© 2009 Maplesoft, a division of Waterloo Maple Inc. Computer Algebra vs. Reality Erik Postma and Elena Shmoylova Maplesoft June 25, 2009

© 2009 Maplesoft, a division of Waterloo Maple Inc. 2 Outline Introduction How to apply computer algebra techniques to real world problems? Example Open discussion

© 2009 Maplesoft, a division of Waterloo Maple Inc. 3 Introduction Computer algebra is based on symbolic computations Benefit: Result is a nice closed form solution Drawback: Problem itself should be nice too

© 2009 Maplesoft, a division of Waterloo Maple Inc. 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

© 2009 Maplesoft, a division of Waterloo Maple Inc. 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.

© 2009 Maplesoft, a division of Waterloo Maple Inc. 6 How to apply computer algebra techniques to real-world problems?

© 2009 Maplesoft, a division of Waterloo Maple Inc. 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

© 2009 Maplesoft, a division of Waterloo Maple Inc. 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

© 2009 Maplesoft, a division of Waterloo Maple Inc. 9 Floating Point Numbers Replace with rational numbers

© 2009 Maplesoft, a division of Waterloo Maple Inc. 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

© 2009 Maplesoft, a division of Waterloo Maple Inc. 11 Simple Example DAEs ICs

© 2009 Maplesoft, a division of Waterloo Maple Inc. 12 Identifying Mode (I) From constraint Do not know what branch to choose Index reduction can be performed on both branches Hidden constraint

© 2009 Maplesoft, a division of Waterloo Maple Inc. 13 Identifying Mode (II) Check which branch of the hidden constraint is satisfied mode is active

© 2009 Maplesoft, a division of Waterloo Maple Inc. 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

© 2009 Maplesoft, a division of Waterloo Maple Inc. 15 Open Discussion: How to apply computer algebra techniques to real-world problems?