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Introduction
Contents
ALIAS-C++
A C++ Algorithms Library of Interval Analysis for equation Systems
Version 2.7
September 2012
The COPRIN project
Download ALIAS here!
Introduction
How to read this manual
Solving with Interval Analysis
Introduction
Interval Analysis
Mathematical background
Implementation
Problems with the interval-valuation of an expression
Dealing with infinity
Non 0-dimensional system
General purpose solving algorithm
Mathematical background
Principle
Managing the bisection and ordering
An alternative: the single bisection
Solutions and Distinct solutions
The 3B method
Simplification procedure
Implementation
Number of unknowns and functions
Type of the functions
Interval Function
The order
Storage
Accuracy
Distinct solutions
Return code
Debugging
Examples and Troubleshooting
Example 1
Example 2
Example 3
Example 4
General comments
General purpose solving algorithm with Jacobian
Mathematical background
Using the monotonicity
Improving the evaluation using the Jacobian and centered form
Single bisection mode
Implementation
Return code
Jacobian matrix
Evaluation procedure using the Jacobian
Storage
Examples
Example 1
Example 2
Example 3
Example 4
General comments
General purpose solving algorithm with Jacobian and Hessian
Mathematical background
Single bisection mode
Implementation
Hessian procedure
Storage
Improvement of the function evaluation and of the Jacobian
Return code and debug
Examples
Example 2
Example 3
Example 4
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
Mathematical background
Implementation
Return value
Functions
Systematic use of Newton
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
Mathematical background
Implementation
Examples
Solving systems with linear and non-linear terms: the simplex method
Mathematical background
Implementation without gradient
The
NonLinear
procedure
The
CoeffLinear
procedure
Using an expansion
Example
Implementation with gradient
The
GradientNonLinear
procedure
Solving systems with determinants
Solving systems of distance equations
Principle
Implementation
Return code
Inflation and Newton scheme
Choosing the right set of equations and variables
Initial domain and simplification procedures
Filtering a system of equation
Analyzing systems of equations
Introduction
Moore theorem
Mathematical background
Implementation
Kantorovitch theorem
Mathematical background
Implementation
Return code
Rouche theorem
Mathematical background
Implementation
Interval Newton
Miranda theorem
Mathematical background
Implementation
Inflation
Mathematical background
Implementation
Analyzing trigonometric equations
Introduction
Number of roots of trigonometric equation
Mathematical background
Implementation
Example
Bound on the roots of trigonometric equation
Implementation
Example
Utilities for trigonometric equation
Inclusion in an angle interval
Distance between two angles
Generalized inverse trigonometric functions
Analyzing univariate polynomials
Introduction
Finding bounds on the roots
First Cauchy theorem
Mathematical background
Implementation
Example
Second Cauchy theorem
Mathematical background
Implementation
Example
Third Cauchy theorem
Mathematical background
Implementation
Lagrange-MacLaurin theorem
Mathematical background
Implementation
Example
Laguerre method
Mathematical background
Implementation
Example
Laguerre second method
Mathematical background
Implementation
Newton method
Mathematical background
Implementation
Newton theorem
Mathematical background
Implementation
Joyal bounds
Mathematical background
Implementation
Pellet method
Mathematical background
Implementation
Global implementation
Example
Kantorovitch theorem
Implementation
Example
Bounds on the product and sum of roots
Newton relations
Mathematical background
Implementation
Viète relations
Mathematical background
Implementation
Maximum number of real roots
Number of real roots
Descartes Lemma
Mathematical background
Implementation
Budan-Fourier method
Mathematical background
Implementation
Example
Sturm method
Mathematical background
Implementation
Example
Du Gua-Huat-Euler theorem
Mathematical background
Implementation
Separation between the roots
Rump theorem
Mathematical background
Implementation
Example
Analyzing the real roots
Analyzing the real part of the roots
Utilities
Addition of two polynomials
Multiplication of two polynomials
Evaluation of a polynomial
Evaluation in centered form
Safe evaluation of a polynomial
Sign of a polynomial
Implementation
Derivative of a polynomial
Euclidian division
Expansion of
Centered form
Unitary polynomial
Safe evaluation of a vector
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
Implementation
Rohn simplification procedure
Mathematical background
Implementation
Regularity of matrix with linear elements
Mathematical background
Implementation
Characteristic polynomial
Spectral radius
Optimization
Definition of a minimum and a maximum
Methods
Implementation
Optimization with function evaluation
Return code
Dealing with inequalities on the same function
Optimization with function and jacobian evaluation
Return code
Order
General principle
The variable table
Examples
Example 1
Example 2
Continuation for one dimensional system
Continuation 1D
Mathematical background
Implementation
Certified Newton
Procedure for following branches
Full continuation procedure
Missed branches
Example
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
Principle
Practical implementation
The generic analyzer
Dealing with inequalities
Pre-processing and dealing with parametric equations
Pre-processing
Parametric equations
Errors and Debug
Examples
Non algebraic equations
Using the generic analyzer in a program
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
Example 2
Example 3
Example 4
Examples of applications of interval analysis
A geometrical example
A robot kinematics example
A robot control problem
Robot singularity analysis
Robot workspace analysis
Robot synthesis example
A quantum mechanics example
Troubleshooting:
ALIAS
does not work!
Compilation problems
Execution problems
Interval valuation problems
Wrong results
Running out of memory
Large computation time
Changing the formulation of the problem
Choosing the right heuristics
Crash and bugs
Setting the debug option
Contents
Bibliography
About this document ...
Jean-Pierre Merlet 2012-12-20
