eBook

Matrices, Moments and Quadrature with Applications

Golub, Gene H, Meurant, Gérard

2009,2010

Overview

This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms. The book bridges different mathematical areas to obtain algorithms to estimate bilinear forms involving two vectors and a function of the matrix. The first part of the book provides the necessary mathematical background and explains the theory. The second part describes the applications and gives numerical examples of the algorithms and techniques developed in the first part. Applications addressed in the book include computing elements of functions of matrices; obtaining estimates of the error norm in iterative methods for solving linear systems and computing parameters in least squares and total least squares; and solving ill-posed problems using Tikhonov regularization. This book will interest researchers in numerical linear algebra and matrix computations, as well as scientists and engineers working on problems involving computation of bilinear forms.

• Publisher: Princeton University Press
• Subject: Algebra & number theory / Algorithm / Basis (linear algebra) / Biconjugate gradient method / Bidiagonal matrix / Bilinear form / Calculation / Characteristic polynomial / Coefficient / Computation / COMPUTERS / Computer Science / Condition number / Conjugate gradient method / Conjugate transpose / Cross-validation (statistics) / Determinant / Diagonal matrix / Dimension (vector space) / Eigenvalues and eigenvectors / Equation / Estimation / Factorization / Function (mathematics) / Function of a real variable / Gaussian quadrature / General Topics for Engineers / Hankel matrix / Hessenberg matrix / Hilbert matrix / Holomorphic function / Identity matrix / Inverse iteration / Inverse problem / Invertible matrix / Iteration / Iterative method / Jacobi matrix / Krylov subspace / Laguerre polynomials / Lanczos algorithm / Linear differential equation / Linear regression / Machine epsilon / MATHEMATICS / MATHEMATICS / Algebra / General / MATHEMATICS / Applied / MATHEMATICS / General / Matrices / Matrix function / Matrix polynomial / Meromorphic function / Moment (mathematics) / Moment matrix / Monic polynomial / Newton's method / Numerical analysis / Numerical integration / Orthogonal polynomials / Orthogonal transformation / Orthogonalization / Orthonormal basis / Partial fraction decomposition / Polynomial / QR algorithm / QR decomposition / Quadratic form / Rate of convergence / Recurrence relation / Regularization (mathematics) / Rotation matrix / Singular value / Square (algebra) / Symmetric matrix / Technology / Tikhonov regularization / Trace (linear algebra) / Tridiagonal matrix / Upper and lower bounds / Variable (mathematics) / Weight function
• ISBN: 9781400833887, 1400833884, 0691143412, 9780691143415