Iterative Methods for Sparse Linear Systems Second Edition

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x CONTENTS 11.5 Matrix-by-Vector Products . . . . . . . . . . . . . . . . . . . 378 11.5.1 The CSR and CSC Formats . . . . . . . . . . . . 378 11.5.2 Matvecs in the Diagonal Format . . . . . . . . . . 380 11.5.3 The Ellpack-Itpack Format . . . . . . . . . . . . 381 11.5.4 The Jagged Diagonal Format . . . . . . . . . . . 382 11.5.5 The Case of Distributed Sparse Matrices . . . . . 383 11.6 Standard Preconditioning Operations . . . . . . . . . . . . . . 386 11.6.1 Parallelism in Forward Sweeps . . . . . . . . . . 386 11.6.2 Level Scheduling: the Case of 5-Point Matrices . 387 11.6.3 Level Scheduling for Irregular Graphs . . . . . . 388 12 Parallel Preconditioners 393 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 12.2 Block-Jacobi Preconditioners . . . . . . . . . . . . . . . . . . 394 12.3 Polynomial Preconditioners . . . . . . . . . . . . . . . . . . . 395 12.3.1 Neumann Polynomials . . . . . . . . . . . . . . . 396 12.3.2 Chebyshev Polynomials . . . . . . . . . . . . . . 397 12.3.3 Least-Squares Polynomials . . . . . . . . . . . . 400 12.3.4 The Nonsymmetric Case . . . . . . . . . . . . . . 403 12.4 Multicoloring . . . . . . . . . . . . . . . . . . . . . . . . . . 406 12.4.1 Red-Black Ordering . . . . . . . . . . . . . . . . 406 12.4.2 Solution of Red-Black Systems . . . . . . . . . . 407 12.4.3 Multicoloring for General Sparse Matrices . . . . 408 12.5 Multi-Elimination ILU . . . . . . . . . . . . . . . . . . . . . . 409 12.5.1 Multi-Elimination . . . . . . . . . . . . . . . . . 410 12.5.2 ILUM . . . . . . . . . . . . . . . . . . . . . . . 412 12.6 Distributed ILU and SSOR . . . . . . . . . . . . . . . . . . . 414 12.7 Other Techniques . . . . . . . . . . . . . . . . . . . . . . . . 417 12.7.1 Approximate Inverses . . . . . . . . . . . . . . . 417 12.7.2 Element-by-Element Techniques . . . . . . . . . 417 12.7.3 Parallel Row Projection Preconditioners . . . . . 419 13 Multigrid Methods 423 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 13.2 Matrices and spectra of model problems . . . . . . . . . . . . 424 13.2.1 Richardson’s iteration . . . . . . . . . . . . . . . 428 13.2.2 Weighted Jacobi iteration . . . . . . . . . . . . . 431 13.2.3 Gauss-Seidel iteration . . . . . . . . . . . . . . . 432 13.3 Inter-grid operations . . . . . . . . . . . . . . . . . . . . . . . 436 13.3.1 Prolongation . . . . . . . . . . . . . . . . . . . . 436 13.3.2 Restriction . . . . . . . . . . . . . . . . . . . . . 438 13.4 Standard multigrid techniques . . . . . . . . . . . . . . . . . . 439 13.4.1 Coarse problems and smoothers . . . . . . . . . . 439 13.4.2 Two-grid cycles . . . . . . . . . . . . . . . . . . 441
13.4.3 V-cycles and W-cycles . . . . . . . . . . . . . . . 443 13.4.4 Full Multigrid . . . . . . . . . . . . . . . . . . . 447 13.5 Analysis for the two-grid cycle . . . . . . . . . . . . . . . . . 451 13.5.1 Two important subspaces . . . . . . . . . . . . . 451 13.5.2 Convergence analysis . . . . . . . . . . . . . . . 453 13.6 Algebraic Multigrid . . . . . . . . . . . . . . . . . . . . . . . 455 13.6.1 Smoothness in AMG . . . . . . . . . . . . . . . . 456 13.6.2 Interpolation in AMG . . . . . . . . . . . . . . . 457 13.6.3 Defining coarse spaces in AMG . . . . . . . . . . 460 13.6.4 AMG via Multilevel ILU . . . . . . . . . . . . . 461 13.7 Multigrid vs Krylov methods . . . . . . . . . . . . . . . . . . 464 14 Domain Decomposition Methods 469 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 14.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . 470 14.1.2 Types of Partitionings . . . . . . . . . . . . . . . 472 14.1.3 Types of Techniques . . . . . . . . . . . . . . . . 472 14.2 Direct Solution and the Schur Complement . . . . . . . . . . . 474 14.2.1 Block Gaussian Elimination . . . . . . . . . . . . 474 14.2.2 Properties of the Schur Complement . . . . . . . 475 14.2.3 Schur Complement for Vertex-Based Partitionings 476 14.2.4 Schur Complement for Finite-Element Partitionings 479 14.2.5 Schur Complement for the model problem . . . . 481 14.3 Schwarz Alternating Procedures . . . . . . . . . . . . . . . . . 484 14.3.1 Multiplicative Schwarz Procedure . . . . . . . . . 484 14.3.2 Multiplicative Schwarz Preconditioning . . . . . . 489 14.3.3 Additive Schwarz Procedure . . . . . . . . . . . . 491 14.3.4 Convergence . . . . . . . . . . . . . . . . . . . . 492 14.4 Schur Complement Approaches . . . . . . . . . . . . . . . . . 497 14.4.1 Induced Preconditioners . . . . . . . . . . . . . . 497 14.4.2 Probing . . . . . . . . . . . . . . . . . . . . . . . 500 14.4.3 Preconditioning Vertex-Based Schur Complements 500 14.5 Full Matrix Methods . . . . . . . . . . . . . . . . . . . . . . . 501 14.6 Graph Partitioning . . . . . . . . . . . . . . . . . . . . . . . . 504 14.6.1 Basic Definitions . . . . . . . . . . . . . . . . . . 504 14.6.2 Geometric Approach . . . . . . . . . . . . . . . . 505 14.6.3 Spectral Techniques . . . . . . . . . . . . . . . . 506 14.6.4 Graph Theory Techniques . . . . . . . . . . . . . 507 References 514 Index 535
Page 1: Iterative Methods for Sparse Linear
Page 4 and 5: vi CONTENTS 2.1.2 The Convection Di
Page 6 and 7: viii CONTENTS 6.11.4 Convergence of
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Page 12 and 13: xiv PREFACE underlying architecture
Page 14 and 15: xvi PREFACE The rest could be skipp
Page 16 and 17: xviii PREFACE
Page 18 and 19: xx PREFACE Apart from two recent vo
Page 21 and 22: Chapter 1 BACKGROUND IN LINEAR ALGE
Page 23 and 24: 1.2. SQUARE MATRICES AND EIGENVALUE
Page 25 and 26: 1.3. TYPES OF MATRICES 5 • Unitar
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1.13. BASIC CONCEPTS IN LINEAR SYST
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Chapter 2 DISCRETIZATION OF PDES Pa
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2.1. PARTIAL DIFFERENTIAL EQUATIONS
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2.2. FINITE DIFFERENCE METHODS 51 f
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2.2. FINITE DIFFERENCE METHODS 53 u
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2.2. FINITE DIFFERENCE METHODS 55 B
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2.2. FINITE DIFFERENCE METHODS 57
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2.2. FINITE DIFFERENCE METHODS 59 i
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2.2. FINITE DIFFERENCE METHODS 61 T
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2.3. THE FINITE ELEMENT METHOD 63 o
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2.3. THE FINITE ELEMENT METHOD 65 F
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2.3. THE FINITE ELEMENT METHOD 67 i
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2.4. MESH GENERATION AND REFINEMENT
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2.5. FINITE VOLUME METHOD 71 3 (a)
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2.5. FINITE VOLUME METHOD 73 This g
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Chapter 3 SPARSE MATRICES As descri
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3.2. GRAPH REPRESENTATIONS 77 Figur
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3.3. PERMUTATIONS AND REORDERINGS 7
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3.4. STORAGE SCHEMES 93 containing
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3.5. BASIC SPARSE MATRIX OPERATIONS
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3.6. SPARSE DIRECT SOLUTION METHODS
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3.7. TEST PROBLEMS 99 3 4 1 4 ✻ a
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3.7. TEST PROBLEMS 101 Figure 3.13:
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3.7. TEST PROBLEMS 103 P-3.8 Consid
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Chapter 4 BASIC ITERATIVE METHODS T
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4.1. JACOBI, GAUSS-SEIDEL, AND SOR
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4.2. CONVERGENCE 115 4.2.1 General
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4.2. CONVERGENCE 117 Then, the eige
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4.2. CONVERGENCE 119 From this and
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4.2. CONVERGENCE 121 cannot be an i
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4.2. CONVERGENCE 123 4.2.5 Property
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4.2. CONVERGENCE 125 Consider now a
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4.3. ALTERNATING DIRECTION METHODS
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Chapter 5 PROJECTION METHODS Most o
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5.1. BASIC DEFINITIONS AND ALGORITH
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5.2. GENERAL THEORY 137 5.2 General
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5.2. GENERAL THEORY 139 ✻✯ ❥
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5.2. GENERAL THEORY 141 Thus, an n-
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5.3. ONE-DIMENSIONAL PROJECTION PRO
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5.3. ONE-DIMENSIONAL PROJECTION PRO
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5.4. ADDITIVE AND MULTIPLICATIVE PR
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Chapter 6 KRYLOV SUBSPACE METHODS P
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6.2. KRYLOV SUBSPACES 159 Proof. Th
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6.3. ARNOLDI’S METHOD 161 which s
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6.3. ARNOLDI’S METHOD 163 from a
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6.4. ARNOLDI’S METHOD FOR LINEAR
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6.5. GMRES 171 6. Fortunately, an i
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6.5. GMRES 173 6.5.2 The Householde
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6.5. GMRES 175 Define the rotation
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6.5. GMRES 177 3. The residual vect
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6.5. GMRES 179 6.5.4 Breakdown of G
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6.5. GMRES 181 note that if ¯ Hm i
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6.5. GMRES 183 then b − Axm2 = |
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6.5. GMRES 185 Matrix Iters Kflops
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6.5. GMRES 187 Proposition 6.12 Ass
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6.5. GMRES 189 Proof. The following
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6.5. GMRES 191 In the situation whe
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6.5. GMRES 193 QMRES/DQGMRES. The k
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6.6. THE SYMMETRIC LANCZOS ALGORITH
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6.7. THE CONJUGATE GRADIENT ALGORIT
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6.8. THE CONJUGATE RESIDUAL METHOD
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6.9. GCR, ORTHOMIN, AND ORTHODIR 20
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6.10. THE FABER-MANTEUFFEL THEOREM
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6.11. CONVERGENCE ANALYSIS 209 One
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6.11. CONVERGENCE ANALYSIS 211 Ther
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6.11. CONVERGENCE ANALYSIS 213 over
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6.11. CONVERGENCE ANALYSIS 215 Ther
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ℑm(z) 6.11. CONVERGENCE ANALYSIS
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6.12. BLOCK KRYLOV METHODS 219 Kryl
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6.12. BLOCK KRYLOV METHODS 221 Now
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6.12. BLOCK KRYLOV METHODS 223 wher
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6.12. BLOCK KRYLOV METHODS 225 P-6.
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6.12. BLOCK KRYLOV METHODS 227 is p
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Chapter 7 KRYLOV SUBSPACE METHODS P
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7.1. LANCZOS BIORTHOGONALIZATION 23
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7.1. LANCZOS BIORTHOGONALIZATION 23
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7.3. THE BCG AND QMR ALGORITHMS 235
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7.4. TRANSPOSE-FREE VARIANTS 241 AL
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7.4. TRANSPOSE-FREE VARIANTS 243 th
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7.4. TRANSPOSE-FREE VARIANTS 245 Ac
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7.4. TRANSPOSE-FREE VARIANTS 247 4.
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7.4. TRANSPOSE-FREE VARIANTS 249 Ne
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7.4. TRANSPOSE-FREE VARIANTS 251 Th
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7.4. TRANSPOSE-FREE VARIANTS 253 15
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7.4. TRANSPOSE-FREE VARIANTS 255 P-
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7.4. TRANSPOSE-FREE VARIANTS 257 A
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Chapter 8 METHODS RELATED TO THE NO
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8.2. ROW PROJECTION METHODS 261 the
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8.2. ROW PROJECTION METHODS 263 ALG
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8.2. ROW PROJECTION METHODS 265 in
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8.3. CONJUGATE GRADIENT AND NORMAL
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8.4. SADDLE-POINT PROBLEMS 269 with
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8.4. SADDLE-POINT PROBLEMS 271 Apar
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8.4. SADDLE-POINT PROBLEMS 273 a. S
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Chapter 9 PRECONDITIONED ITERATIONS
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9.2. PRECONDITIONED CONJUGATE GRADI
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9.3. PRECONDITIONED GMRES 283 Somet
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9.3. PRECONDITIONED GMRES 285 9.3.3
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9.4. FLEXIBLE VARIANTS 287 In most
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9.4. FLEXIBLE VARIANTS 289 Proposit
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9.5. PRECONDITIONED CG FOR THE NORM
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9.6. THE CONCUS, GOLUB, AND WIDLUND
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9.6. THE CONCUS, GOLUB, AND WIDLUND
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Chapter 10 PRECONDITIONING TECHNIQU
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10.2. JACOBI, SOR, AND SSOR PRECOND
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10.3. ILU FACTORIZATION PRECONDITIO
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10.4. THRESHOLD STRATEGIES AND ILUT
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10.5. APPROXIMATE INVERSE PRECONDIT
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10.6. REORDERING FOR ILU 351 become
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10.6. REORDERING FOR ILU 353 Rows 1
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10.7. BLOCK PRECONDITIONERS 355 A b
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10.8. PRECONDITIONERS FOR THE NORMA
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Chapter 11 PARALLEL IMPLEMENTATIONS
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11.3. TYPES OF PARALLEL ARCHITECTUR
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11.4. TYPES OF OPERATIONS 377 among
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11.5. MATRIX-BY-VECTOR PRODUCTS 379
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11.6. STANDARD PRECONDITIONING OPER
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Chapter 12 PARALLEL PRECONDITIONERS
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12.3. POLYNOMIAL PRECONDITIONERS 39
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12.4. MULTICOLORING 407 22 10 23 11
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12.5. MULTI-ELIMINATION ILU 409 D1
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12.5. MULTI-ELIMINATION ILU 411 as
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12.5. MULTI-ELIMINATION ILU 413 1.
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12.6. DISTRIBUTED ILU AND SSOR 415
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12.7. OTHER TECHNIQUES 417 As in th
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12.7. OTHER TECHNIQUES 419 The prec
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12.7. OTHER TECHNIQUES 421 P-12.3 S
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Chapter 13 MULTIGRID METHODS The co
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13.2. MATRICES AND SPECTRA OF MODEL
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13.3. INTER-GRID OPERATIONS 437 and
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13.4. STANDARD MULTIGRID TECHNIQUES
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13.5. ANALYSIS FOR THE TWO-GRID CYC
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13.5. ANALYSIS FOR THE TWO-GRID CYC
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13.6. ALGEBRAIC MULTIGRID 455 to sa
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13.6. ALGEBRAIC MULTIGRID 457 It is
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13.6. ALGEBRAIC MULTIGRID 459 all k
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13.6. ALGEBRAIC MULTIGRID 461 • T
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13.6. ALGEBRAIC MULTIGRID 463 1. f
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13.7. MULTIGRID VS KRYLOV METHODS 4
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13.7. MULTIGRID VS KRYLOV METHODS 4
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Chapter 14 DOMAIN DECOMPOSITION MET
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14.1. INTRODUCTION 471 in Figure 14
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14.1. INTRODUCTION 473 4. Subdomain
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14.2. DIRECT SOLUTION AND THE SCHUR
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14.3. SCHWARZ ALTERNATING PROCEDURE
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14.4. SCHUR COMPLEMENT APPROACHES 4
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14.4. SCHUR COMPLEMENT APPROACHES 4
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14.5. FULL MATRIX METHODS 501 inter
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14.5. FULL MATRIX METHODS 503 Assum
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14.6. GRAPH PARTITIONING 505 14.6.2
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14.6. GRAPH PARTITIONING 507 The ma
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14.6. GRAPH PARTITIONING 509 ALGORI
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14.6. GRAPH PARTITIONING 511 Figure
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14.6. GRAPH PARTITIONING 513 foo wi
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516 BIBLIOGRAPHY [16] O. AXELSSON,
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518 BIBLIOGRAPHY [48] , An iterativ
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520 BIBLIOGRAPHY [83] N. CHRISOCHOI
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522 BIBLIOGRAPHY [117] H. C. ELMAN
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524 BIBLIOGRAPHY [153] G. H. GOLUB
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526 BIBLIOGRAPHY [188] R. KETTLER,
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528 BIBLIOGRAPHY [222] C. C. PAIGE,
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530 BIBLIOGRAPHY [255] Y. SAAD AND
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532 BIBLIOGRAPHY [289] H. A. VAN DE
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534 BIBLIOGRAPHY [324] L. ZHOU AND
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536 INDEX block GMRES, 222-223 mult
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538 INDEX Element-By-Element precon
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540 INDEX ILU, 301-332 Crout varian
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542 INDEX normal, 4, 21 orthogonal,
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544 INDEX by ILQ, 359 EBE, 417 inco
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546 INDEX permutation and reorderin
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