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Solving Least Squares Problems

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by Charles L. Lawson and Richard J. Hanson


1995 / xii + 337 pages / Softcover / ISBN: 978-0-898713-56-5 / List Price $68.50 / SIAM Member Price $47.95 / Order Code CL15

An accessible text for the study of numerical methods for solving least squares problems remains an essential part of a scientific software foundation. This book has served this purpose well. Numerical analysts, statisticians, and engineers have developed techniques and nomenclature for the least squares problems of their own discipline. This well-organized presentation of the basic material needed for the solution of least squares problems can unify this divergence of methods. Mathematicians, practicing engineers, and scientists will welcome its return to print.

The material covered includes Householder and Givens orthogonal transformations, the QR and SVD decompositions, equality constraints, solutions in nonnegative variables, banded problems, and updating methods for sequential estimation. Both the theory and practical algorithms are included. The easily understood explanations and the appendix providing a review of basic linear algebra make the book accessible for the non-specialist.

This Classic edition includes a new appendix which summarizes the major developments since the book was originally published in 1974. The additions are organized in short sections associated with each chapter. An additional 230 references have been added bringing the bibliography to over 400 entries. Appendix C has been edited to reflect changes in the associated software package and software distribution method. The software has been upgraded to conform to the FORTRAN 77 standard and a new subroutine has been added in FORTRAN 90 for the solution of the bounded variables least squares problem (BVLS). The codes are available from netlib via the internet.

This book is intended to be used as both a text and a reference for persons who are investigating the solutions of linear least squares problems. Specifically, it will be of interest to numerical analysts, statisticians working with regression models, and engineers studying such topics as parameter estimation, filtering, or process identification.

Preface to the Classics Edition; Preface; Chapter 1: Introduction; Chapter 2: Analysis of the Least Squares Problem; Chapter 3: Orthogonal Decomposition by Certain Elementary Orthogonal Transformations; Chapter 4: Orthogonal Decomposition by Singular Value Decomposition; Chapter 5: Perturbation Theorems for Singular Values; Chapter 6: Bounds for the Condition Number of a Triangular Matrix; Chapter 7: The Pseudoinverse; Chapter 8: Perturbation Bounds for the Pseudoinverse; Chapter 9: Perturbation Bounds for the Solution of Problem LS; Chapter 10: Numerical Computations Using Elementary Orthogonal Transformations; Chapter 11: Computing the Solution for the Overdetermined or Exactly Determined Full Rank Problem; Chapter 12: Computation of the Covariance Matrix of the Solution Parameters; Chapter 13: Computing the Solution for the Underdetermined Full Rank Problem; Chapter 14: Computing the Solution for Problem LS with Possibly Deficient Pseudorank; Chapter 15: Analysis of Computing Errors for Householder Transformations; Chapter 16: Analysis of Computing Errors for the Problem LS; Chapter 17: Analysis of Computing Errors for the Problem LS Using Mixed Precision Arithmetic; Chapter 18: Computation of the Singular Value Decomposition and the Solution of Problem LS; Chapter 19: Other Methods for Least Squares Problems; Chapter 20: Linear Least Squares with Linear Equality Constraints Using a Basis of the Null Space; Chapter 21: Linear Least Squares with Linear Equality Constraints by Direct Elimination; Chapter 22: Linear Least Squares with Linear Equality Constraints by Weighting; Chapter 23: Linear Least Squares with Linear Inequality Constraints; Chapter 24: Modifying a QR Decomposition to Add or Remove Column Vectors; Chapter 25: Practical Analysis of Least Squares Problems; Chapter 26: Examples of Some Methods of Analyzing a Least Squares Problem; Chapter 27: Modifying a QR Decomposition to Add or Remove Row Vectors with Application to Sequential Processing of Problems Having a Large or Banded Coefficient Matrix; Appendix A: Basic Linear Algebra Including Projections; Appendix B: Proof of Global Quadratic Convergence of the QR Algorithm; Appendix C: Description and Use of FORTRAN Codes for Solving Problem LS; Appendix D: Developments from 1974 to 1995; Bibliography; Index. 

ISBN: 9780898713565

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