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Numerical Methods for Least Squares Problems

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by Åke Bjöšrck


1996 / xviii + 408 pages / Softcover / ISBN: 978-0-898713-60-2 / List Price $91.50 / Member Price $64.05 / Order Code OT51

"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 Mathematics Review, Vol. 50, No. 2, February 1997

"A comprehensive and up-to-date treatment that includes many recent developments."--- Arnold M. Osterbee, The American Mathematical Monthly, January 1997

The method of least squares was discovered by Gauss in 1795. It has since become the principal tool to reduce 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.

Mathematicians working in numerical linear algebra, computational scientists and engineers, statisticians, and electrical engineers. The book can also be used in upper-level undergraduate and beginning graduate courses in scientific computing and applied sciences.

Chapter 1: Mathematical and Statistical Properties of Least Squares Solutions; Chapter 2: Basic Numerical Methods; Chapter 3: Modified Least Squares Problems; Chapter 4: Generalized Least Squares Problems; Chapter 5: Constrained Least Squares Problems; Chapter 6: Direct Methods for Sparse Least Squares Problems; Chapter 7: Iterative Methods for Least Squares Problems; Chapter 8: Least Squares With Special Bases; Chapter 9: Nonlinear Least Squares Problems; References; Bibliography.

ISBN: 9780898713602

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