Contents

• Introduction
  • How to read this manual
  
  
• Solving with Interval Analysis
  • Introduction
    • Interval Analysis
  • Non 0-dimensional system
  • General purpose solving algorithm
    • Mathematical background
    • The 3B method
    • Simplification procedure
    • Implementation
    • Examples and Troubleshooting
  • General purpose solving algorithm with Jacobian
    • Mathematical background
    • Implementation
    • Examples
    • General comments
  • General purpose solving algorithm with Jacobian and Hessian
    • Mathematical background
    • Implementation
    • Examples
  • 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
    • Systematic use of Newton
  • Krawczyk method for solving systems of equations
    • Mathematical background
    • Implementation
  • Solving univariate polynomial with interval analysis
    • Mathematical background
    • Implementation
  • 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
    • Implementation with gradient
  • Solving systems with determinants
  • Solving systems of distance equations
    • Principle
    • Implementation
  • Filtering a system of equation
  
  
• Analyzing systems of equations
  • Introduction
    • Moore theorem
    • Kantorovitch theorem
    • Rouche theorem
    • Interval Newton
    • Miranda theorem
    • Inflation
  
  
• Analyzing trigonometric equations
  • Introduction
  • Number of roots of trigonometric equation
    • Mathematical background
    • Implementation
    • Example
  • Bound on the roots of trigonometric equation
    • Implementation
  • 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
    • Second Cauchy theorem
    • Third Cauchy theorem
    • Lagrange-MacLaurin theorem
    • Laguerre method
    • Laguerre second method
    • Newton method
    • Newton theorem
    • Joyal bounds
    • Pellet method
    • Global implementation
    • Kantorovitch theorem
  • Bounds on the product and sum of roots
    • Newton relations
    • Implementation
    • Viète relations
    • Implementation
  • Maximum number of real roots
  • Number of real roots
    • Descartes Lemma
    • Budan-Fourier method
    • Sturm method
    • Du Gua-Huat-Euler theorem
  • Separation between the roots
    • Rump theorem
  • Analyzing the real roots
  • Analyzing the real part of the roots
  • Utilities
    • Addition of two polynomials
    • Multiplication of two polynomials
    • Evaluation of a polynomial
    • Sign of a polynomial
    • Derivative of a polynomial
    • Euclidian division
    • Expansion of $(x-a)^n$
    • 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
    • Regularity of matrix with linear elements
  • Characteristic polynomial
  • Spectral radius
  
  
• Optimization
  • Definition of a minimum and a maximum
  • Methods
  • Implementation
    • Optimization with function evaluation
    • Optimization with function and jacobian evaluation
    • Return code
    • Order
    • The variable table
  • Examples
    • Example 1
    • Example 2
  
  
• Continuation for one dimensional system
  • Continuation 1D
    • Mathematical background
    • Implementation
    • 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
    • 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
    • Dealing with inequalities
    • Pre-processing and dealing with parametric equations
    • Errors and Debug
    • Examples
  • 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
    • Crash and bugs
  • Setting the debug option
  
  
• Bibliography