Saad Y. — Numerical Methods for Large Eigenvalue Problems :: Электронная библиотека попечительского совета мехмата МГУ

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Saad Y. — Numerical Methods for Large Eigenvalue Problems

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Название: Numerical Methods for Large Eigenvalue Problems

Автор: Saad Y.

Аннотация:

A detailed view of the numerical methods used to solve large matrix eigenvalue problems that arise in various engineering and scientific applications. The emphasis is on the more difficult nonsymmetric problems, but much of the important material for symmetric problems is also covered. The text contains a solid theoretical section, and also describes some of the important techniques developed in recent years together with a few computer programs. Co-published with Manchester U. Press. Annotation copyright Book News, Inc. Portland, Or.

Язык:

Рубрика: Математика/Численные методы/Численная линейная алгебра/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1992

Количество страниц: 358

Добавлена в каталог: 25.02.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

a-posteriori error bounds      76
Addition of matrices      3
Algebraic multiplicity      14
Angle between a vector and a subspace      62 130
Angle between vectors      62
Approximate problem      127 170
ARNINV      263
ARNIT      271
ARNLS      270
Arnoldi — Chebyshev iteration      226
Arnoldi’s method      172 263
Arnoldi’s method as a purification process      226
Arnoldi’s method with implicit deflation      179
Arnoldi’s method with modified Gram — Schmidt      176
Arnoldi’s method, breakdown of      174
Arnoldi’s method, convergence      204—213
Arnoldi’s method, iterative version      179
Arnoldi’s method, practical implementations      176
Banded matrices      6
Bandwidth of a matrix      6
Basis of a subspace      11
Bauer — Fike theorem      77
Best uniform approximation in C      205
Bidiagonal matrices      6
Bifurcation      315
Bifurcation analysis      315
Bifurcation, Hopf      316
Bifurcation, real bifurcation point      315
Bifurcation, turning point      315
Biorthogonal vectors      64 188
Block Arnoldi algorithm      196
Block Arnoldi algorithm, Ruhe’s variant      197
Block diagonal matrices      7
Block Gram — Schmidt      197
Block Krylov Methods      168 195
Block Lanczos      304
Block-tridiagonal matrices      7
Breakdown in the Lanczos algorithm      192—194
Breakdown in the Lanczos algorithm, incurable      193
Breakdown in the Lanczos algorithm, serious      193
Breakdown in the Lanczos algorithm, ‘lucky’      192
Brusselator model      317
Cancellations      177
Canonical forms of matrices      14—25
Canonical forms of matrices, diagonal      15
Canonical forms of matrices, Jordan      15
Canonical forms of matrices, Schur      23
Canonical forms of matrices, triangular      16
Cauchy — Schwartz inequality      8
Characteristic polynomial      4 170
Characteristic polynomial in Krylov methods      170
Chebsyshev — Subspace iteration      237
Chebyshev bases      243
Chebyshev iteration      220
Chebyshev iteration with Arnodi’s method      226
Chebyshev iteration, algorithm      223
Chebyshev iteration, basic scheme      220
Chebyshev iteration, convergence      224
Chebyshev iteration, convergence ratio      224
Chebyshev iteration, damping coefficients      224
Chebyshev iteration, optimal ellipse in      228
Chebyshev polynomials      141—148
Chebyshev polynomials, asymptotic optimality      148
Chebyshev polynomials, complex      143—144
Chebyshev polynomials, optimality      146
Chebyshev polynomials, real      142—143
Chebyshev polynomials, relation with ellipses      144
Chemical reaction example      50 267
Chemical reactions      316
Condition number      93
Condition number of an eigenvalue      93
Condition number of an eigenvector      96
Condition number of an invariant subspace      100
Configuration interaction method      313
Conjugate gradient method      47
Consistent matrix norms      9
Coordinate storage scheme      40
Courant characterization      32 132
Cramer’s Rule      66
Critical points      206
CSC storage format      42
CSR storage format      42
Damping      305—307
Davidson’s method      272—276 313
Davidson’s method, convergence      275
Defective eigenvalue      15
Deflated Arnoldi — Chebyshev algorithm      236
Deflation techniques      117 235 292
Deflation techniques with several vectors      122
Derogatory      15
Determinant      3
Diagonal form of matrices      16
Diagonal matrices      6
Diagonal storage format      43
Diagonalizable matrices      16
Direct sum of subspaces      11 60
Distances between subspaces      63
Double orthogonalization      177
Double shift approach      261
Dunford integral      67
Dynamical systems      313
Dynamical systems, locally stable solutions      314
Eigenspace      12
Eigenvalue      3
Eigenvalue, averages, analyticity      73
Eigenvalue, branches      74
Eigenvalue, index      17
Eigenvalue, pair      284
Eigenvector      4
Eigenvector, left      5
Eigenvector, right      5
Electrical networks      311
Ellipses for Chebyshev iteration      222
Ellpack — Itpack storage format      43
Enhanced initial vector      227
Equivalent pencils      286
Error bounds      76
Essential convergence      153
Essential singularities      66
Exponential propagation operator      277
Field of values      28
First resolvent equality      67
Frobenius norm      9
Galerkin condition      127
Galerkin process      268
Gap between subspaces      63
Generalized Arnoldi’s method      276
Generalized eigenvalue      283—284
Generalized eigenvalue problem      258 282 300
Generalized eigenvector      17
Geometric multiplicity      15
Gerschgorin discs      103
Gerschgorin’s theorem      102
Grade of a vector      169 226
Gram matrices      244
Gram — Schmidt procedure      12
Haar conditions      205 207
HARWELL library      47
Harwell — Boeing collection      47 52
Hausdorff’s convex hull theorem      28
Hermitian definite matrix pairs      295
Hermitian matrices      5 29
Hessenberg matrices      6
Holder norms      8
Hopf bifurcation      316
Hotelling’s deflation      119
Householder orthogonalization      177
Idempotent      11
Implicit deflation      179
Indefinite inner product      193

Index of an eigenvalue      17
Indirect addressing      40
Instability, in power systems      312
Invariant subspace      11 128
Inverse iteration      114
Inverse power method      114
Iterative Arnoldi method      179
Iterative Arnoldi method, example      271
Jacobian matrix      314
Jordan block      18
Jordan box      19
Jordan canonical form      17
Jordan curve      67
Jordan submatrix      19
Joukowski mapping      144
Kahan, Jiang, Parlett theorem      86—87
Kahan, Parlett, Jiang error bound      79
Kato — Temple’s theorem      81
Kernel      11
Kernel polynomials      248
Krylov Subspace Methods      168—217
Krylov Subspace Methods, characteristic property      171
Krylov subspaces      168
Lanczos algorithm      183—184 198 296
Lanczos algorithm and orthogonal polynomials      185
Lanczos algorithm for matrix pairs      297
Lanczos algorithm, breakdown      191
Lanczos algorithm, convergence      198—204.
Lanczos algorithm, Hermitian case      183
Lanczos algorithm, incurable breakdown      193
Lanczos algorithm, look-ahead version      192
Lanczos algorithm, loss of orthogonality      185
Lanczos algorithm, modified Gram — Schmidt version      184
Lanczos algorithm, non-Hermitian case      186
Lanczos algorithm, partial reorthogonalization      185
Lanczos algorithm, practical implementation      192
Lanczos algorithm, selective reorthogonalization      185
Lanczos algorithm, serious breakdown in      193
Least squares Arnoldi algorithm      239 251
Least squares polynomials      240
Least squares polynomials, Gram matrices      244
Least squares preconditioning      268
Left eigenvector      5 286
Left subspace      126 138
Leontiev’s model      318
Linear mappings      3
Linear perturbations of a matrix      71
Linear shifts for matrix pairs      162 293
Linear span      11
Linear stability      314
Localization of eigenvalues      101
Locking technique      160
Locking vectors      160
Look-ahead Lanczos algorithm      192—194
Lower triangular matrices      6
Lucky breakdown      192
MA28 package      47
Macro-economics      318
Markov chain models      319
Matrices      3
Matrix exponentials      277
Matrix pair      283
Matrix pencil      260 283
Matrix reduction      14
Mechanical vibrations      305
Min-max problem      221
Min-max theorem      30
Modified Gram — Schmidt      176
Moment matrix      243
Moment matrix in Lanczos procedure      193
Moment matrix in least squares approach      244
MSR storage format      42
Multiple eigenvalue      15
Multiplication of matrices      3
NASTRAN      304
Neuman series expansion      66
Newton’s law of motion      306
Nilpotent matrix      21—22
Nonnegative matrices      5 33
Normal matrices      5 26
Norms of matrices      9
Null space      11 60—61 292
Oblique projection method      138 186
Oblique projector      63 139
Optimal ellipse      228
Optimal polynomial      246
Orthogonal complement      14 60—61
Orthogonal matrix      6
Orthogonal projection methods      127
Orthogonal projector      14 60 129
Orthogonality      12
Orthogonality between vectors      12
Orthogonality of a vector to a subspace      14
Orthogonalization      12
Orthonormal      12
Oscillatory solutions      311
Outer product matrices      6
Partial reorthogonalization      195
Partial Schur decomposition      24 123
Permutation matrices      7
Perron — Frobenius theorem      319 321
Petrov — Galerkin condition      138
Petrov — Galerkin method      126
Polynomial acceleration      220
Polynomial iteration      220
Polynomial preconditioning      267
Positive definite matrix      32
Positive real matrices      47
Positive semi-definite      32
Power method      110 152 162 178
Power method, convergence      112
Power method, example      112
Power systems      312
Preconditioning      163 257 272
Principal vector      17
Projection method      126 170
Projection method for matrix pairs      294
Projection method, Hermitian case      131
Projection method, oblique      126
Projection method, orthogonal      126
Projection operators      129
Projector      11 60
QR decomposition      13
Quadratic eigenvalue problem      282 299
Quantum chemistry      312
Quasi-Schur form      24 124
Random walk example      48
RANGE      11
Rank      11
Rayleigh quotient      28 30
Rayleigh quotient iteration      116
Rayleigh — Ritz procedure      128
Real Chebyshev polynomials      142
Real Schur form      24
Reduced resolvent      95
Reducible      33
Reduction of matrices      14
Regular matrix pair      285
Residual norm      176
Resolvent      66
Resolvent equalities      67
Resolvent operator      66
Resonance phenomena      308
Right eigenvector      286
Right subspace      126 138
Ritz eigenvalues      175
Ritz values      175 187
RQI (Rayleigh Quotient Iteration)      116
Schrodinger’s equation      313
Schur form      23—25

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