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| Saad Y. — Iterative Methods for Sparse Linear Systems |
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| Предметный указатель |
Additive projection procedure 136
ADI 116
ADI, Peaceman — Rachford algorithm 117
Adjacency graph 71
Adjacency graph of PDE matrices 71
Adjoint of a matrix 7
Algebraic multiplicity 15
Alternating Direction Implicit see “ADI”
Angle between a vector and a subspace 130
Anisotropic medium 47
Approximate inverse preconditioners 297
Approximate inverse preconditioners for improving a preconditioner 308
Approximate inverse preconditioners, column-oriented 300
Approximate inverse preconditioners, global iteration 298
Approximate inverse techniques 375
Arnoldi’s method 146—157
Arnoldi’s method for linear systems 151
Arnoldi’s method with Householder orthogonalization 149
Arnoldi’s method with modified Gram — Schmidt 148
Arnoldi’s method, basic algorithm 146
Arnoldi’s method, breakdown of 148
Arnoldi’s method, lucky breakdown 148
Arnoldi’s method, practical implementation 148
Arrow — Hurwicz’s algorithm 241
Assembled matrix 60
Assembly process 59
Banded matrices 5
Bandwidth of a bus 327
Bandwidth of a matrix 5
Basis of a subspace 10
BCG 209—213
BCG, algorithm 210
BCG, transpose-free variants 213—226
BICGSTAB 216
Biconjugate Gradient see “BCG”
Bidiagonal matrices 5
Bilinear form 56
Biorthogonal bases 35
Biorthogonal vectors 35 205
Biorthogonalization 204
Bipartite graph 82 112
Block Arnoldi, algorithm 196
Block Arnoldi, Ruhe’s variant 197
Block diagonal matrices 5
Block FOM 199
Block Gaussian elimination 385—388
Block Gaussian elimination, algorithm 388
Block GMRES 199—200
Block GMRES, multiple right-hand sides 199
Block Gram — Schmidt 197
Block Jacobi 102
Block Jacobi as a preconditioner 353
Block Krylov subspace methods 144 196–200
Block preconditioners 309
Block relaxation 98
Block tridiagonal matrices 5 309
Block tridiagonal matrices, preconditioning 309
boundary conditions 45 46
Boundary conditions, Dirichlet 46
Boundary conditions, mixed 46
Boundary conditions, Neumann 46
Cache memory 327
Canonical form 15
Canonical form, Jordan 16
Canonical form, Schur 17
Cauchy — Schwartz inequality 6 8
Cayley — Hamilton theorem 144
Cell-centered scheme 64
Cell-vertex scheme 64
Centered difference approximation 48
Centered difference formula 48
Centerpoint 415
CG algorithm see “Conjugate gradient algorithm”
CG for normal equations 236 237
CGNE 237
CGNE, algorithm 238
CGNE, optimality 238
CGNR 236
CGNR, algorithm 236
CGNR, optimality 236
cgs 214—216
CGS, algorithm 216
Characteristic polynomial 3
Chebyshev polynomials 186—192 194 356—364
Chebyshev polynomials and ellipses 188
Chebyshev polynomials for preconditioning 356
Chebyshev polynomials, complex 188 203
Chebyshev polynomials, optimality 189—191
Chebyshev polynomials, real 187
Chebyshev, acceleration 358
Cimmino’s method 233
Circuit switching 328
Coarse-grain 353
Coefficient matrix 95
Coloring vertices 81
Column reordering 74
Compressed Sparse Column storage see “CSC”
Compressed Sparse Row storage see “CSR”
Concus, Golub and Widlund algorithm 260
Condition number 40
Condition number for normal equation systems 230
Condition numbers and CG 180
Conjugate gradient algorithm 174—181
Conjugate gradient algorithm for the normal equations 236
Conjugate gradient algorithm, algorithm 178
Conjugate gradient algorithm, alternative formulations 178
Conjugate gradient algorithm, convergence 191 192
Conjugate gradient algorithm, derivation 174 177
Conjugate gradient algorithm, eigenvalue estimates 180
Conjugate gradient algorithm, preconditioned 244
Conjugate gradient squared see “CGS”
Conjugate Residual algorithm 181
Consistent matrix norms 8
Consistent orderings 112—116
Control volume 63
Convection-diffusion equation 47
Convergence of GMRES 193
Convergence of relaxation methods 104
Convergence of Schwarz procedures 402
Convergence of the Minimal Residual method 135
Convergence, factor 105
Convergence, factor, general 105
Convergence, factor, specific 105
Convergence, rate 105
COO storage scheme 84
Coordinate storage format see “COO”
Courant characterization 26
Craig’s method 238
CRAY T3D 329
CSC storage format 85
CSC storage format, matvecs in 335
CSR storage format 85 272
CSR storage format, matvecs in 335
Cut-edges 416
Cuthill — McKee ordering 77
Data coherence 327
Data-parallel 326
Defective eigenvalue 15
Derogatory 15
Determinant 3
DIA storage format 85 338
DIA storage format, matvecs in 338
Diagonal compensation 285
Diagonal dominance 108 109
Diagonal form of matrices 16
Diagonal matrices 5
Diagonal storage format see “DIA”
Diagonalizable matrix 16
Diagonally dominant matrix 109
Diagonally structured matrices 85
| Diameter of a graph 417
Diameter of a triangle 58
DIOM 154—157 175
DIOM, algorithm 156
Direct IOM see “DIOM”
Direct sum of subspaces 10 33
Directed graph 71
Dirichlet boundary conditions 45 46
Distributed, computing 325
Distributed, ILU 372
Distributed, memory 328
Distributed, sparse matrices 341 373
Divergence of a vector 46
Divergence operator 46
Domain decomposition and direct solution 387
Domain decomposition, convergence 402
Domain decomposition, full matrix methods 411
Domain decomposition, induced preconditioners 407
Domain decomposition, Schur complement approaches 406
Domain decomposition, Schwarz alternating procedure 394
Domain sweep 396
Double orthogonalization 148
Double-striping 418
DQGMRES 168—172 258
DQGMRES, algorithm 169
EBE preconditioner 376
EBE regularization 377
Edge in a graph 71
Eigenspace 10
Eigenvalues 3
Eigenvalues from CG iteration 180
Eigenvalues of an orthogonal projector 37
Eigenvalues, definition 3
Eigenvalues, index 16 17
Eigenvector 3
Eigenvector, left 4
Eigenvector, right 4
Eisenstat’s implementation 248 263
Eisenstat’s trick see “Eisenstat’s implementation”
Element-By-Element preconditioner see “EBE preconditioner”
ELL storage format 86
ELL storage format, matvecs in 339
Elliptic operators 44
Ellpack-Itpack storage format see “ELL storage format”
Energy norm 32 236 238
Error projection methods 129
Euclidean inner product 6
Euclidean norm 7
Faber — Manteuffel theorem 184
Factored approximate inverse 306
Fast solvers 47 383
FGMRES 255—258
FGMRES, algorithm 256
Fictitious domain methods 387
Fiedler vector 416
Field of values 23
Fill-in elements 275
Fine-grain algorithms 353
Finite difference scheme 47
Finite difference scheme for 1-D problems 50
Finite difference scheme for 2-D problems 54
Finite difference scheme for the Laplacean 49
Finite difference scheme, upwind schemes 51
Finite element method 44 55
Finite volume method 63
Flexible GMRES see “FGMRES”
Flexible iteration 255
Flux vector 63
FOM 151
FOM with restarting 153
FOM, algorithm 152
Frobenius norm 8
Frontal methods 60 376
Full matrix methods 411–413
Full Orthogonalization Method see “FOM”
Galerkin conditions 124
Gastinel’s method 139
Gather operation 336
Gauss — Seidel iteration 95
Gauss — Seidel iteration for normal equations 231
Gauss — Seidel iteration for normal equations in parallel 378
Gauss — Seidel iteration, backward 97
Gauss — Seidel iteration, symmetric 97
Gaussian elimination 60 176 269—273 278 282 283 285—287 368 369 383
Gaussian elimination in IOM and DIOM 156
Gaussian elimination in Lanczos process 176
Gaussian elimination in skyline format 295
Gaussian elimination, block 385
Gaussian elimination, frontal methods 376
Gaussian elimination, IKJ variant 271
Gaussian elimination, parallel 409
Gaussian elimination, parallelism in 71
Gaussian elimination, reordering in 75
Gaussian elimination, sparse 70
GCR 182—184
Generalized Conjugate Residual see “GCR”
Geometric multiplicity 15
Gershgorin discs 110
Gershgorin’s theorem 109
Global iteration 298—300 305
Global reduction operations 332
gmres 157—172 184 193—196
GMRES with polynomial preconditioning 363
GMRES with restarting 167
GMRES, algorithm 158
GMRES, block algorithm 199
GMRES, breakdown 163 164
GMRES, convergence 193
GMRES, flexible variant 250 255—258
GMRES, Householder version 158
GMRES, lucky breakdown 164
GMRES, parallel implementation 331
GMRES, practical implementation 160
GMRES, relation with FOM 164 166
GMRES, stagnation 167
GMRES, truncated 168
Grade of a vector 144
Gram — Schmidt algorithm 11—12 314
Gram — Schmidt algorithm, block 197
Gram — Schmidt algorithm, cancellations in 148
Gram — Schmidt algorithm, modified 11
Gram — Schmidt algorithm, standard 11
Graph 71
Graph, bipartite 82
Graph, coloring 81
Graph, directed 71
Graph, edges 71
Graph, Laplacean of a 416
Graph, partitioning 382 413
Graph, partitioning, geometric 414
Graph, partitioning, graph theory techniques 417
Graph, partitioning, spectral techniques 416
Graph, partitioning, type 384
Graph, undirected 71
Graph, vertices 71
Hankel matrix 208
Harmonic functions 46
Harwell — Boeing collection 89 90
Hausdorff’s convex hull theorem 23
Heap-sort, in ILUT 291
Hermitian inner product 6
Hermitian matrices 4 24
Hermitian positive definite 31
Hessenberg matrices 5
Holder norms 7
Householder algorithm 12
Householder orthogonalization in Arnoldi’s method 149
Householder reflectors 12
HPD see “Hermitian positive definite”
Hypercube 329
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