computer algebra vs. reality
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© 2009 Maplesoft, a division of Waterloo Maple Inc.
Computer Algebra vs. Reality
Erik Postma and Elena ShmoylovaMaplesoft
June 25, 2009
© 2009 Maplesoft, a division of Waterloo Maple Inc. 2
Outline
• Introduction• How to apply computer algebra techniques to
real world problems?• Example• Open discussion
© 2009 Maplesoft, a division of Waterloo Maple Inc. 3
Introduction• Computer algebra is based on symbolic
computations• Benefit: Result is a nice closed form solution• Drawback: Problem itself should be nice too
© 2009 Maplesoft, a division of Waterloo Maple Inc. 4
Computer Algebra Methods• Polynomial solvers for polynomial systems with
coefficients in a rational extension field• Differential Groebner basis for polynomial DEs with
coefficients in a rational extension field• Functional decomposition for multi- or univariate
polynomials over a rational extension field• Index reduction for continuous and in some cases
piecewise-continuous models
© 2009 Maplesoft, a division of Waterloo Maple Inc. 5
Common Elements of Real-World Problems
• Floating point numbers and powers• Trigonometric and other special functions• Lookup tables• Piecewise functions• Numerical differentiators• Compiled numerical procedures (“black-box”
functions)• Delay elements• Random noise terms• etc.
© 2009 Maplesoft, a division of Waterloo Maple Inc. 6
How to apply computer algebra techniques to real-world problems?
© 2009 Maplesoft, a division of Waterloo Maple Inc. 7
Convert One Type of Difficulty into Another
• Look-up tables into piecewise• Almost anything into black-box function• Approximate functions by their Taylor or Padé
series• Smooth piecewise functions, e.g. using radial
basis functions• Floating point numbers into rationals
© 2009 Maplesoft, a division of Waterloo Maple Inc. 8
Remove Difficulty from Model
• If a difficulty can be combined into a subsystem, remove the subsystem from the model– View its arguments as outputs of the model– View its result as inputs into the model– Use symbolic technique on the model
• Limited to techniques that can deal with arbitrary external inputs
© 2009 Maplesoft, a division of Waterloo Maple Inc. 9
Floating Point Numbers
• Replace with rational numbers
© 2009 Maplesoft, a division of Waterloo Maple Inc. 10
Initial Conditions for Hybrid DAE Models
• Problem:– User does not provide all initial conditions, need
to find remaining initial conditions
• Difficulty:– High-order DAEs have hidden constraints that may
be needed to find initial conditions
© 2009 Maplesoft, a division of Waterloo Maple Inc. 11
Simple Example• DAEs
• ICs
1
6.1
75.0
2
2
1
x
x
x
01)1(
040
01)1(
01
2
12
221
122
21
12
122
11
xxx
xxx
xx
xxx
xx
© 2009 Maplesoft, a division of Waterloo Maple Inc. 12
Identifying Mode (I)• From constraint
• Do not know what branch to choose• Index reduction can be performed on both
branches• Hidden constraint
08.0
02.1
1
11 x
xx
0)1(
00
12211
12211
xxxxx
xxxxx
© 2009 Maplesoft, a division of Waterloo Maple Inc. 13
Identifying Mode (II)
• Check which branch of the hidden constraint is satisfied
• mode is active•
00
05.2
0)1(
0
1
1
12211
12211
x
x
xxxxx
xxxxx
01 x
8.01 x
© 2009 Maplesoft, a division of Waterloo Maple Inc. 14
Initial Conditions for Hybrid DAEs
• To find ICs, hidden constraints are needed• To find hidden constraints, index reduction
should be performed• It is infeasible to perform index reduction for
all modes separately, need to know what mode system is in
• To find mode of system, need to know the values of all variables, i.e. ICs
© 2009 Maplesoft, a division of Waterloo Maple Inc. 15
Open Discussion:How to apply computer algebra
techniques to real-world problems?
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