introduction to numerical analysis i
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Introduction to Numerical Analysis I. Interpolation. MATH/CMPSC 455. Chapter 3. Interpolation. A function is said to interpolate a set of data points if it passes through those points . Definition: The function interpolates the data sets if . - PowerPoint PPT PresentationTRANSCRIPT
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Introduction to Numerical Analysis I
MATH/CMPSC 455
Interpolation
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CHAPTER 3. INTERPOLATION
A function is said to interpolate a set of data points if it passes through those points
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Definition: The function interpolates the data sets if
Note that is required to be a function!
Restriction on the data set:
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Main theorem of Polynomial interpolation:If are distinct, there is a unique polynomial of degree such that
How to find this polynomial?
INTERPOLATION POLYNOMIALMathematical Problem: (Interpolate points)Given n+1 points , we seek a polynomial of degree such that Mathematical Problem: (Interpolate a function)A function , assuming its values are known or computable at a set of n+1 points. we seek a polynomial of degree such that ,
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LAGRANGE INTERPOLATION
For a data set , the Lagrange form of the interpolation polynomial is
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Example:
x 5 -7y 1 -23
Example:
xy
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HOW TO?Method 1: Solving a linear system
Determine coefficients
Method 2: Lagrange Form of Interpolation
Determine basis
Method 3: Newton Form of InterpolationUse another basis which is easy to get, and has similar property as the basis for Lagrange form, and determine the coefficient easily.
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forms a basis of
Newton form of interpolation polynomial:
Determine the coefficients
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NEWTON’S DIVIDED DIFFERENCES
Definition:
Example:
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NEWTON FORM OF THE INTERPOLATION POLYNOMIAL
Nested Form:
Definition:
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Example:
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Example:
x 0 2 3f(x) 1 2 4