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Watkins D. — Fundamentals of matrix computations
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: en

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
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Предметный указатель
$LDL^T$ decomposition      84
$MD^{-1}M^T$ 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 ($R_\infty(G)$)      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, $LDL^T$      84 479
Decomposition, $MD^{-1}^M^T$      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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