Computer Algebra vs. Reality

1. 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

3. 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

5. 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.

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

7. 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

8. 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

9. Floating Point Numbers • Replace with rational numbers

10. 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

11. Simple Example • DAEs • ICs

12. Identifying Mode (I) • From constraint • Do not know what branch to choose • Index reduction can be performed on both branches • Hidden constraint

13. Identifying Mode (II) • Check which branch of the hidden constraint is satisfied • mode is active

14. 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

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