# Numerical Linear Approximation in C

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Illustrating the relevance of linear approximation in a variety of fields, Numerical Linear Approximation in C presents a unique collection of linear approximation algorithms that can be used to analyze, model, and compress discrete data. Developed by the lead author, the algorithms have been successfully applied to several engineering projects at the National Research Council of Canada. Basing most of the algorithms on linear programming techniques, the book begins with an introductory section that covers applications, the simplex method, and matrices. The next three parts focus on various L1, Chebyshev, and least squares approximations, including one-sided, bounded variables, and piecewise. The final section presents the solution of underdetermined systems of consistent linear equations that are subject to different constraints on the elements of the unknown solution vector. Except in the preliminary section, all chapters include the C functions of the algorithms, along with drivers that contain numerous test case examples and results. The accompanying CD-ROM also provides the algorithms written in C code as well as the test drivers. To use the software, it is not required to understand the theory behind each function.show more

## Product details

- Electronic book text | 968 pages
- 19 May 2008
- Taylor & Francis Ltd
- Chapman & Hall/CRC
- London, United Kingdom
- 24 Illustrations, black and white
- 1584889799
- 9781584889793

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## Table of contents

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 Lâ 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 An Introduction appears at the beginning of each chapter.show more