Iterative Methods for Sparse Linear Systems Second Edition

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2 CHAPTER 1. BACKGROUND IN LINEAR ALGEBRA • Multiplication by another matrix: C = AB, where A ∈ C n×m , B ∈ C m×p , C ∈ C n×p , and cij = m k=1 aikbkj. Sometimes, a notation with column vectors and row vectors is used. The column vector a∗j is the vector consisting of the j-th column of A, a∗j = ⎛ ⎜ ⎝ a1j a2j . anj ⎞ ⎟ ⎠ . Similarly, the notation ai∗ will denote the i-th row of the matrix A ai∗ = (ai1, ai2, . . .,aim). For example, the following could be written or A = (a∗1, a∗2, . . .,a∗m), ⎛ ⎜ A = ⎜ ⎝ The transpose of a matrix A in C n×m is a matrix C in C m×n whose elements are defined by cij = aji, i = 1, . . .,m, j = 1, . . .,n. It is denoted by A T . It is often more relevant to use the transpose conjugate matrix denoted by A H and defined by a1∗ a2∗ . . an∗ ⎞ ⎟ ⎠ . A H = ĀT = A T , in which the bar denotes the (element-wise) complex conjugation. Matrices are strongly related to linear mappings between vector spaces of finite dimension. This is because they represent these mappings with respect to two given bases: one for the initial vector space and the other for the image vector space, or range of A.
1.2. SQUARE MATRICES AND EIGENVALUES 3 1.2 Square Matrices and Eigenvalues A matrix is square if it has the same number of columns and rows, i.e., if m = n. An important square matrix is the identity matrix I = {δij}i,j=1,...,n, where δij is the Kronecker symbol. The identity matrix satisfies the equality AI = IA = A for every matrix A of size n. The inverse of a matrix, when it exists, is a matrix C such that CA = AC = I. The inverse of A is denoted by A −1 . The determinant of a matrix may be defined in several ways. For simplicity, the following recursive definition is used here. The determinant of a 1 × 1 matrix (a) is defined as the scalar a. Then the determinant of an n × n matrix is given by det(A) = n (−1) j+1 a1jdet(A1j), j=1 where A1j is an (n − 1) × (n − 1) matrix obtained by deleting the first row and the j-th column of A. A matrix is said to be singular when det(A) = 0 and nonsingular otherwise. We have the following simple properties: • det(AB) = det(A)det(B). • det(A T ) = det(A). • det(αA) = α n det(A). • det( Ā) = det(A). • det(I) = 1. From the above definition of determinants it can be shown by induction that the function that maps a given complex value λ to the value pA(λ) = det(A − λI) is a polynomial of degree n; see Exercise 8. This is known as the characteristic polynomial of the matrix A. Definition 1.1 A complex scalar λ is called an eigenvalue of the square matrix A if a nonzero vector u of C n exists such that Au = λu. The vector u is called an eigenvector of A associated with λ. The set of all the eigenvalues of A is called the spectrum of A and is denoted by σ(A). A scalar λ is an eigenvalue of A if and only if det(A − λI) ≡ pA(λ) = 0. That is true if and only if (iff thereafter) λ is a root of the characteristic polynomial. In particular, there are at most n distinct eigenvalues. It is clear that a matrix is singular if and only if it admits zero as an eigenvalue. A well known result in linear algebra is stated in the following proposition.
Page 1: Iterative Methods for Sparse Linear
Page 4 and 5: vi CONTENTS 2.1.2 The Convection Di
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Page 12 and 13: xiv PREFACE underlying architecture
Page 14 and 15: xvi PREFACE The rest could be skipp
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Page 18 and 19: xx PREFACE Apart from two recent vo
Page 21: Chapter 1 BACKGROUND IN LINEAR ALGE
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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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Iterative Methods for Sparse Linear Systems Second Edition

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