Lawson C.L., Hanson R.J. Solving Least Squares Problems

Society for Industrial and Applied Mathematics, 1982. — 352 p.This book brings together a body of information on solving least squares problems whose practical development has taken place mainly during the past decade. This information is valuable to the scientist, engineer, or student who must analyze and solve systems of linear algebraic equations. These systems may be overdetermined, underdetermined, or exactly determined and may or may not be consistent. They may also include both linear equality and inequality constraints.
Practitioners in specific fields have developed techniques and nomenclature for the least squares problems of their own discipline. The material presented in this book can unify this divergence of methods.
Essentially all real problems are nonlinear. Many of the methods of reaching an understanding of nonlinear problems or computing using nonlinear models involve the local replacement of the nonlinear problem by a linear one. Specifically, various methods of analyzing and solving the nonlinear least squares problem involve solving a sequence of linear least squares problems. One essential requirement of these methods is the capability of computing solutions to (possibly poorly determined) linear least squares problems which are reasonable in the context of the nonlinear problem.
For the reader whose immediate concern is with a particular application we suggest first reading Chapter
25. Following this the reader may find that some of the Fortran programs and subroutines of Appendix C have direct applicability to his problem.Analysis of the Least Squares Problem
Orthogonal Decomposition by Certain Elementary Orthogonal Transformations
Orthogonal Decomposition by Singular Value Decomposition
Perturbation Theorems for Singular Values
Bounds for the Condition Number of a Triangular Matrix
The Pseudoinverse
Perturbation Bounds for the Pseudoinverse
Perturbation Bounds for the Solution of Problem LS
Numerical Computations Using Elementary Orthogonal Transformations
Computing the Solution for the Overdetermined or Exactly Determined Full Rank Problem
Computation of the Covariance Matrix or the solution Parameters
Computing the Solution for the Underdetermined Full Rank Problem
Computing the Solution for Problem LS with Possibly Deficient Pseudorank
Analysis of Computing Errors for Householder Transformations
Analysis of Computing Errors for the Problem LS
Analysis of Computing Errors for the Problem LS Using Mixed Precision Arithmetic
Computation of the Singular Value Decomposition and the Solution of Problem LS
Other Methods for Least Squares Problems
Linear Least Squares with linear Equality Constraints Using a Basis of the Null Space
Linear Least Squares with linear Equality Constraints by Direct Elimination
Linear Least Squares with linear Equality Constraints by Weighting
Linear Least Squares with Linear Inequality Constraints
Modifying a QR Decomposition to Add or Remove Column Vector
Practical Analysis of Least Squares Problems
Examples of Some Methods of Analyzing a Least Squares Problem
Modifying a QR Decomposition to Add or Remove Row Vectors with Application to Sequential Processing of Problems Having a Large or Banded Coefficient Matrix
A: Bask Linear Algebra Including Projections
B: Proof of Global Quadratic Convergence of the QR Algorithm
C: Description and Use of FORTRAN Codes for Solving Problem LS
D: Developments from 1974 to 1995