© 2009 Maplesoft, a division of Waterloo Maple Inc.

Computer Algebra vs. Reality

Erik Postma and Elena ShmoylovaMaplesoft

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

1

6.1

75.0

2

2

1

x

x

x

01)1(

040

01)1(

01

2

12

221

122

21

12

122

11

xxx

xxx

xx

xxx

xx

© 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

08.0

02.1

1

11 x

xx

0)1(

00

12211

12211

xxxxx

xxxxx

© 2009 Maplesoft, a division of Waterloo Maple Inc. 13

Identifying Mode (II)

• Check which branch of the hidden constraint is satisfied

• mode is active•

00

05.2

0)1(

0

1

1

12211

12211

x

x

xxxxx

xxxxx

01 x

8.01 x

© 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?