Numerical methods for least squares problems.

*(English)*Zbl 0847.65023
Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. xvii, 408 p. (1996).

Today the method of least squares (discovered by Gauss in 1795) is applied in a great number of scientific and practical areas. This monograph covers the full spectrum of relevant problems and up-to-date methods in least squares. To appreciate this we give the detailed contents:

1. Mathematical and statistical properties of least squares solutions (Introduction; The singular value decompositions (SVD); The QR decomposition; Sensitivity of least squares solutions).

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).

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).

4. Generalized least squares problems (Generalized QR decompositions; The generalized SVD; General linear models and Generalized least squares; Weighted least squares problems; Minimizing the \(\ell_p\)-norm; Total least squares).

5. Constrained least squares problems (Linear equality constraints; Linear inequality constraints; Quadratic constraints).

6. Direct methods for sparse problems (Introduction; Banded least squares problems; Block angular least squares problems; Tools for general problems; Fill minimizing column orderings; The numerical Cholesky and QR-decompositions; Special topics; Sparse constrained problems; Software and test results).

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).

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).

9. Nonlinear least squares problems (The nonlinear least squares problem; Gauss-Newton-type methods; Newton-type methods; Separable and constrained problems).

Bibliography. Index.

The substantial bibliography contains 860 references. (Unfortunately, not each reference seems to be cited from its context). Overlooking the FORTRAN subroutines in the book of C. L. Lawson and R. J. Hanson [(*) Solving least squares problems, Classics Appl. Math., SIAM, 2nd edition 1995 (first published 1974; MR 51.2270)] this book now supersedes the standard work (*) for a long time and is a milestone in numerical linear algebra as well as the books of R. S. Varga [Matrix iterative analysis, Prentice Hall (1963; Zbl 0133.08602)], J. H. Wilkinson [The algebraic eigenvalue problem, Oxford: Clarendon Press (1965; Zbl 0258.65037)] and G. H. Golub and C. F. van Loan [Matrix computations (1983; Zbl 0559.65011)] still are.

1. Mathematical and statistical properties of least squares solutions (Introduction; The singular value decompositions (SVD); The QR decomposition; Sensitivity of least squares solutions).

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).

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).

4. Generalized least squares problems (Generalized QR decompositions; The generalized SVD; General linear models and Generalized least squares; Weighted least squares problems; Minimizing the \(\ell_p\)-norm; Total least squares).

5. Constrained least squares problems (Linear equality constraints; Linear inequality constraints; Quadratic constraints).

6. Direct methods for sparse problems (Introduction; Banded least squares problems; Block angular least squares problems; Tools for general problems; Fill minimizing column orderings; The numerical Cholesky and QR-decompositions; Special topics; Sparse constrained problems; Software and test results).

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).

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).

9. Nonlinear least squares problems (The nonlinear least squares problem; Gauss-Newton-type methods; Newton-type methods; Separable and constrained problems).

Bibliography. Index.

The substantial bibliography contains 860 references. (Unfortunately, not each reference seems to be cited from its context). Overlooking the FORTRAN subroutines in the book of C. L. Lawson and R. J. Hanson [(*) Solving least squares problems, Classics Appl. Math., SIAM, 2nd edition 1995 (first published 1974; MR 51.2270)] this book now supersedes the standard work (*) for a long time and is a milestone in numerical linear algebra as well as the books of R. S. Varga [Matrix iterative analysis, Prentice Hall (1963; Zbl 0133.08602)], J. H. Wilkinson [The algebraic eigenvalue problem, Oxford: Clarendon Press (1965; Zbl 0258.65037)] and G. H. Golub and C. F. van Loan [Matrix computations (1983; Zbl 0559.65011)] still are.

Reviewer: H.Späth (Oldenburg)

##### MSC:

65F20 | Numerical solutions to overdetermined systems, pseudoinverses |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65C99 | Probabilistic methods, stochastic differential equations |

62J05 | Linear regression; mixed models |

65J10 | Numerical solutions to equations with linear operators (do not use 65Fxx) |