ALIAS

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ALIAS-C++
A C++ Algorithms Library of Interval Analysis for equation Systems
Version 2.5
September 2007
The COPRIN project
Download ALIAS here!

Introduction
- How to read this manual
Solving with Interval Analysis
- Introduction
  - Interval Analysis
- Non 0-dimensional system
- General purpose solving algorithm
- General purpose solving algorithm with Jacobian
- General purpose solving algorithm with Jacobian and Hessian
- Stopping the general solving procedures
- Ridder method for solving one equation
  - Mathematical background
  - Implementation
- Brent method for solving one equation
  - Mathematical background
  - Implementation
- Newton method for solving systems of equations
- Krawczyk method for solving systems of equations
  - Mathematical background
  - Implementation
- Solving univariate polynomial with interval analysis
  - Mathematical background
  - Implementation
    - Example
- Solving univariate polynomial numerically
- Solving trigonometric equation
- Solving systems with linear and non-linear terms: the simplex method
- Solving systems with determinants
- Solving systems of distance equations
  - Principle
  - Implementation
- Filtering a system of equation
Analyzing systems of equations
- Introduction
Analyzing trigonometric equations
- Introduction
- Number of roots of trigonometric equation
- Bound on the roots of trigonometric equation
  - Implementation
    - Example
- Utilities for trigonometric equation
Analyzing univariate polynomials
- Introduction
- Finding bounds on the roots
- Bounds on the product and sum of roots
- Maximum number of real roots
- Number of real roots
- Separation between the roots
  - Rump theorem
- Analyzing the real roots
- Analyzing the real part of the roots
- Utilities
Parametric polynomials and eigenvalues of parametric matrices
- Minimal and maximal real roots of a parametric polynomial
- Possible parameters values for a given range on the real roots
  - Approximation of the set of solutions
  - Largest square enclosed in the regions
- Condition number
- Kharitonov polynomials
  - Implementation
- Gerschgorin circles
  - Mathematical background
  - Implementation
- Cassini ovals
  - Mathematical background
  - Implementation
- Routh
- Weyl filter
  - Mathematical background
  - Implementation
- Coefficient of the characteristic polynomial
Linear algebra
- Calculating determinant
  - Scalar and interval matrix
  - Polynomial matrix
- Matrix inverse
- Solving systems of linear equations
  - Mathematical background
  - Implementation
- Regularity of parametric interval matrices
- Characteristic polynomial
- Spectral radius
Optimization
- Definition of a minimum and a maximum
- Methods
- Implementation
- Examples
  - Example 1
  - Example 2
Continuation for one dimensional system
- Continuation 1D
Integration
- Definite integrals
  - Integral with one variable
  - Integral with multiple variable
Miscellaneous procedures
- Management of boxes list
  - Adding boxes to a list
  - Freeing boxes in a list
- Void procedures
- Bisection procedures
- 3B procedures
- Volume of a box
- Filtering
  - Square adjustment
- Box reduction
- Binomial
Parser, Generic Solver and Analyzer
- The ALIAS parser
  - Using the ALIAS parser in a program
    - Evaluating a single formula
    - Evaluating multiple equations
  - Example of use of the parser
- The generic Solver
  - Dealing with inequalities
  - Dealing with parametric system
- MAPLE library for the Interval Solver
- The generic analyzer
- Using the parser programs
Parallel processing
- Parallelizing ALIAS programs
- Stopping a procedure and miscellaneous utilities
How to install and use ALIAS
- Installing ALIAS
- Header and linking
- Reserved variables
- Changes with the previous version
Examples of application of ALIAS-C++
- Examples presented in this documentation
- Examples of applications of interval analysis
Troubleshooting: ALIAS does not work!
- Compilation problems
- Execution problems
- Setting the debug option
Contents
Bibliography
Index
About this document ...

Jean-Pierre Merlet