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Watkins D. — Fundamentals of matrix computations

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Название: Fundamentals of matrix computations

Автор: Watkins D.

Аннотация:

A significantly revised and improved introduction to a critical aspect of scientific computation
Matrix computations lie at the heart of most scientific computational tasks. For any scientist or engineer doing large-scale simulations, an understanding of the topic is essential. Fundamentals of Matrix Computations, Second Edition explains matrix computations and the accompanying theory clearly and in detail, along with useful insights.
This Second Edition of a popular text has now been revised and improved to appeal to the needs of practicing scientists and graduate and advanced undergraduate students. New to this edition is the use of MATLAB for many of the exercises and examples, although the Fortran exercises in the First Edition have been kept for those who want to use them. This new edition includes:
* Numerous examples and exercises on applications including electrical circuits, elasticity (mass-spring systems), and simple partial differential equations
* Early introduction of the singular value decomposition
* A new chapter on iterative methods, including the powerful preconditioned conjugate-gradient method for solving symmetric, positive definite systems
* An introduction to new methods for solving large, sparse eigenvalue problems including the popular implicitly-restarted Arnoldi and Jacobi-Davidson methods
With in-depth discussions of such other topics as modern componentwise error analysis, reorthogonalization, and rank-one updates of the QR decomposition, Fundamentals of Matrix Computations, Second Edition will prove to be a versatile companion to novice and practicing mathematicians who seek mastery of matrix computation.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: second edition

Год издания: 2002

Количество страниц: 618

Добавлена в каталог: 20.07.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
Предметный указатель
 decomposition      84 decomposition      85 Abel, N.H.      310 478 Absolute value of a matrix      159 176 acosh      450 Airfoil      69 Algebraic multiplicity      306 417 Ammar — Dillo conjecture      604 Angle between two vectors      186 Arnoldi process      435 Arnoldi process, implicitly restarted (IRA)      451 Arrowhead matrix      63 Augmented matrix      71 Back substitution      29 backslash      15 Backward error analysis      145 Backward stability      146 Backward stability and residual      146 463 Backward stability of Cholesky's method      164 Backward stability of forward substitution      159 Backward stability of Gaussian elimination      162 Backward stability of matrix multiplication      211 Backward stability of rotators and reflectors      206 211 Backward stability, componentwise      161 179 Backward stability, normwise      161 Band width      54 Banded matrix      54 Basis (of a subspace)      224 Bauer — Fike theorem      463 Bauer's Theorem      173 Best approximation theorem      243 583 bicg      600 BICGSTAB      599 Biconjugate-gradient (BiCG) algorithm      598 Bidiagonal matrix      402 Bidiagonal matrix, proper      405 Biorthogonal vectors      465 599 Bisection method      483 BLAS      102 Block Gaussian elimination      90 Block iteration      541 bucky      64 Bucky ball      64 Bulge-chasing algorithm      375 380 407 486 513 Bunch — Kaufman algorithm      106 Cache      10 Cancellation      144 Cancellation in Gaussian elimination      151 Catastrophic cancellation      144 Cauchy —Schwarz inequality      113 186 cgs      599 Characteristic equation      305 505 Characteristic polynomial      305 505 Chebyshev acceleration      550 Chebyshev polynomials      448 Chebyshev polynomials and convergence of CG algorithm      594 Chebyshev polynomials and convergence of Lanczos process      450 chol      39 Cholesky decomposition      34 Cholesky decomposition, downdating      253 255 257 Cholesky decomposition, updating      251 Cholesky's method, backward stability      164 Cholesky's method, block      45 Cholesky's method, bordered form      43 Cholesky's method, inner-product form      36 Cholesky's method, outer-product form      42 choline      581 Cofactor method      103 Column scaling      171 Column space      242 Companion matrix      309 Complete pivoting      104 Componentwise backward stability      179 Computer      148 cond      122 condeig      468 condest      132 Condition number      121 Condition number of nonsquare matrix      232 Condition number of simple eigenvalue      466 Condition number of simple eigenvector      471 Condition number of square matrix      121 Condition number, control of      284 Condition number, estimation of      131 Condition number, geometric view      123 128 Condition number, Skeel      177 Conductance      50 Conformable partition      9 Congruence      483 Congruence of matrix pairs      509 conjugate      208 Conjugate (vectors)      583 Conjugate transpose      49 Conjugate-gradient (CG) method      576 Conjugate-gradient (CG) method and Lanczos process      588 590 597 Conjugate-gradient (CG) method as acceleration technique      550 Conjugate-gradient (CG) method, convergence of      590 Conjugate-gradient (CG) method, derivation      581 Conjugate-gradient (CG) method, preconditioned      578 Contraction number      315 Convection-diffusion operator      107 109 Convergence rate, asymptotic ()      548 Convergence rate, of classical methods      549 Convergence ratio      315 Convergence, cubic      328 Convergence, linear      315 Convergence, quadratic      328 Convergence, slower than linear      315 cosh      450 cputime      5 Cramer's rule      105 Crout reduction      81 Cubic convergence      328 Cuppen's algorithm      491 Cuppen's algorithm, secular equation      494 Curse of dimensionality      529 Cuthill — McKee ordering, reverse      64 Damped iteration      553 Damped Jacobi method      544 Damped pendulum      299 Davidson's method      458 Decomposition,       84 479 Decomposition,       85 Decomposition, Cholesky      34 Decomposition, LDV      84 Decomposition, LR      482 Decomposition, LU      80 Decomposition, QR      190 Decomposition, Schur      337 Decomposition, singular value      262 Decomposition, spectral      339 Decomposition, twisted      489 Defective matrix      308 Deflation in the QR algorithm      361 delsq      68 demo      5 Departure from normality      468 Descent method      560 Descent method, CG      576 Descent method, Gauss — Seidel      561 Descent method, preconditioned CG      578 Descent method, SOR      562 Descent method, steepest      562 DET      104 Determinant, computation of      104 diag      220 Diagonal matrix      84 Diagonalization theorem      335 diary      5 Differential equations      289 502 Differential equations, asymptotically stable      298 Differential equations, homogeneous linear      291 Differential equations, nonhomogeneous linear      294 Differential equations, nonlinear      298 Differential equations, unstable      298 Dimension (of a subspace)      224 Direct sum (of subspaces)      241 Direct vs. iterative methods      310 Distance to nearest rank-deficient matrix      270 Distance to nearest singular matrix      274 Divide-and-conquer algorithm      491 Domain decomposition      549 Dominant eigenvalue      314 Doolittle reduction      81 Downdating QR, deleting a column      254 Downdating QR, deleting a row      252 Dqds algorithm      487 Duality in subspace iteration      428 EIG      300 504 Eigenpair      305 Eigenspace      414 Eigenvalue      305 505 Eigenvalue, condition number      466 Eigenvalues of normal matrices      345 Eigenvector      305 505 Eigenvector, condition number      471 Eigenvector, left      331 465 Eigenvector, right      465 eigs      452 Elementary row operation      71 Elimination, Gauss — Jordan      105 Elimination, Gaussian      71 elmat      70 Energy inner product and norm      582 Envelope of a matrix      57 EPS      169 Equivalence of linear systems      70 Equivalence of matrices      262 Equivalence of matrix pairs (pencils)      507 Equivalence, orthogonal      262 401 Error analysis, a priori vs. a posteriori      137 Error analysis, backward vs. forward      145 Error analysis, componentwise      175 Error, absolute vs. relative      141 Euclidean norm      112 Exact shift      323 367 Exceptional shift      367 eye      170 Fast Poisson solver      529 Fast rotators      194 Fill-in      63 Finite difference method      18 522 524 Five-point stencil      525 Floating-point, arithmetic      139 Floating-point, IEEE standard      140 FLOP      3 FORMAT      153 format, long      153 format, long e      500 format, short e      272 Fortran-90      101 Forward substitution      24 Forward substitution, backward stability      159 Forward substitution, block      30 Forward substitution, column oriented      27 Forward substitution, row oriented      24 Frobenius norm      115 full      322 Full rank      213 Fundamental subspaces      241 264 Galerkin condition      447 598 Galerkin condition, Petrov      599 gallery      70 Gauss transform      89 210 Gauss — Jordan elimination      105 Gauss — Seidel method      533 Gauss — Seidel method, as a descent method      561 Gauss — Seidel method, as smoother for multigrid      544 Gauss — Seidel method, block      541 Gauss — Seidel method, convergence rate      549 Gauss — Seidel method, line      542 Gauss — Seidel method, orderings      536 Gauss — Seidel method, red-black      537 Gauss — Seidel method, symmetric      538 Gaussian elimination, backward stability      162 Gaussian elimination, block      90 102 Gaussian elimination, column oriented      100 Gaussian elimination, generic      91 Gaussian elimination, inner-product formulation      81 Gaussian elimination, row oriented      100 Gaussian elimination, sparse      106 Gaussian elimination, with pivoting      93 Gaussian elimination, with small pivot      154 Gaussian elimination, without pivoting      71 Generalized eigenvalue problem      502 505 Generalized inverse, Moore — Penrose      277 Geometric multiplicity      342 417 Gerschgorin disk      473 Givens rotator      192 gmres      597 Gradient      560 Gram — Schmidt process, and QR decomposition      227 Gram — Schmidt process, classical      226—227 Gram — Schmidt process, modified      229 Gram — Schmidt process, reorthogonalization      233 Harmonic Ritz value      461 help      5 Hermitian matrix      49 337 hess      355 Hessenberg matrix      329 349 Hessenberg matrix, j-Hessenberg      460 Hessenberg matrix; 2-Hessenberg      379 Hessenberg matrix; proper (unreduced)      365 Hessenberg-triangular form      510 hilb      123 Hilbert matrix      123 150 248 Householder transformation      196 Hyperbolic transformation      256 Hyperplane      196 Ideal computer      142 Idempotent      195 Identity matrix      12 IEEE floating-point standard      140 Ill-conditioned matrix      121 Implicit-Q theorem      381 408 514 Implicitly restarted Amoldi (IRA)      451 Inductance      290 Induction, mathematical      31 Inertia      483 inf      148 Inner product      186 581—582 Inner product, complex      206 Inner product, energy      582 Inner product, induced by H      582 Inner product, of functions      247 INV      59 104 Invariant subspace      414 Inverse matrix      12 Inverse matrix, computation of      103 Inverse matrix, of a sparse matrix      59 Inverse matrix, pseudoinverse      277 Inverse power method (inverse iteration)      319 Invertible matrix      13 IRA      451 isieee      148 Isometric matrix (isometry)      221 issparse      67 Iterative refinement      178 Iterative vs. direct methods      310 Jacobi rotator      477 495
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