PostmaShmoylova.ppt 1301KB Jun 23 2011 12:12:12 PM

Computer Algebra vs. Reality

Erik Postma and Elena Shmoylova Maplesoft

Outline

• _Introduction

• _{How to apply computer algebra techniques to}

real world problems?

• _Example

Introduction

• _{Computer algebra is based on symbolic}

computations

• _{Benefit: Result is a nice closed form solution}

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}

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}

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

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}

Floating Point Numbers

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}

Identifying Mode (I)

• _{From constraint}

• _{Do not know what branch to choose}

• _{Index reduction can be performed on both}

branches

• _{Hidden constraint}

  

 

 

0 8

. 0

0 2

. 1

1 1 1

Identifying Mode (II)

• _{Check which branch of the hidden constraint is}

satisfied

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}

Open Discussion: