Computer Algebra vs. Reality

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Computer Algebra vs. Reality

• Erik Postma and Elena Shmoylova
• Maplesoft
• June 25, 2009

2
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

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

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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.

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

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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

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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

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Simple Example

• DAEs
• ICs

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Identifying Mode (I)

• From constraint
• Do not know what branch to choose
• Index reduction can be performed on both branches
• Hidden constraint

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Identifying Mode (II)

• Check which branch of the hidden constraint is
  satisfied
• mode is active

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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

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Open DiscussionHow to apply computer algebra
techniques to real-world problems?