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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
  • 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
  • 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
  • Replace with rational numbers
initial conditions for hybrid dae models
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
identifying mode i
Identifying Mode (I)
  • 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)
  • Check which branch of the hidden constraint is satisfied
  • mode is active
initial conditions for hybrid daes
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