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Numerical Methods for Least Squares ProblemsNumerical Methods for Least Squares Problems

Åke Björck

"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. Introduction; The Singular Value Decomposition; The QR Decomposition and Least Squares Problem; 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 Problems; Estimating Condition Numbers and Errors; Iterative Refinement; Chapter 3: Modified Least Squares Problems. Introduction; Updating the Full QR Decomposition; Modifying the Gram Schmidt Decomposition; Downdating the Cholesky Factorization; Modifying the Singular Value Decomposition; Updating Rank Revealing QR Decompositions; Chapter 4: Generalized Least Squares Problems. Generalized QR Decompositions; The Generalized SVD; The General Linear Models and Generalized Least Squares; Weighted Least Squares Problems; Minimizing the $l_p$ Norm; Total Least Squares and Linear Orthogonal Regression; Chapter 5: Constrained Least Squares Problems. Linear Equality Constraints; Linear Inequality Constraints; Quadratic Constraints; Chapter 6: Direct Methods for Sparse Least Squares Problems. Introduction; Banded Least Squares Problems; Block Angular Least Squares Problems; Tools for General Sparse Problems; Fill Minimizing Ordering Methods; QR Decompositions for Sparse Problems; Special Topics; Sparse Constrained Problems; 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; Discrete Ill-posed Problems; Chapter 8: Least Squares With Special Bases. Least Squares Approximations and Orthogonal Systems; Polynomial Approximation; Discrete Fourier Analysis; Toeplitz Least Squares Approximations; Problems of Kronecker Structure; Chapter 9: Nonlinear Least Squares Problems. The Nonlinear Least Squares Problem; Gauss Newton Type Methods; Newton Type Methods; Methods for Separable Problems; Constrained Problems; References; Bibliography.

1996 / xviii + 408 pages / Softcover / ISBN-13: 978-0-898713-60-2 / ISBN-10: 0-89871-360-9 /
List Price $83.00 / Member Price $58.10 / Order Code OT51
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Banner art adapted from a figure by Hinke M. Osinga and Bernd Krauskopf (University of Auckland, NZ.)