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Higham N.J. — Accuracy and Stability of Numerical Algorithms
Higham N.J. — Accuracy and Stability of Numerical Algorithms



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Название: Accuracy and Stability of Numerical Algorithms

Автор: Higham N.J.

Аннотация:

What is the most accurate way to sum floating point numbers? What are the advantages of IEEE arithmetic? How accurate is Gaussian elimination and what were the key breakthroughs in the development of error analysis for the method? The answers to these and many related questions are included here.

This book gives a thorough, up-to-date treatment of the behavior of numerical algorithms in finite precision arithmetic. It combines algorithmic derivations, perturbation theory, and rounding error analysis. Software practicalities are emphasized throughout, with particular reference to LAPACK and MATLAB. The best available error bounds, some of them new, are presented in a unified format with a minimum of jargon. Because of its central role in revealing problem sensitivity and providing error bounds, perturbation theory is treated in detail.

Historical perspective and insight are given, with particular reference to the fundamental work of Wilkinson and Turing, and the many quotations provide further information in an accessible format.

The book is unique in that algorithmic developments and motivations are given succinctly and implementation details minimized, so that attention can be concentrated on accuracy and stability results. Here, in one place and in a unified notation, is error analysis for most of the standard algorithms in matrix computations. Not since Wilkinson's Rounding Errors in Algebraic Processes (1963) and The Algebraic Eigenvalue Problem (1965) has any volume treated this subject in such depth. A number of topics are treated that are not usually covered in numerical analysis textbooks, including floating point summation, block LU factorization, condition number estimation, the Sylvester equation, powers of matrices, finite precision behavior of stationary iterative methods, Vandermonde systems, and fast matrix multiplication.

Although not designed specifically as a textbook, this volume is a suitable reference for an advanced course, and could be used by instructors at all levels as a supplementary text from which to draw examples, historical perspective, statements of results, and exercises (many of which have never before appeared in textbooks). The book is designed to be a comprehensive reference and its bibliography contains more than 1100 references from the research literature.

Audience

Specialists in numerical analysis as well as computational scientists and engineers concerned about the accuracy of their results will benefit from this book. Much of the book can be understood with only a basic grounding in numerical analysis and linear algebra.

About the Author

Nicholas J. Higham is a Professor of Applied Mathematics at the University of Manchester, England. He is the author of more than 40 publications and is a member of the editorial boards of the SIAM Journal on Matrix Analysis and Applications and the IMA Journal of Numerical Analysis. His book Handbook of Writing for the Mathematical Sciences was published by SIAM in 1993.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2

Год издания: 2002

Количество страниц: 680

Добавлена в каталог: 19.06.2009

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
GEPP      see “Gaussian elimination partial
Geuder, James C.      186
Ghavimi, Ali R.      315—317
Gill, Philip E.      210 226 227 229 413 476
Gill, S.      83
Givens rotation      365
Givens rotation, disjoint rotations      367—368 379
Givens rotation, fast      376
Givens, Wallace J.      29 61
Gluchowska, J.      404
GNU MP library      502
Godunov, S.K.      285
Gohberg, L.      128 129 428
Goldberg, David      35 53 85 532
Goldberg, I. Bennett      57
Goldstine, Herman H.      1 29 48 183 184 188 259 260 515
Golub, Gene H.      xxviii 11 24 29 115 132 133 172 187 208 231 256 257 282 299 307 308 308 323 345 375 377 380—382 389 399 402—405 429
Goodman, R.      57
Goodnight, James H.      282
Gould, Nicholas I.M.      170 189
Govaerts, Willy J.F.      241
Gradual underflow      38 42 45 56
Gragg, W.B.      295 376
Graham, Ronald L.      79 518
Gram — Schmidt method      369—373
Gram — Schmidt method reorthogonalization      376
Gram — Schmidt method, classical      369
Gram — Schmidt method, classical, error analysis      371 373
Gram — Schmidt method, modified      370
Gram — Schmidt method, modified, connection with Householder QR factorization      353 371—372 376
Gram — Schmidt method, modified, error analysis      371—373
Gram — Schmidt method, modified, error analysis for application to LS problem      386
Gram — Schmidt method, modified, stability      24
Greenbaum, Anne      324 337
Gregory, Robert T.      29 512 522
Griewank, Andreas      469
Grimes, Roger G.      188 219 227 302 524
Grosse, Eric      573
Group inverse      331
Growth factor      165—173
Growth factor for banded matrix      173
Growth factor for block $\textrm{LDL}^\mathrm{T}$ factorization      216 218 221 226
Growth factor for complete pivoting      169—170
Growth factor for diagonally dominant matrix      172
Growth factor for partial pivoting      166—173
Growth factor for partial pivoting large growth in practical problems      167
Growth factor for random matrices      189
Growth factor for rook pivoting      170
Growth factor for tridiagonal matrix      173
Growth factor for upper Hessenberg matrix      172
Growth factor maximization by direct search      472—473
Growth factor numerical maximization for complete pivoting      170 189
Growth factor, a posteriori estimates for      180—181
Growth factor, define using exact or computed quantities      165 189
Growth factor, lower bound for      167
Growth factor, statistical model of      168
Gu, Ming      346 404 406 534
Guard digit      44
Guard digit, test for      52
Gudmundsson, Thorkell      298
Guggenheimer, Heinrich W.      284
Gulliksson, Marten      405
Gustafson, John L.      446
Gustavson, F.G.      187
Haar distribution, random orthogonal matrix from      517—518
Hadamard condition number      279 282 284
Hadamard matrix      116 168 170 193
Hadamard's inequality      284
Hager, William W.      292 301
Hall, Jr., Marshall      168
Halley's method      508
Halmos, Paul R.      511
Hamada, Hozumi      49
Hammarling, Sven J.      317 322 323 376 397 500
Hamming, R.W.      53 431
Hansen, Per Christian      133 377 448
Hanson, Richard J.      375 376 399 402 403 414 499
Harter, Richard      446
Hartfiel, D.J.      132
Harwell — Boeing sparse matrix collection      524
Hauser, John R.      57 59
Hearon, John Z.      311
Heath, Michael T.      259 388
Hedayat, A.S.      168
Hein, Piet      31 115
Helvin, Marie      153
Henderson, Harold V.      317 487
Hennessy, John L.      35 53
Henrici, Peter      48 49 61 77 344
Henson, Van Emden      456
Heron's formula      45
Heroux, Michael      446 569
Herzberger, Jiirgen      483
Hessenberg matrix Gaussian elimination      24—25 30
Hessenberg matrix, determinant of      24—25 30 280
Hessenberg matrix, growth factor for      172
Hewer, Gary      318
Hewlett — Packard HP 48G calculator condition estimator      302
Hewlett — Packard HP 48G calculator exhausting its range and precision      15—16
Hidden bit      41
Higham, Desmond J.      114 128 129 136 167 187 339 342 394 429
Higham, Nicholas J.      65 90 128 129 132 151—153 167 187 190 210 219 227 228 240—242 249—253 256 257 262 269 278 281 292 294 295 298 301—304 317 324 347 349 351 362 375 377 378 380 394—396 398 400 402—405 409 413 414 429 431 446—448 486 505 523 525 564 569 571
Hilbert matrix      512—515 523—524
Hilbert matrix determinant      513
Hilbert matrix, Cholesky factor      513
Hilbert matrix, inverse      513
Hilbert, David      523
Hildebrand, F.B.      28 35
Hodel, A. Scottedward      317
Hodges, Andrew      xxix
Hoelder inequality      106 107
Hoffman, A.J.      377
Hoffmann, Christoph M.      29
Hoffmann, W.      281 282 376
Hooper, Judith A.      501
Horn, Roger A.      107 115 137 152 306 342 543 547 551 554
Horner's method      94—104
Horner's method for derivatives      96—99
Horner's method for rational function      26
Horner's method running error bound      95—96 103
Horning, Jim      489
Hotelling, Harold      183 433
Hough's underflow story      506—507
Hough, David      43 506 507
Hough, Patricia D.      395
Householder transformation      354—355
Householder transformation error analysis      357—363
Householder transformation in QR factorization      355—357
Householder transformation, aggregated (WY representation)      363—365
Householder transformation, block      375
Householder transformation, choice of sign in      355
Householder transformation, history of      374—375
Householder, Alston S.      2 105 115 147 161 374 403 560
Hull, T.E.      48 50 485 503 528
Hunt, Julian      54
Huskey, H.D.      184
Hyman's method      30 280 285
Hyman, M.A.      280
IBM, ESSL library      436 438
IEEE arithmetic      39 41—43
IEEE arithmetic $\infty$      42 490 492
IEEE arithmetic double rounding      43 58
IEEE arithmetic exception handling      41 491—492
IEEE arithmetic exceptions      41—42
IEEE arithmetic exploiting in software      490—493
IEEE arithmetic extended formats      42
IEEE arithmetic gradual underflow      42
IEEE arithmetic implementation using formal methods      55—56
IEEE arithmetic NaN      41—42 490 492
IEEE arithmetic parameters      371 41
IEEE arithmetic recommended auxiliary functions      492—493
IEEE arithmetic rounding modes      41
IEEE arithmetic signed zeros      42
IEEE arithmetic Standard 754      41
IEEE arithmetic Standard 854      43
IEEE arithmetic subnormal numbers      42 492
Ikebe, Yasuhiko      300 302 304
Ikramov, Khakim D.      209 525
Incertis, F.      509
Index of a matrix      331
Inertia      214
Inner product error analysis      62—65
Inner product in extended precision      64
Inner product, reducing constant in error bound      63
Intel Itanium chip      46
Intel Pentium chip, division bug      55
Internet      574
Interval analysis      41 190 481—484
Interval analysis, dependencies      482
Interval analysis, fallibility      483—484
Interval analysis, Gaussian elimination      482
Interval analysis, super-accurate inner product      483
Interval arithmetic      see “Interval analysis”
INTLAB      483
Inverse iteration      24
Inverse matrix      259—285
Inverse matrix bound using diagonal dominance      154
Inverse matrix in solving Ax=b stability      260
Inverse matrix left and right residuals      261
Inverse matrix perturbation theory      127—128
Inverse matrix, Csanky's method for      278
Inverse matrix, error analysis for Gauss — Jordan elimination      273—277
Inverse matrix, error analysis for LU factorization      267—271
Inverse matrix, error analysis for triangular matrix      262—267
Inverse matrix, high-accuracy computation of      281
Inverse matrix, Newton's method for      278
Inverse matrix, Schulz iteration      278
Inverse matrix, Strassen's method for      448—449 478—479
Inverse matrix, times for computation on early computers      272
Inverse matrix, triangular, bounds for      147—149
Inverse matrix, triangular, methods for      262—267
Inverse matrix, why not to compute      260
Involutary matrix      519
Ipsen, Use C.F.      123 129 133 376 377
Iri, Masao      49 485
Irreducible matrix      127
Isaacson, Eugene      186 257
Itanium chip      46
Iterative methods      see also “Stationary iterative methods”
Iterative methods dates of publication      322
Iterative methods error analysis      325—335
Iterative methods error analysis, survey of      323—324
Iterative methods stopping criteria      335—337 467—468
Iterative refinement backward error analysis      235—239 463
Iterative refinement for least squares problem      388—391 403
Iterative refinement for square linear system      27 184 231—243 462—463
Iterative refinement for Vandermonde system      427—428
Iterative refinement forward error analysis      232—235 463
Iterative refinement in fixed precision      179 234 494
Iterative refinement in mixed precision      234
Iterative refinement LAPACK convergence test      241
Iterative refinement practical issues      241—242
Iterative refinement with QR factorization      368—369
Iterative refinement, condition number estimate from      242
Jacobi method, forward error analysis      328—329
Jalby, William      376
Jankowski, M.      85 240 241
Jansen, Paul      483
Jennings, A.      152
Jennings, L.S.      413
Johnson, Charles R.      107 115 137 152 284 306 342 543 547 551 554
Johnson, Samuel      667
Jones, Mark T.      227
Jones, William B.      505
Jordan canonical form      340
Jordan, Camille      281
Jordan, T.L.      402
Jordan, Wilhelm      281
Kagstroem, Bo      302 315 316 318
Kahan matrix      149 205
Kahan matrix second smallest singular value      154
Kahan, William M. (Velvel)      1 26 29 30 32 42 43 44 45 49 55 58 59 60 69 78 83—86 88 103 111 114 124 149 152 157 226 242 404 483 486 491 494—496 498 499 504
Kahaner, David K.      381
Kailath, T.      429
Kala, R.      318
Kaniel, S.      227
Kaporin, Igor      447
Karasalo, Ilkka      152
Karatsuba, A.      447
Karlin, Samuel      520
Karlson, Rune      392 404 406
Karney, David L.      512 522
Karp, A.      187
Karpinski, Richard      52 495
Kato, Tosio      115
Kaufman, Linda      213 216 226 227 376
Kaufmann, Matt      56
Kearfott, R. Baker      483
Keiper, Jerry B.      31
Keller, Herbert Bishop      186 257 321
Kelley, C.T.      468 476
Kennedy, Jr., William J.      29
Kenney, Charles S.      298 318 523
Kerr, Thomas H.      210
Kielbasihski, Andrzej      77 85 209 241 393 404 410 414
Kincaid, D.R.      499
Kiriljuk, O.P.      285
Kittaneh, Fuad      523
Knight, Philip A.      324 347 349 351 447 448
Knuth, Donald E.      xxiii xxviii 49 53 54 61 79 84 85 104 447 471 489 518 523 532
Kocak, Hiiseyin      29
Koltracht, L.      128 129
Koq, O.K.      103
Korner, T.W.      451
Kornerup, Peter      49
Kosowski, Przemyslaw      525
Kostin, V.I.      285
Kostlan, Eric      516 524
Kovarik, Z.V.      132
Kowalewski, G.      428
Krasny, Robert      502
Kreczmar, Antoni      446
Kreiss matrix theorem      346
Krogh, F.T.      499
Kronecker product      306 317
Krueckeberg, F.      191
Kubota, Koichi      485
Kucherov, Andrey B.      209
Kuczyriski, J.      298
Kuki, H.      56
Kulisch, Ulrich W.      483
Kuperman, I.B.      132
La Porte, M.      486
La Touche, Mrs.      79 80
Laderman, Julian      437 446
Laeuchli, Peter      379
Lagrange, Joseph Louis      215
Lancaster, Peter      317 468
LANCELOT      170 189
Lanczos, Cornelius      486
Langlois, Philippe      484
LAPACK      579—581
LAPACK, $2 \times 2$ problems, solving      497—498
LAPACK, block $\textrm{LDL}^\mathrm{T}$ factorization (of symmetric matrix)      228
LAPACK, block and partitioned LU factorization      257
LAPACK, Cholesky factorization      210—211
LAPACK, condition number estimation      292—294 303
LAPACK, forward error bound for linear systems      131
LAPACK, iterative refinement      241—242
LAPACK, least squares problem      405
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