Loading in 5 sec....

Computer Algebra vs. RealityPowerPoint Presentation

Computer Algebra vs. Reality

- 92 Views
- Uploaded on
- Presentation posted in: General

Download Presentation
## PowerPoint Slideshow about 'Computer Algebra vs. Reality ' - aileen

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

Outline

- Introduction
- How to apply computer algebra techniques to real world problems?
- Example
- Open discussion

Introduction

- Computer algebra is based on symbolic computations
- Benefit: Result is a nice closed form solution
- Drawback: Problem itself should be nice too

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

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.

Convert One Type of Difficulty into Another problems?

- 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

Remove Difficulty from Model problems?

- 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

Floating Point Numbers problems?

- Replace with rational numbers

Initial Conditions for Hybrid DAE Models problems?

- 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

Simple Example problems?

- DAEs
- ICs

Identifying Mode (I) problems?

- From constraint
- Do not know what branch to choose
- Index reduction can be performed on both branches
- Hidden constraint

Identifying Mode (II) problems?

- Check which branch of the hidden constraint is satisfied
- mode is active

Initial Conditions for Hybrid DAEs problems?

- 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

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