuous amortization via integral bounds Amortized root separation bounds Amortized evaluation bounds Cluster analysis
Major Open Problems How to characterize local complexity? How to extend to higher dimensions
![Page 573: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/573.jpg)
Summary of Lecture 3
Complexity Analysis of Adaptivity at infancy
Analysis Techniques we have seen so far:
Continuous amortization via integral bounds Amortized root separation bounds Amortized evaluation bounds Cluster analysis
Major Open Problems How to characterize local complexity? How to extend to higher dimensions
![Page 574: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/574.jpg)
Summary of Lecture 3
Complexity Analysis of Adaptivity at infancy
Analysis Techniques we have seen so far:
Continuous amortization via integral bounds Amortized root separation bounds Amortized evaluation bounds Cluster analysis
Major Open Problems How to characterize local complexity? How to extend to higher dimensions
![Page 575: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/575.jpg)
Summary of Lecture 3
Complexity Analysis of Adaptivity at infancy
Analysis Techniques we have seen so far:
Continuous amortization via integral bounds Amortized root separation bounds Amortized evaluation bounds Cluster analysis
Major Open Problems How to characterize local complexity? How to extend to higher dimensions
![Page 576: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/576.jpg)
Summary of Lecture 3
Complexity Analysis of Adaptivity at infancy
Analysis Techniques we have seen so far:
Continuous amortization via integral bounds Amortized root separation bounds Amortized evaluation bounds Cluster analysis
Major Open Problems How to characterize local complexity? How to extend to higher dimensions
![Page 577: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/577.jpg)
Summary of Lecture 3
Complexity Analysis of Adaptivity at infancy
Analysis Techniques we have seen so far:
Continuous amortization via integral bounds Amortized root separation bounds Amortized evaluation bounds Cluster analysis
Major Open Problems How to characterize local complexity? How to extend to higher dimensions
![Page 578: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/578.jpg)
Summary of Lecture 3
Complexity Analysis of Adaptivity at infancy
Analysis Techniques we have seen so far:
Continuous amortization via integral bounds Amortized root separation bounds Amortized evaluation bounds Cluster analysis
Major Open Problems How to characterize local complexity? How to extend to higher dimensions
![Page 579: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/579.jpg)
Summary of Lecture 3
Complexity Analysis of Adaptivity at infancy
Analysis Techniques we have seen so far:
Continuous amortization via integral bounds Amortized root separation bounds Amortized evaluation bounds Cluster analysis
Major Open Problems How to characterize local complexity? How to extend to higher dimensions
![Page 580: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/580.jpg)
Summary of Lecture 3
Complexity Analysis of Adaptivity at infancy
Analysis Techniques we have seen so far:
Continuous amortization via integral bounds Amortized root separation bounds Amortized evaluation bounds Cluster analysis
Major Open Problems How to characterize local complexity? How to extend to higher dimensions
![Page 581: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/581.jpg)
Summary of Tutorial
MANY advantages in numerical/analytic approaches to algebraicand geometric problems
practical, adaptive, easy to implement
New ingredient we seek: a priori guarantees and exactness
Zero problems is the locus of our investigation
Exact Numerical Computation (ENC) is a suitable computationalmodel
The explicitization problems are central for ENC
Analysis of adaptive algorithms is wide open
![Page 582: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/582.jpg)
Summary of Tutorial
MANY advantages in numerical/analytic approaches to algebraicand geometric problems
practical, adaptive, easy to implement
New ingredient we seek: a priori guarantees and exactness
Zero problems is the locus of our investigation
Exact Numerical Computation (ENC) is a suitable computationalmodel
The explicitization problems are central for ENC
Analysis of adaptive algorithms is wide open
![Page 583: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/583.jpg)
Summary of Tutorial
MANY advantages in numerical/analytic approaches to algebraicand geometric problems
practical, adaptive, easy to implement
New ingredient we seek: a priori guarantees and exactness
Zero problems is the locus of our investigation
Exact Numerical Computation (ENC) is a suitable computationalmodel
The explicitization problems are central for ENC
Analysis of adaptive algorithms is wide open
![Page 584: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/584.jpg)
Summary of Tutorial
MANY advantages in numerical/analytic approaches to algebraicand geometric problems
practical, adaptive, easy to implement
New ingredient we seek: a priori guarantees and exactness
Zero problems is the locus of our investigation
Exact Numerical Computation (ENC) is a suitable computationalmodel
The explicitization problems are central for ENC
Analysis of adaptive algorithms is wide open
![Page 585: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/585.jpg)
Summary of Tutorial
MANY advantages in numerical/analytic approaches to algebraicand geometric problems
practical, adaptive, easy to implement
New ingredient we seek: a priori guarantees and exactness
Zero problems is the locus of our investigation
Exact Numerical Computation (ENC) is a suitable computationalmodel
The explicitization problems are central for ENC
Analysis of adaptive algorithms is wide open
![Page 586: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/586.jpg)
Summary of Tutorial
MANY advantages in numerical/analytic approaches to algebraicand geometric problems
practical, adaptive, easy to implement
New ingredient we seek: a priori guarantees and exactness
Zero problems is the locus of our investigation
Exact Numerical Computation (ENC) is a suitable computationalmodel
The explicitization problems are central for ENC
Analysis of adaptive algorithms is wide open
![Page 587: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/587.jpg)
Summary of Tutorial
MANY advantages in numerical/analytic approaches to algebraicand geometric problems
practical, adaptive, easy to implement
New ingredient we seek: a priori guarantees and exactness
Zero problems is the locus of our investigation
Exact Numerical Computation (ENC) is a suitable computationalmodel
The explicitization problems are central for ENC
Analysis of adaptive algorithms is wide open
![Page 588: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/588.jpg)
Summary of Tutorial
MANY advantages in numerical/analytic approaches to algebraicand geometric problems
practical, adaptive, easy to implement
New ingredient we seek: a priori guarantees and exactness
Zero problems is the locus of our investigation
Exact Numerical Computation (ENC) is a suitable computationalmodel
The explicitization problems are central for ENC
Analysis of adaptive algorithms is wide open
![Page 589: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/589.jpg)
Summary of Tutorial
MANY advantages in numerical/analytic approaches to algebraicand geometric problems
practical, adaptive, easy to implement
New ingredient we seek: a priori guarantees and exactness
Zero problems is the locus of our investigation
Exact Numerical Computation (ENC) is a suitable computationalmodel
The explicitization problems are central for ENC
Analysis of adaptive algorithms is wide open
![Page 590: Tutorial: Exact Numerical Computation in Algebra and Geometry · Many problems in Computational Science & Engineering (CS&E) are defined on the continuum. Standard algorithms for](https://reader033.vdocument.in/reader033/viewer/2022050307/5f6fe99980e70d59a05d8d58/html5/thumbnails/590.jpg)
Thank you!
Website http://cs.nyu.edu/exact/
Download Papers
Download Core Library
GENERAL REFERENCE
Chee K. Yap.
Theory of Real Computation according to EGC.
In P. Hertling, Ch.M. Hoffmann, W. Luther, and N.Revol, editors, ReliableImplementation of Real Number Algorithms: Theory and Practice,No. 5045 in LNCS, pp. 193–237. Springer, 2008.
Yap (NYU) Tutorial: Exact Numerical Computation ISSAC, July 2009 113 / 113