Numerical Methods for Least Square Problems
Will be published approx. on 31. December 1996
Book
Paperback/Softback
425 pages
978-0-89871-360-2 (ISBN)
Description
The method of least squares was discovered by Gauss in 1795. It has since become the principal tool for reducing the influence of errors when fitting models to given observations. Today, applications of least squares arise in a great number of scientific areas, such as statistics, geodetics, signal processing, and control.
In the last 20 years there has been a great increase in the capacity for automatic data capturing and computing. Least squares problems of large size are now routinely solved. Tremendous progress has been made in numerical methods for least squares problems, in particular for generalized and modified least squares problems and direct and iterative methods for sparse problems. Until now there has not been a monograph that covers the full spectrum of relevant problems and methods in least squares.
This volume gives an in-depth treatment of topics such as methods for sparse least squares problems, iterative methods, modified least squares, weighted problems, and constrained and regularized problems. The more than 800 references provide a comprehensive survey of the available literature on the subject.
Special Features:
Discusses recent methods, many of which are still described only in the research literature.
Provides a comprehensive up-to-date survey of problems and numerical methods in least squares computation and their numerical properties.
Collects recent research results and covers methods for treating very large and sparse problems with both direct and iterative methods.
Covers updating of solutions and factorizations as well as methods for generalized and constrained least squares problems.
A solid understanding of numerical linear algebra is needed for the more advanced sections. However, many of the chapters are more elementary and because basic facts and theorems are given in an introductory chapter, the book is partly self-contained.
In the last 20 years there has been a great increase in the capacity for automatic data capturing and computing. Least squares problems of large size are now routinely solved. Tremendous progress has been made in numerical methods for least squares problems, in particular for generalized and modified least squares problems and direct and iterative methods for sparse problems. Until now there has not been a monograph that covers the full spectrum of relevant problems and methods in least squares.
This volume gives an in-depth treatment of topics such as methods for sparse least squares problems, iterative methods, modified least squares, weighted problems, and constrained and regularized problems. The more than 800 references provide a comprehensive survey of the available literature on the subject.
Special Features:
Discusses recent methods, many of which are still described only in the research literature.
Provides a comprehensive up-to-date survey of problems and numerical methods in least squares computation and their numerical properties.
Collects recent research results and covers methods for treating very large and sparse problems with both direct and iterative methods.
Covers updating of solutions and factorizations as well as methods for generalized and constrained least squares problems.
A solid understanding of numerical linear algebra is needed for the more advanced sections. However, many of the chapters are more elementary and because basic facts and theorems are given in an introductory chapter, the book is partly self-contained.
Reviews / Votes
'...This book, which was written by a celebrated person in the field, has become a monumental work ... Altogether, very clearly written and a must for everyone who is interested in least squares, as well as all mathematics libraries.' Bob Matheij, ITW Nieuws '...Until now there has not been a monograph covering the full spectrum of methods and applications in least squares. ...This book is a great masterpiece that will serve as a reference for many years. It must be on the shelves of every numerical analyst, computational scientist and engineer, statistician and electrical engineer.' Claude Brejinski, Numerical Algorithms 'This monograph covers the full spectrum of relevant problems and up-to-date methods in least squares ... a milestone in numerical linear algebra ...' H. Spaeth, Zentrallblatt fuer Mathematik ' ... The author has done a superb job of including the most recent research results, many of which are presented here for the first time in book form. The bibliography is comprehensive and contains more than eight hundred entries. The emphasis of the book is on linear least squares problems, but it also contains a chapter on surveying numerical methods for nonlinear problems.' Hongyuan Zha, Mathematical Reviews 'This book gives a very broad coverage of linear least squares problems. Detailed descriptions are provided for the best algorithms to use and the current literature, with some identification of software availability. No examples are given, and there are few graphs, but the detailed information about methods and algorithms makes this an excellent book ... If you are going to solve a least squares problem of any magnitude, you need Numerical Methods for Least Squares Problems ...' B. A. Finlayson, Applied Mechanics Review "A comprehensive and up-to-date treatment that includes many recent developments. In addition to basic methods, it covers methods for modified and generalized least squares problems, and direct and iterative methods for sparse problems." -Arnold M. Ostebee, The American Mathematical Monthly, January 1997. "Bjorck is an expert on least squares problems....This volume surveys numerical methods for these problems. ...its strength is in the detailed discussion of least squares problems and of their various solution techniques." -B. Borchers, CHOICE, Vol. 34, No. 3, November 1996.More details
Language
English
Place of publication
New York
United States
Target group
Professional and scholarly
College/higher education
Product notice
Paperback (trade)
Unsewn / adhesive bound
Dimensions
Height: 251 mm
Width: 175 mm
Thickness: 23 mm
Weight
748 gr
ISBN-13
978-0-89871-360-2 (9780898713602)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Content
Preface
Chapter 1: MATHEMATICAL AND STATISTICAL PROPERTIES OF LEAST SQUARES SOLUTIONS. Introduction
The Singular Value Decomposition
The QR Decomposition
Sensitivity of Least Squares Solutions
Chapter 2: BASIC NUMERICAL METHODS. Basics of Floating Point Computation
The Method of Normal Equations
Elementary Orthogonal Transformations
Methods Based on the QR Decomposition
Methods Based on Gaussian Elimination
Computing the SVD
Rank Deficient and Ill-Conditioned Problems
Estimating Condition Numbers and Errors
Iterative Refinement
Chapter 3: MODIFIED LEAST SQUARES PROBLEMS. Introduction
Modifying the Full QR Decomposition
Downdating the Cholesky Factorization
Modifying the Singular Value Decomposition
Modifying Rank Revealing QR Decompositions
Chapter 4: GENERALIZED LEAST SQUARES PROBLEMS. Generalized QR Decompositions
The Generalized SVD
General Linear Models and Generalized Least Squares
Weighted Least Squares Problems
Minimizing the $l_p$ Norm
Total Least Squares
Chapter 5: CONSTRAINED LEAST SQUARES PROBLEMS. Linear Equality Constraints
Linear Inequality Constraints
Quadratic Constraints
Chapter 6: DIRECT METHODS FOR SPARSE PROBLEMS. Introduction
Banded Least Squares Problems
Block Angular Least Squares Problems
Tools for General Sparse Problems
Fill Minimizing Column Orderings
The Numerical Cholesky and QR-Decompositions
Special Topics
Sparse Constrained Problems
Software and Test Results
Chapter 7: ITERATIVE METHODS FOR LEAST SQUARES PROBLEMS. Introduction
Basic Iterative Methods
Block Iterative Methods
Conjugate Gradient Methods
Incomplete Factorization Preconditioners
Methods Based on Lanczos Bidiagonalization
Methods for Constrained Problems
Chapter 8: LEAST SQUARES PROBLEMS WITH SPECIAL BASES. Least Squares Approximation and Orthogonal Systems
Polynomial Approximation
Discrete Fourier Analysis
Toeplitz Least Squares Problems
Kronecker Product Problems
Chapter 9: NONLINEAR LEAST SQUARES PROBLEMS. The Nonlinear Least Squares Problem
Gauss--Newton-Type Methods
Newton Type Methods
Separable and Constrained Problems
Bibliography
Index.
Chapter 1: MATHEMATICAL AND STATISTICAL PROPERTIES OF LEAST SQUARES SOLUTIONS. Introduction
The Singular Value Decomposition
The QR Decomposition
Sensitivity of Least Squares Solutions
Chapter 2: BASIC NUMERICAL METHODS. Basics of Floating Point Computation
The Method of Normal Equations
Elementary Orthogonal Transformations
Methods Based on the QR Decomposition
Methods Based on Gaussian Elimination
Computing the SVD
Rank Deficient and Ill-Conditioned Problems
Estimating Condition Numbers and Errors
Iterative Refinement
Chapter 3: MODIFIED LEAST SQUARES PROBLEMS. Introduction
Modifying the Full QR Decomposition
Downdating the Cholesky Factorization
Modifying the Singular Value Decomposition
Modifying Rank Revealing QR Decompositions
Chapter 4: GENERALIZED LEAST SQUARES PROBLEMS. Generalized QR Decompositions
The Generalized SVD
General Linear Models and Generalized Least Squares
Weighted Least Squares Problems
Minimizing the $l_p$ Norm
Total Least Squares
Chapter 5: CONSTRAINED LEAST SQUARES PROBLEMS. Linear Equality Constraints
Linear Inequality Constraints
Quadratic Constraints
Chapter 6: DIRECT METHODS FOR SPARSE PROBLEMS. Introduction
Banded Least Squares Problems
Block Angular Least Squares Problems
Tools for General Sparse Problems
Fill Minimizing Column Orderings
The Numerical Cholesky and QR-Decompositions
Special Topics
Sparse Constrained Problems
Software and Test Results
Chapter 7: ITERATIVE METHODS FOR LEAST SQUARES PROBLEMS. Introduction
Basic Iterative Methods
Block Iterative Methods
Conjugate Gradient Methods
Incomplete Factorization Preconditioners
Methods Based on Lanczos Bidiagonalization
Methods for Constrained Problems
Chapter 8: LEAST SQUARES PROBLEMS WITH SPECIAL BASES. Least Squares Approximation and Orthogonal Systems
Polynomial Approximation
Discrete Fourier Analysis
Toeplitz Least Squares Problems
Kronecker Product Problems
Chapter 9: NONLINEAR LEAST SQUARES PROBLEMS. The Nonlinear Least Squares Problem
Gauss--Newton-Type Methods
Newton Type Methods
Separable and Constrained Problems
Bibliography
Index.