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Meurant G. — The Lanczos and conjugate gradient algorithms: from theory to finite precision computations
Meurant G. — The Lanczos and conjugate gradient algorithms: from theory to finite precision computations



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Название: The Lanczos and conjugate gradient algorithms: from theory to finite precision computations

Автор: Meurant G.

Аннотация:

"No present book comes near this one in the range and depth of treatment of these two extremely important methods—the Lanczos algorithm and the method of conjugate gradients." Chris Paige, School of Computer Science, McGill University. The Lanczos and conjugate gradient (CG) algorithms are fascinating numerical algorithms. This book presents the most comprehensive discussion to date of the use of these methods for computing eigenvalues and solving linear systems in both exact and floating point arithmetic. The author synthesizes the research done over the past 30 years, describing and explaining the "average" behavior of these methods and providing new insight into their properties in finite precision. Many examples are given that show significant results obtained by researchers in the field. The author emphasizes how both algorithms can be used efficiently in finite precision arithmetic, regardless of the growth of rounding errors that occurs. He details the mathematical properties of both algorithms and demonstrates how the CG algorithm is derived from the Lanczos algorithm. Loss of orthogonality involved with using the Lanczos algorithm, ways to improve the maximum attainable accuracy of CG computations, and what modifications need to be made when the CG method is used with a preconditioner are addressed. This book is intended for applied mathematicians, computational scientists, engineers, and physicists who have an interest in linear algebra, numerical analysis, and partial differential equations. It will be of interest to engineers and scientists using the Lanczos algorithm to compute eigenvalues and the CG algorithm to solve linear systems, and to researchers in Krylov subspace methods for symmetric matrices, especially those concerned with floating point error analysis. Moreover, it can be used in advanced courses on iterative methods or as a comprehensive presentation of a well-known numerical method in finite precision arithmetic. Contents Preface; Chapter 1: The Lanczos algorithm in exact arithmetic; Chapter 2: The CG algorithm in exact arithmetic; Chapter 3: A historical perspective on the Lanczos algorithm in finite precision; Chapter 4: The Lanczos algorithm in finite precision; Chapter 5: The CG algorithm in finite precision; Chapter 6: The maximum attainable accuracy; Chapter 7: Estimates of norms of the error in finite precision; Chapter 8: The preconditioned CG algorithm; Chapter 9: Miscellaneous; Appendix; Bibliography; Index.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2006

Количество страниц: 382

Добавлена в каталог: 23.10.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$L_{2}$ norm      xiv 56 65 68 69 86 88 93 187 242 252 257 261 263 265 268 272 283 292 311
A-norm      xiii xiv 23 45 58 61 62 64—66 68 69 72 77 78 80 130 187 242 244 257 260 265 268 272 283 291 292 301 310
Approximate inverse      315
Backward error analysis      83 121
Cauchy      1 18 20
Characteristic polynomial      1 15 22
Chebyshev polynomials      41 78—80 94 131
Chebyshev recurrence      132
Chebyshev series      131
cholesky      xiii 5 9—11 46 50 52 55 114 115 158 191 236 258 260 284 288 291 292 309 315 316
Christoffel — Darboux relation      12 29
Condition number      xiv 1 45 54 57 77 78 136 187 191 223 224 244 301 310
Determinant      9—11 15 151 158 208
Diagonal dominance      56
Diagonally dominant matrix      86
Double reorthogonalization      109 142—144 177 183 236 325
Eigenvalues      xi xii 1 2 4 6—13 15 16 18—20 25—29 31—34 36 37 39 41—43 45 51 57 58 60 62 78 80 81 89 93 94 96—99 101 102 108 113 121 122 126 127 130 135 139 140 142 151—153 162 174 177 180 206—208 236 244 260 264 281 282 288 290 292 298 301 306 310 311 316 325 326 332 335 338 341 344
Eigenvectors      xi 2 8 9 12 16 19—22 26—28 34 39 43 57 58 60 61 63 73 96 98 105 108 113 122—124 126 139 140 142 149 152 153 157 174 175 182 183 198 200 203 204 210 264 282 283 286 298 301 305 306 310 326
Finite precision arithmetic      xi xiii xiv 30 50 81 82 94 99 100 131 143 187 201 214 223 257 258 263—267 272 275 281 284 310
Forward analysis      82 105 157
Forward error      84
Frobenius norm      97 302
Full reorthogonalization      108 117
Gauss lower bound      259 260 268
Gauss quadrature      xiii 12 45 58 60 62 263
Gauss rule      59 257
Gaussian elimination      121 187 252 314 316
Gram — Schmidt algorithm      1
Gram — Schmidt orthogonalization      2 81
Gram — Schmidt process      xii 6 81
IEEE arithmetic      xiii 83
Indefinite linear systems      xiv
Indefinite matrix      310
Inner product      62 75 82—86 93 95 98 111 215 217 266 281 282 284 286 302
Irreducible      171
Iterative method      xiv xv 304
Iterative methods      77 314
Lanczos polynomial      11 13 15 27 56 59 60 73 107 142 143 152
Laplace      1
Level of orthogonality      82 113 114 117 118
Local orthogonality      xiv 54 75 79 80 90 94 102 116 187 215 265 304
Loss of orthogonality      xiii 82 90 98 101 102 105 108 113 139 187
LU decomposition      11 71 300
Machine epsilon      83
Matrix-vector product      xiv 75 86 93 239 260 304 305
Maximum attainable accuracy      xiv 188 239 242 249 272 292
Maximum norm      249 251
Minres      313
Multilevel preconditioners      315 316
Newton's method      43
Norm of the error      62 65 126 131 188 239 241 244 260 261 263 264 268 291
Norm of the residual      55 57 64 66 191 202 214 224 229 239 264 268 286 291 306 310 313 315
Norms of the error      272
Orthogonal basis      xi 1 45 324
Orthogonal matrix      xi 2 16 201 261
Orthogonal polynomials      xii 1 139
Orthogonality      xi—xiii 3 62 74 75 81 82 97 104 108 109 124 139 223 263 266 282 325
Orthogonalization      1 143
Parallel computation      75
Partial differential equations      xiii
Partial reorthogonalization      117
Periodic reorthogonalization      109
Poisson equation      140 182 191 236 241 242 272 288 311 313
Poisson model problem      236 250
Positive definite      xii 9 45 46 49 50 53—55 58 86 112 115 136 191 244 248 258 281
QR algorithm      43
QR factorization      xii 2—5 50 70 114 261 268 311
Reorthogonalization      1 81 90 94 108 109 118 130 142—144 236
Residual smoothing      302
Ritz polynomial      15 32
Ritz value      xii 8 9 13 15 25 28—31 34 37 41–43 45 60 73 81 82 94 96 100 102 103 113 123 125 129—133 135 136 139 142—144 147 150 152 159 160 162 166 176 181 182 184 186 198—201 204 206—208 210 261 264 283 288 290 292 308 310 316 332 333 335
Ritz vector      8 15 28 64 82 98 101 122—125
Roundoff error      xiii 139 140 197 198 214 241 243 249
Selective orthogonalization      117
Semiorthogonality      82 116
Semiorthogonalization      117
Spectral decomposition      16 26 33 62 64 73 97 99 123 129 210 282
SYMMLQ algorithm      50 53 311
Three-term recurrence      xiii 1 9 11 12 41 74 75 82 152 155 168 172 174 195—198 204 207 209 214 275 286 298 300 305
Two-term recurrence      xiv 75 198 210 275
UL decomposition      11 51
Unit roundoff      82
Variable precision      142 176 177 193 204 230 236
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