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