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Computer Algebra vs. Reality. Erik Postma and Elena Shmoylova Maplesoft June 25, 2009. Outline. Introduction How to apply computer algebra techniques to real world problems? Example Open discussion. Introduction. Computer algebra is based on symbolic computations

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Computer algebra vs reality

Computer Algebra vs. Reality

Erik Postma and Elena Shmoylova

Maplesoft

June 25, 2009


Outline
Outline

  • Introduction

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

  • Example

  • Open discussion


Introduction
Introduction

  • Computer algebra is based on symbolic computations

  • Benefit: Result is a nice closed form solution

  • Drawback: Problem itself should be nice too


Computer algebra methods
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


Common elements of real world problems
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.



Convert one type of difficulty into another
Convert One Type of Difficulty into Another problems?

  • 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


Remove difficulty from model
Remove Difficulty from Model problems?

  • 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


Floating point numbers
Floating Point Numbers problems?

  • Replace with rational numbers


Initial conditions for hybrid dae models
Initial Conditions for Hybrid DAE Models problems?

  • 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


Simple example
Simple Example problems?

  • DAEs

  • ICs


Identifying mode i
Identifying Mode (I) problems?

  • From constraint

  • Do not know what branch to choose

  • Index reduction can be performed on both branches

  • Hidden constraint


Identifying mode ii
Identifying Mode (II) problems?

  • Check which branch of the hidden constraint is satisfied

  • mode is active


Initial conditions for hybrid daes
Initial Conditions for Hybrid DAEs problems?

  • 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


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


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