×

Matrices, moments and quadrature with applications. (English) Zbl 1217.65056

Princeton Series in Applied Mathematics. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14341-5/hbk; 978-1-400-83388-7/ebook). xi, 363 p. (2010).
This book describes and explains the beautiful mathematical relationships between matrices, moments, orthogonal polynomials, quadrature rules and the Lanczos and conjugate gradient algorithms. The main idea is to apply Gauss quadrature rules to compute estimates or bounds of a Riemann-Stieltjes integral. These orthogonal polynomials associated with Gauss quadratures can be generated by the Lanczos algorithm or its variants in a beautiful way. Many topics from different areas are brought together. The book consists of two parts.
The first part provides the necessary mathematical background and explains the theory. It consists of nine chapters and its content can be briefly described as follows. Chapter 2 recalls the properties of orthogonal polynomials used in the subsequent chapters. Since tridiagonal matrices play a prominent role in the algorithms described in this book, Chapter 3 recalls properties of these matrices. Chapter 4 briefly describes the well-known Lanczos and conjugate gradient algorithms. Chapter 5 deals with the computation of the tridiagonal matrices containing the coefficients of the three-term recurrences satisfied by orthogonal polynomials. The subject of Chapter 6 is Gauss quadrature to obtain approximations or bounds for Riemann-Stieltjes integrals. Chapter 7 can be regarded as a summary of the previous chapters, and the techniques summarized are extended in Chapter 8 to the case of a nonsymmetric matrix. Chapter 9 is devoted to solving secular equations.
The second part consists of six chapters describing applications and giving numerical examples of the algorithms and techniques developed in the first part. Many interesting topics are discussed. For example, the computation of Gauss quadrature rules, the relationships of the conjugate gradient algorithm with Gauss quadratures, the least squares fit of some given data by polynomials, total least squares, and the determination of the Tikhonov regularization parameter for ill-posed problems. The book is useful for people interested in matrix computations. It can also be of interest for scientists and engineers solving problems in which the computation of bilinear forms arises naturally.

MSC:

65D32 Numerical quadrature and cubature formulas
65F22 Ill-posedness and regularization problems in numerical linear algebra
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F20 Numerical solutions to overdetermined systems, pseudoinverses
65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
15-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra
65F10 Iterative numerical methods for linear systems
65D10 Numerical smoothing, curve fitting
41A55 Approximate quadratures
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
PDFBibTeX XMLCite