Applied Numerical Linear Algebra
SIAM, 01.08.1997 - 430 Seiten
Designed for first-year graduate students from a variety of engineering and scientific disciplines, this comprehensive textbook covers the solution of linear systems, least squares problems, eigenvalue problems, and the singular value decomposition. The author, who helped design the widely used LAPACK and ScaLAPACK linear algebra libraries, draws on this experience to present state-of-the-art techniques for these problems, including recommending which algorithms to use in various practical situations. Algorithms are derived in a mathematically illuminating way, including condition numbers and error bounds. Direct and iterative algorithms, suitable for dense and sparse matrices, are discussed. Algorithm design for modern computer architectures, where moving data is often more expensive than arithmetic operations, is discussed in detail, using LAPACK as an illustration. There are many numerical examples throughout the text and in the problems at the ends of chapters, most of which are written in MATLAB and are freely available on the Web.
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accurate approximate backward error bidiagonal BLAS block called Cholesky choose compute condition number convergence cost discussed divide-and-conquer eigendecomposition eigenproblem eigenvalue problem eigenvalues and eigenvectors error bound example factor Figure floating point floating point number flops Gaussian elimination GEPP graph grid points high relative accuracy IEEE implementation inverse iteration iterative methods Jacobi's method Krylov subspace Lanczos algorithm Lanczos step Lanczos vectors LAPACK LAPACK routine least squares problem Lemma linear system Matlab matrix minimizes model problem multigrid multiplication nodes nonsingular nonsymmetric nonzero norm operations orthogonal orthogonal matrix perturbation pivoting plots Poisson's equation polynomial positive definite Proof QR algorithm QR decomposition QR iteration QUESTION Rayleigh quotient iteration residual Ritz values Ritz vectors Schur form SIAM singular values smallest eigenvalue ſº solution solving Aa sparse symmetric matrix theorem tridiagonal matrix upper Hessenberg upper triangular write zero