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Stewart G.W. — Matrix algorithms. Volume 2: Eigensystems
Stewart G.W. — Matrix algorithms. Volume 2: Eigensystems

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Название: Matrix algorithms. Volume 2: Eigensystems

Автор: Stewart G.W.


This book is the second volume in a projected five-volume survey of numerical linear algebra and matrix algorithms. This volume treats the numerical solution of dense and large-scale eigenvalue problems with an emphasis on algorithms and the theoretical background required to understand them. Stressing depth over breadth, Professor Stewart treats the derivation and implementation of the more important algorithms in detail. The notes and references sections contain pointers to other methods along with historical comments.

The book is divided into two parts: dense eigenproblems and large eigenproblems. The first part gives a full treatment of the widely used QR algorithm, which is then applied to the solution of generalized eigenproblems and the computation of the singular value decomposition. The second part treats Krylov sequence methods such as the Lanczos and Arnoldi algorithms and presents a new treatment of the Jacobi-Davidson method.

The volumes in this survey are not intended to be encyclopedic. By treating carefully selected topics in depth, each volume gives the reader the theoretical and practical background to read the research literature and implement or modify new algorithms. The algorithms treated are illustrated by pseudocode that has been tested in MATLAB implementations.

Язык: en

Рубрика: Computer science/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1

Год издания: 2001

Количество страниц: 469

Добавлена в каталог: 23.10.2010

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Предметный указатель
$A\leq B$, A < B, etc. (componentwise comparison)      423
$A^{-1}$ (inverse of A)      422
$A^{-H}$ (conjugate transpose inverse of A)      422
$A^{-T}$ (transpose inverse of A)      422
$A^{H}$ (conjugate transpose of A)      422
$A^{T}$ (transpose of A)      422
$diag(\delta_{1},..., \delta_{n})$ (diagonal matrix)      423
$dim(\mathcal{X})$ (dimension of $\mathcal{X}$)      424
$e_{i}$ (ith unit vector)      423
$K_{k}$ (A,u) (Krylov matrix)      267
$span(\mathcal{X})$ (the space spanned by $\mathcal{X}$)      424
$\infty$-norm      see “Norm”
$\Lambda(A)$ (the spectrum of A)      2
$\mathbb{C}$ (complex numbers)      421
$\mathbb{C}^{m\times n}$ (space of complex $m\times n$ matrices)      421
$\mathbb{C}^{n}$ (complex n-space)      421
$\mathbb{R}$ (real numbers)      421
$\mathbb{R}^{m\times n}$ (space of real $m\times n$ matrices)      421
$\mathbb{R}^{n}$ (real n-space)      421
$\mathcal{K}_{k}$ (A,u) (Krylov subspace)      267
$\mathcal{N}(X)$ (null space of X)      424
$\mathcal{R}(X)$ (column space of X)      424
$\overline{A}$ (conjugate of A)      422
$\sigma_{i}(X)$ (the ith singular value of X)      204
$\Theta(\mathcal{X},\mathcal{Y})$ (canonical angle matrix)      248
$\varepsilon_{M}$ (rounding unit)      27
1-norm      see “Norm”
2-norm      see “Norm”
Abel, N.H.      55
Absolute norm      see “Norm”
Accumulation of transformations      76
Aitken, A.C.      69 110
Algebraic multiplicity      see “Eigenvalue”
angle      see “Canonical angle”
Approximate Newton method      396
Approximate Newton method and inexact Newton method      419
Approximate Newton method, analysis      400—402
Approximate Newton method, constant shift approximation      403
Approximate Newton method, correction equation      397 420
Approximate Newton method, correction equation, inexact solution      403—404
Approximate Newton method, correction equation, solution      398
Approximate Newton method, correction formula      397
Approximate Newton method, derivation      396—397
Approximate Newton method, diagonally dominant A      403
Approximate Newton method, drawbacks      404—405
Approximate Newton method, equivalence of correction formula and equation      398
Approximate Newton method, error recurrence      402
Approximate Newton method, inner and outer iterations      403
Approximate Newton method, local convergence      402
Approximate Newton method, local convergence, convergence rates      402
Approximate Newton method, natural approximation      403
Approximate Newton method, orthogonal correction      399
Approximate Newton method, orthogonal correction, nomenclature      399
Arnoldi decomposition      297 299 315
Arnoldi decomposition, alternate form      300
Arnoldi decomposition, Arnoldi factorization      315
Arnoldi decomposition, computation      see “Arnoldi method”
Arnoldi decomposition, computation of residuals      336
Arnoldi decomposition, hat convention      300
Arnoldi decomposition, QR factorization of Krylov sequence      298
Arnoldi decomposition, Rayleigh quotient      301
Arnoldi decomposition, reduced      299 302
Arnoldi decomposition, termination of Krylov sequence      300
Arnoldi decomposition, uniqueness      300
Arnoldi decomposition, uniqueness of starting vector      301
Arnoldi factorization      see “Arnoldi decomposition”
Arnoldi method      117 344
Arnoldi method, Arnoldi process      302 303—304
Arnoldi method, block      344
Arnoldi method, comparison of implicit and Krylov — Schur restarting      332
Arnoldi method, computation of residuals      336
Arnoldi method, convergence criteria      334—336 347
Arnoldi method, convergence criteria, choice of tolerance      341—342
Arnoldi method, convergence criteria, shift-and-invert enhancement      336
Arnoldi method, deflation      337—340 347
Arnoldi method, deflation, advantages      337
Arnoldi method, deflation, in Arnoldi decomposition      340
Arnoldi method, deflation, in Krylov — Schur method      340
Arnoldi method, deflation, nonorthogonal bases      339—340
Arnoldi method, deflation, residual norm      337—338
Arnoldi method, deflation, stability      338—339
Arnoldi method, equivalence of implicit and Krylov — Schur restarting      331
Arnoldi method, filter polynomial      317 345
Arnoldi method, implicit restarting      318 345
Arnoldi method, implicit restarting, contraction phase      318—322
Arnoldi method, implicit restarting, double shift method      320
Arnoldi method, implicit restarting, exact (Rayleigh quotient) shifts      324—325
Arnoldi method, implicit restarting, example      323
Arnoldi method, implicit restarting, forward instability      325 346
Arnoldi method, implicit restarting, operation count      320—323
Arnoldi method, implicit restarting, overview      318
Arnoldi method, implicit restarting, the cycle      322
Arnoldi method, implicit restarting, truncation index      320
Arnoldi method, Krylov — Schur restarting      325—326 346
Arnoldi method, Krylov — Schur restarting, exchanging eigenblocks      326—328 me
Arnoldi method, Krylov — Schur restarting, operation count      329
Arnoldi method, Krylov — Schur restarting, the cycle      328—329
Arnoldi method, loss of orthogonality      117
Arnoldi method, operation count      304
Arnoldi method, reorthogonalization      303
Arnoldi method, restarted      316 344—345 implicit “Arnoldi Krylov
Arnoldi method, restarted, rationale      316
Arnoldi method, shift-and-invert enhancement      305
Arnoldi method, shift-and-invert enhancement, convergence criteria      336
Arnoldi method, shift-and-invert enhancement, stability      334 347
Arnoldi method, shift-and-invert enhancement, stagnation of Krylov subspace      305—306 316
Arnoldi method, stability      332—333 346—347
Arnoldi method, stability, shift-and-invert enhancement      334 347
Arnoldi method, storage requirements      304—305 316
Arnoldi, W.E.      315 344
ARPACK      325 345
ARPACK, convergence criteria      341
Arrowhead matrix      201
B inner product      371
B inner product, Cauchy inequality      371
B inner product, norm      371
B inner product, orthogonal matrix      372
B inner product, orthogonality      230 372
B inner product, orthogonalization      372—373
B inner product, orthogonalization, economizing B products      372—373
B inner product, orthogonalization, maintaining orthogonality      373
B inner product, orthonormal matrix      372
B inner product, symmetry      371
B-Arnoldi method      373
B-Arnoldi method, deflation      374
B-Arnoldi method, orthogonalization      373
B-Arnoldi method, residual norms      373—374
B-Arnoldi method, restarting      374
B-Arnoldi method, Ritz pairs      373
B-Lanczos method      374—375
B-Lanczos method, periodic reorthogonalization      375
B-Lanczos method, restarted      375
Backward error and unitary similarities      10
Backward error as paradigm      70
Backward error from residual      61 70 196 253 265 334—335 369
Backward error in convergence testing      10
Backward error, convergence criteria      62
Backward error, inverse power method      69
Backward stability and complex eigenvalues of real matrices      126
Backward stability, Arnoldi method      332—333 346—347
Backward stability, band tridiagonalization      189
Backward stability, bidiagonal QR step      220
Backward stability, deflation in Arnoldi method      338—339
Backward stability, divide-and-conquer algorithm for the spectral decomposition      183
Backward stability, double shift QR algorithm      126
Backward stability, eigenvector computation      102
Backward stability, Hessenberg QR algorithm      94
Backward stability, Householder’s reduction to tridiagonal form      162 170
Backward stability, implicit tridiagonal QR step      167
Backward stability, inertia of a tridiagonal matrix      192—193
Backward stability, Lanczos algorithm with full orthogonalization      351
Backward stability, QZ algorithm      152
Backward stability, reduction to bidiagonal form      217
Backward stability, reduction to Hessenberg-triangular form      147
Backward stability, spectral decomposition updating      179
Backward stability, stability in the ususal sense      87
Backward stability, subspace iteration      388
Backward stability, Wilkinson’s contribution      111
Bai, Z.      23 129 346 395
Balancing      107—108 112
Balancing and grading      110
Balancing, matrix pencil      152—153 156
Balancing, operation count      108
Battels, R.H.      24
Bau, D.      23
Bauer, F.L.      394
Bellman, R      23
Beltrami, E.      226
Bhatia, R.      52
Bi-Lanczos algorithm      283 315 367
Bi-Lanczos algorithm, look-ahead recurrence      367
Bidiagonal matrix      215
Bidiagonal matrix, complex bidiagonal matrix to real      217
Bidiagonal matrix, reduction to      215—217 227
Bidiagonal matrix, reduction to, first column of the transformation      217
Bidiagonal matrix, reduction to, operation count      217
Bidiagonal matrix, reduction to, stability      217
Bidiagonal matrix, relative stability of singular values      217 223
Bidiagonal QR algorithm      217 227—228
Bidiagonal QR algorithm, combined with QR decomposition      226 228
Bidiagonal QR algorithm, combined with QR decomposition, operation count      226
Bidiagonal QR algorithm, combined with QR decomposition, pivoting      228
Bidiagonal QR algorithm, deflation      224
Bidiagonal QR algorithm, detecting negligible superdiagonal elements      223—224 227
Bidiagonal QR algorithm, graded matrices      224 227—228
Bidiagonal QR algorithm, QR step      219—220
Bidiagonal QR algorithm, QR step, operation count      220
Bidiagonal QR algorithm, QR step, stability      220
Bidiagonal QR algorithm, shift computation      222—223
Bidiagonal QR algorithm, zero shift      227
Biorthogonal bases      245
Bjoerck, A.      264
Block Krylov subspace      see “Krylov subspace”
Block triangularization of nearly block triangular matrix      255
Braman, K      129
Bunch, J.R.      201
Byers, R.      129
Canonical angle      248 264
Canonical angle of a combination of subspaces      250—251
Canonical angle of a vector and a subspace      250
Canonical angle, angles between right and left eigenspaces      251
Canonical angle, between two vectors      44
Canonical angle, between two vectors, computation      49
Canonical angle, characterization of largest angle      249
Canonical angle, computation      249 264
Canonical angle, subspaces of unequal dimensions      249—250
Cauchy, A.      23
Cayley, A.      23
Chan, T.F.      228
Chandrasekaran, S.      235
Characteristic equation      4 24
Characteristic polynomial      4 24
Characteristic polynomial and companion matrix      55
Characteristic polynomial and terminating Krylov sequences      279
Characteristic polynomial in Lanczos’s method      315
Characteristic polynomial, matrix of order two      4—5
Characteristic value      23
Chatelin, F.      23 279
Chebyshev polynomial      271 280
Chebyshev polynomial as filter polynomial      317
Chebyshev polynomial in complex plane      280
Chebyshev polynomial in subspace iteration      392
Cholesky algorithm      157
Cholesky decomposition      231 426
Cholesky decomposition, pivoted      234
Cholesky decomposition, updating      171
Chordal distance      138 155
Clint, M.      394
Column space (’1Z(X))      424
Column- and row-oriented algorithms      102 162
Companion matrix      55
Complete system of eigenvectors      2
Complete system of eigenvectors and diagonalization      8—9
Condition number      see “Condition”
Condition, condition number      48
Condition, condition number, limitations      48
Condition, eigenvalue      48 53
Condition, eigenvalue, condition number      48
Condition, eigenvalue, Hermitian matrix      42—43 51
Condition, eigenvector      48—51 53
Condition, eigenvector, condition number      50
Condition, eigenvector, Hermitian matrix      51—52
Condition, generalized eigenvalue      140
Condition, generalized eigenvector      143
Condition, ill conditioning      48
Condition, S/PD generalized eigenvalue      231
Condition, simple eigenblock      260
Condition, simple eigenspace      261
Condition, singular values      206
Condition, singular vector      209—210
Condition, singular vector, condition number      210
Condition, well conditioning      48
Congruence transformation      190
Conjugate gradient method      279
Consistent norm      see “Norm”
Convergence ratios      34
Convergence, normwise and componentwise      30
Crawford, C.R.      236
Cross-product algorithm for the S VD      210—211
Cross-product algorithm for the S VD, assessment      214 227
Cross-product algorithm for the S VD, attractive features      211
Cross-product algorithm for the S VD, inaccuracy in the left singular vectors      214
Cross-product algorithm for the S VD, inaccuracy in the right singular vectors      212—213
Cross-product algorithm for the S VD, inaccuracy in the singular values      211—212
Cross-product algorithm for the S VD, refined Ritz vector      291—292 296
Cross-product algorithm for the S VD, use by statisticians      227
Cross-product matrix      205
CS decomposition      236—237
CS decomposition, algorithms      237
Cullum, J.      280 367
Cuppen, J.J.M.      201
Curtis, A.R.      156
Datta, B.N.      23
Davidson, E.R.      419
Davidson’s algorithm      419
Davis, C.      53 265
Defectiveness      384
Defectiveness and Krylov subspaces      276
Defectiveness, dependence of eigenvectors      8
Defectiveness, dominant eigenvalues      34
Defectiveness, eigenvalue      7
Defectiveness, matrix      7 15
Defectiveness, sensitivity of eigenvalues      7—8 38
Definite generalized eigenvalue problem      235—236
Definite generalized eigenvalue problem, rotating to make B positive definite      236
Deflation      12 344 “Power “QR “Etc.”)
Deflation by inspection      106—107
Deflation by inspection, operation count      107
Deflation, complex eigenvector      114
Dembo, R.S.      419
Demmel, J.W.      23 129 227 346
Dennis, J.E.      265 418
Det(A) (determinant of A)      422
Dhillon, I.S.      202
Diagonalization, block      15 19—20
Diagonalization, block, assessment      25
Diagonalization, block, uniqueness      20
Diagonalization, complete system of eigenvectors      8—9
Diagonalization, distinct eigenvalues      20
Divide-and-conquer algorithm for the spectral decomposition      181—183 201
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