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Computer Algebra vs. RealityPowerPoint Presentation

Computer Algebra vs. Reality

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Computer Algebra vs. Reality

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Computer Algebra vs. Reality

Erik Postma and Elena Shmoylova

Maplesoft

June 25, 2009

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

- Computer algebra is based on symbolic computations
- Benefit: Result is a nice closed form solution
- Drawback: Problem itself should be nice too

- 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

- 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.

- 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

- 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

- Replace with rational numbers

- 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

- DAEs
- ICs

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

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

- 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