Numerical Linear Approximation in C

Preliminaries and Tutorials 
Applications of Linear Approximation 
Applications to social sciences and economics
Applications to industry
Applications to digital images
Preliminaries 
Discrete linear approximation and solution of overdetermined linear equations
Comparison between the L₁, the L₂, and the L_∞ norms by a practical example
Error tolerances in the calculation
Representation of vectors and matrices in C
Outliers and dealing with them
Linear Programming and the Simplex Algorithm 
Notations and definitions
The simplex algorithm
The simplex tableau
The two-phase method
Duality theory in linear programming
Degeneracy in linear programming and its resolution
Linear programming and linear approximation
Stability of the solution in linear programming
Efficient Solutions of Linear Equations 
Vector and matrix norms and relevant theorems
Elementary matrices
Gauss LUdecomposition with complete pivoting
Orthogonal factorization methods
Gauss–Jordan method
Rounding errors in arithmetic operations
The L₁ Approximation 
Linear L₁ Approximation 
Linear programming formulation of the problem
Description of the algorithm
The dual simplex method
Modification to the algorithm
Occurrence of degeneracy
A significant property of the L₁ approximation
Triangular decomposition of the basis matrix
Arithmetic operations count
Numerical results and comments
C source code
One-Sided L₁ Approximation 
A special problem of a general constrained one
Linear programming formulation of the problem
Description of the algorithm
Numerical results and comments
C source code
L₁ Approximation with Bounded Variables 
A special problem of a general constrained one
Linear programming formulation of the problem
Description of the algorithm
Numerical results and comments
C source code
L₁ Polygonal Approximation of Plane Curves 
Approaches to polygonal approximation
The L₁ approximation problem
Description of the algorithm
Linear programming technique
Numerical results and comments
C source code
Piecewise L₁ Approximation of Plane Curves 
Characteristics of the piecewise approximation
The discrete linear L₁ approximation problem
Description of the algorithms
Numerical results and comments
C source code
The Chebyshev Approximation 
Linear Chebyshev Approximation 
Linear programming formulation of the problem
Description of the algorithm
A significant property of the Chebyshev approximation
Numerical results and comments
C source code
One-Sided Chebyshev Approximation 
A special problem of a general constrained one
Linear programming formulation of the problem
Description of the algorithm
Numerical results and comments
C source code
Chebyshev Approximation with Bounded Variables 
A special problem of a general constrained one
Linear programming formulation of the problem
Description of the algorithm
Numerical results and comments
C source code
Restricted Chebyshev Approximation 
A special problem of general constrained algorithms
Linear programming formulation of the problem
Description of the algorithm
The triangular decomposition method
Arithmetic operations count
Numerical results and comments
C source code
Strict Chebyshev Approximation 
The problem as presented by Descloux
Linear programming analysis of the problem
Numerical results and comments
C source code
Piecewise Chebyshev Approximation 
Characteristic properties of piecewise approximation
The discrete linear Chebyshev approximation problem
Description of the algorithms
Numerical results and comments
C source code
Solution of Linear Inequalities 
Pattern classification problem
Solution of the system of linear inequalities Ca > 0
Linear one-sided Chebyshev approximation algorithm
Linear one-sided L₁ approximation algorithm
Numerical results and comments
C source code
The Least Squares Approximation 
Least Squares and Pseudo-Inverses of Matrices 
Least squares solution of linear equations
Factorization of matrix A
Explicit expression for the pseudo-inverse
The singular value decomposition (SVD)
Practical considerations in computing
Linear spaces and the pseudo-inverses
Multicollinearity, collinearity, or the ill-conditioning of matrix A
Principal components analysis (PCA)
Partial least squares method (PLS)
Ridge equation
Numerical results and comments
C source code
Piecewise Linear Least Squares Approximation 
Characteristics of the approximation
The discrete linear least squares approximation problem
Description of the algorithms
Numerical results and comments
Updating and downdating techniques
C source code
Solution of Ill-Posed Linear Systems 
Solution of ill-posed linear systems
Estimation of the free parameter
Description of the new algorithm
Optimum value of the rank
Use of linear programming techniques
Numerical results and comments
C source code
Solution of Underdetermined Systems Of Linear Equations 
L₁ Solution of Underdetermined Linear Equations 
Linear programming formulation of the problem
Description of the algorithm
Numerical results and comments
C source code
Bounded and L₁ Bounded Solutions of Underdetermined Linear Equations 
Linear programming formulation of the two problems
Description of the algorithms
Numerical results and comments
C source code
Chebyshev Solution of Underdetermined Linear Equations 
The linear programming problem
Description of the algorithm
Numerical results and comments
C source code
Bounded Least Squares Solution of Underdetermined Linear Equations 
Quadratic programming formulation of the problems
Solution of problem (E0)
Solution of problem (E)
Numerical results and comments
C source code
Appendix A: References
Appendix B: Main Program 
Appendix C: Constants, Types, and Function Prototypes 
Appendix D: Utilities and Common Functions 
Index