Matrices, Moments and Quadrature with Applications

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...

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Hlavní autoři:	Golub, Gene H, Meurant, Gérard
Médium:	E-kniha Kniha
Jazyk:	angličtina
Vydáno:	Princeton Princeton University Press 2009
Vydání:	1
Edice:	Princeton Series in Applied Mathematics
Témata:	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 Polynomial Identity matrix Linear differential equation Iterative method Eigenvalues and eigenvectors Invertible matrix Inverse iteration Calculation Gaussian quadrature Trace (linear algebra) Biconjugate gradient method Conjugate transpose Meromorphic function Function of a real variable Bidiagonal matrix Rotation matrix Recurrence relation QR algorithm Estimation QR decomposition Determinant Holomorphic function Machine epsilon Algorithm Bilinear form Matrix function Function (mathematics) Dimension (vector space) Computation Regularization (mathematics) Diagonal matrix Weight function Partial fraction decomposition Matrix polynomial Hessenberg matrix Tridiagonal matrix Cross-validation (statistics) Singular value Characteristic polynomial Iteration Jacobi matrix Tikhonov regularization Upper and lower bounds Moment matrix Orthogonalization Square (algebra) Orthogonal polynomials Monic polynomial Quadratic form Rate of convergence Hilbert matrix Conjugate gradient method Coefficient Numerical integration Krylov subspace Linear regression Equation Moment (mathematics) Basis (linear algebra) Factorization Orthogonal transformation Inverse problem Orthonormal basis Variable (mathematics) Hankel matrix Lanczos algorithm Laguerre polynomials Symmetric matrix Condition number Newton's method
ISBN:	0691143412, 9780691143415, 9781400833887, 1400833884
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Popis
Shrnutí:	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.
Bibliografie:	Includes bibliographical reference (p. [335]-359) and index
ISBN:	0691143412 9780691143415 9781400833887 1400833884
DOI:	10.1515/9781400833887