PostmaShmoylova.ppt 1301KB Jun 23 2011 12:12:12 PM

Computer Algebra vs. Reality
Erik Postma and Elena Shmoylova
Maplesoft
June 25, 2009

© 2009 Maplesoft, a division of Waterloo Maple Inc.

Outline
• Introduction
• How to apply computer algebra techniques to
real world problems?
• Example
• Open discussion

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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.

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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.

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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.

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How to apply computer algebra
techniques to real-world problems?

© 2009 Maplesoft, a division of Waterloo Maple Inc.

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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.

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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.

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Floating Point Numbers

• Replace with rational numbers

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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.

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Simple Example
• DAEs
x1 2 x1
x1 0
  x2  1
x2 
  ( x2  1)  1 x1  0
 x12  x22  4
x1 0
0  2
2
 x1  ( x2  1)  1 x1  0

• ICs

x 1 0.75
x2  1.6
x 2  1

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Identifying Mode (I)
• From constraint
 1.2 x1 0
x1 
  0.8 x1  0

• Do not know what branch to choose
• Index reduction can be performed on both
branches
• Hidden constraint
 x1 x 1  x2 x 2
0 
 x1 x 1  ( x2  1) x 2

x1 0
x1  0

© 2009 Maplesoft, a division of Waterloo Maple Inc.

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Identifying Mode (II)
• Check which branch of the hidden constraint is
satisfied
 x1 x 1  x2 x 2

 x1 x 1  ( x2  1) x 2

•
•

x1  0

x1 0  2.5 x1 0

x1  0  0 x1  0

mode is active

x1  0.8

© 2009 Maplesoft, a division of Waterloo Maple Inc.

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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.

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Open Discussion:
How to apply computer algebra
techniques to real-world problems?

© 2009 Maplesoft, a division of Waterloo Maple Inc.

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