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Solving Least Squares Problems

SIAM's Classics in Applied Mathematics series consists of books that were previously allowed to go out of print. These books are republished by SIAM as a professional service because they continue to be important resources for mathematical scientists. Editor-in-Chief Robert E. O'Malley, Jr., University of Washington Editorial Board Richard A. Brualdi, University of Wisconsin-Madison Herbert B. Keller, California Institute of Technology Andrzej Z. Manitius, George Mason University Ingram Olkin, Stanford University Stanley Richardson, University of Edinburgh Ferdinand Verhulst, Mathematisch Instituut, University of Utrecht Classics in Applied Mathematics C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences Johan G. F. Belinfante and Bernard Kolman, A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods James M. Ortega, Numerical Analysis: A Second Course Anthony V. Fiacco and Garth P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques F. H. Clarke, Optimization and Nonsmooth Analysis George F. Carrier and Carl E. Pearson, Ordinary Differential Equations Leo Breiman, Probability R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding Abraham Berman and Robert J. Plemmons, Nonnegative Matrices in the Mathemat- ical Sciences Olvi L. Mangasarian, Nonlinear Programming *Carl Friedrich Gauss, Theory of the Combination of Observations Least Subject to Errors: Part One, Part Two, Supplement. Translated by G. W. Stewart Richard Bellman, Introduction to Matrix Analysis U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations Charles L. Lawson and Richard J. Hanson, Solving Least Squares Problems J. E. Dennis, Jr. and Robert B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations Richard E. Barlow and Frank Proschan, Mathematical Theory of Reliability Cornelius Lanczos, Linear Differential Operators Richard Bellman, Introduction to Matrix Analysis, Second Edition Beresford N. Parlett, The Symmetric Eigenvalue Problem *First time in print.

Classics in Applied Mathematics (continued) Richard Haberman, Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow Peter W. M. John, Statistical Design and Analysis of Experiments Tamer Basar and Geert Jan Olsder, Dynamic Noncooperative Game Theory, Second Edition Emanuel Parzen, Stochastic Processes Petar Kokotovic, Hassan K. Khalil, and John O'Reilly, Singular Perturbation Methods in Control* Analysis and Design Jean Dickinson Gibbons, Ingram Olkin, and Milton Sobel, Selecting and Ordering Populations: A New Statistical Methodology James A. Murdock, Perturbations: Theory and Methods Ivar Ekeland and Roger T6mam, Convex Analysis and Variational Problems Ivar Stakgold, Boundary Value Problems of Mathematical Physics, Volumes I and II J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables David Kinderlehrer and Guido Stampacchia, An Introduction to Variational Inequalities and Their Applications F. Natterer, The Mathematics of Computerized Tomography Avinash C. Kak and Malcolm Slaney, Principles of Computerized Tomographic Imaging R. Wong, Asymptotic Approximations of Integrals O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems: Theory and Computation David R. Brillinger, Time Series: Data Analysis and Theory Joel N. Franklin, Methods of Mathematical Economics: Linear and Nonlinear Programming, Fixed-Point Theorems Philip Hartman, Ordinary Differential Equations, Second Edition Michael D. Intriligator, Mathematical Optimization and Economic Theory Philippe G. Ciarlet, The Finite Element Method for Elliptic Problems Jane K. Cullum and Ralph A. Willoughby, Lanc^os Algorithms for Large Symmetric Eigenvalue Computations, Vol. I: Theory M. Vidyasagar, Nonlinear Systems Analysis, Second Edition Robert Mattheij and Jaap Molenaar, Ordinary Differential Equations in Theory and Practice Shanti S. Gupta and S. Panchapakesan, Multiple Decision Procedures: Theory and Methodology of Selecting and Ranking Populations Eugene L. Allgower and Kurt Georg, Introduction to Numerical Continuation Methods Heinz-Otto Kreiss and Jens Lorenz, Initial-Boundary Value Problems and the Navier- Sto/ces Equations

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Solving Least Squares Problems Charles L. Lawson San Clemente, California Richard J. Hanson Rice University Houston, Texas siam. Society for Industrial and Applied Mathematics Philadelphia

Copyright © 1995 by the Society for Industrial and Applied Mathematics. This SIAM edition is an unabridged, revised republication of the work first published by Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1974. 10 9 8 7 6 5 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. Library of Congress Cataloging-in-Publication Data Lawson, Charles L. Hanson. p. Solving least squares problems / Charles L. Lawson, Richard J. cm. - (Classics in applied mathematics; 15) "This SIAM edition is an unabridged, revised republication of the work first published by Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1974"--T.p. verso. Includes bibliographical references and index. ISBN 0-89871-356-0 (pbk.) 1. Least squares-Data processing. I. Hanson, Richard J., 1938- II. Title. III. Series. QA275.L38 1995 511'.42--dc20 95-35178 Siam. is a registered trademark.

Contents Preface to the Classics Edition .»„.»......»...»....„...«...„..,.„....»...».»........„.„.......»»........... ix xi Preface Chapter 1 Introduction ...........................1 Chapter 2 Analysis of the Least Squares Problem ............................................... 5 Chapter 3 Orthogonal Decomposition by Certain Elementary Orthogonal Transformations................................................................. 9 Chapter 4 Orthogonal Decomposition by Singular Value Decomposition ..................................................... ...—........... 18 Chapter 5 Perturbation Theorems for Singular Values............................... 23 Chapter 6 Bounds for the Condition Number of a Triangular Matrix..... 28 Chapter 7 The Pseudoinverse..........................^...................^...^....^..................... 36 Chapter 8 Perturbation Bounds for the Pseudoinverse............................. 41 Chapter 9 Perturbation Bounds for the Solution of Problem LS................ 49 Chapter 10 Numerical Computations Using Elementary Orthogonal Transformations ....................«..^.».^««..................... 53 Chapter 11 Computing the Solution for the Overdetermined or Exactly Determined Full Rank Problem 63 Chapter 12 Computation of the Covariance Matrix or the solution Parameters »»«...........»«.........................................................67 Chapter 13 Computing the Solution for the Underdetermined Full Rank Problem Chapter 14 Computing the Solution for Problem LS with Possibly Deficient Pseudorank 74 77 Chapter 15 Analysis of Computing Errors for Householder Transformations...................................................... 83 Chapter 16 Analysis of Computing Errors for the Problem LS .................... 90 vii

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