Главная    Ex Libris    Книги    Журналы    Статьи    Серии    Каталог    Wanted    Загрузка    ХудЛит    Справка    Поиск по индексам    Поиск    Форум   
blank
Авторизация

       
blank
Поиск по указателям

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



Обсудите книгу на научном форуме



Нашли опечатку?
Выделите ее мышкой и нажмите Ctrl+Enter


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
blank
Предметный указатель
Second difference matrix      522
Semiconvergent matrix      332
Seminormal equations for least squares problem      391—392
Seminormal equations for under determined system      408
Separation (sep) of two matrices      313
Sha, Xuan-He      437 446
Shampine, Lawrence F.      29 76 86 524
Shannon, Claude E.      56
Shapiro, Alexander      133
Shepherd, David      56
Sherman — Morrison formula      190 487
Sherman — Morrison — Woodbury formula      258 558
Shewchuk, Jonathan Richard      29 91
Shinnerl, Joseph R.      226 229
Shroff, Gautam M.      299
Shub, Michael      133 516
Sierpinski gasket      521
Significance arithmetic      486
Significand      36
Significant digits correct      3—4 28
Significant digits least and most significant      36
Simon, Horst D.      446 447
Singular value decomposition (SVD)      114—115
Skeel, Robert D.      31 123 133 178 186 187 190 231 235 240 482
Skew-symmetric matrix      214 225
Slishman, Gordon      446 569
Sloane, N.J.A.      168
Smale, Steve      2
Smith, David M.      502
Smith, Francis J.      103 427
Smith, Jon M.      32
Smith, Robert L.      500 506
Smith, Roger M.      446 569
Smoktunowicz, Alicja      85 129 136 241 404 525
Snyder, James N.      231 240
Software avoiding underflow and overflow      499—501
Software effects of underflow      501
Software issues in floating point arithmetic      489—509
Software portability      496—499
Software specifying arithmetic parameters      496—497
Software specifying numerical constants      498
Sokolnicka, Jolanta      241
SOR method, forward error analysis      329
Sorensen, Danny C.      180 187 226 228 256
Sorevik, T.      448
Spencer, Herbert      657
Spiefi, J.      446
Spooner, David      485
Square root, of complex number      32
Stability      see “Backward stability” “Forward
Stable algorithms, designing      26—28
Stationary iterative methods      321—337
Stationary iterative methods and powers of a matrix      351
Stationary iterative methods backward error analysis      330—331
Stationary iterative methods forward error analysis      325—329
Stationary iterative methods scale independence      327
Stationary iterative methods singular systems      333—335
Stationary iterative methods stopping criteria      335—337
Stationary iterative methods, Jacobi method      328—329
Stationary iterative methods, singular systems, theory for      331—333
Stationary iterative methods, SOR method      329
Statistics      see also “Variance”
Statistics, computational references      29
Steele, Jr., Guy L.      57
Stegun, Irene A.      30
Sterbenz, Pat H.      29 30 45 53 56 486
Stewart, G.W. (Pete)      xxviii 68 102 107 115 119 126 133 139 152 163 187 190 209 231 240 241 282 295 302 307 318 339 351 373 374 376 377 382 384 397 400 402—404 500 517 525
Stewart, Michael      210
Stewart, William J.      302
Sticky bit      41
Stockmeyer, Larry J.      102
Stoer, Josef      76 107 114 116 187
Stone, Betty Jane      114
Storey, C.      317
Stoutemyer, David R.      486
Strakos, Zdenek      324
Strang, Gilbert      13 112 133 457
Strassen's method      434—436
Strassen's method error analysis      440—443
Strassen's method error versus operation count      14
Strassen's method for inversion      448—449 478—479
Strassen's method implementation issues      446—447
Strassen's method Winograd's variant      435—436 442—443
Strassen's method, accuracy compared with conventional multiplication      441—442
Strassen, Volker      278 434 448 449
Straus, E.G.      126 133
Street, Anne Penfold      168
Stufken, John      168
Stummel, Friedrich      30 76 187
Subdifferential, of a vector norm      289
Subgradient      290
Subnormal numbers      37 42 492
Substitution, back and forward      140—141
Subtraction, exactness in floating point arithmetic      45
Successive overrelaxation method      see “SOR method”
Summation      79—92
Summation, choice of method, summary      89—90
Summation, compensated and applications      83—88
Summation, condition number      91
Summation, criterion for minimizing error      82
Summation, distillation algorithms      88
Summation, doubly compensated      87—88
Summation, error analysis      81—83
Summation, insertion method      80
Summation, pairwise (fan-in)      80
Summation, recursive      80
Summation, recursive, ordering in      17 82—83
Summation, statistical error estimates      88—89
Sun, Ji-guang      107 115 119 126 129 133 181 190 201 209 374 377 382 384 392 400 402 404 406 411 414 429
Sun, Xiaobai      282 376
Sun, Zheng      411 414
SVD (singular value decomposition)      114—115
Swann, W.H.      472
Swartzlander, Jr., Earl E.      49
Sweeney, D.W.      56
Swenson, J.R.      48
Sylvester equation      306
Sylvester equation backward error      308—311
Sylvester equation Bartels — Stewart method      307—308
Sylvester equation generalizations      316—318
Sylvester equation perturbation theory      313—315
Sylvester equation practical error bounds      315—316
Sylvester equation solution methods      307—308
Sylvester, James Joseph      305 317 434
Symbolic manipulation package      6
Symbolic Math Toolbox      3 6 463 514 519
Symmetric indefinite factorization      see “Block $\textrm{LDL}^\mathrm{T}$ factorization (of symmetric matrix)”
Symmetric indefinite matrix      214
Symmetric positive definite matrix      196
Symmetric positive definite matrix and block LU factorization      255—256
Symmetric positive definite matrix, practical test for      210
Symmetric positive semidefinite matrix      201
Symmetric positive semidefinite matrix, determinantal conditions for      211
Symmetric quasidefinite matrix      229
Synthetic division      96
Tablemaker's dilemma      4 50
Tang, Ping Tak Peter      50 496 528
Tang, W.P.      429
Tao, Pham Dinh      116 301
Tartaglia, Niccolo      479
Tasche, Manfred      456
Taussky, Olga      511 514
Test for accuracy of floating point arithmetic      51—52
Test for guard digit      52
Test matrices      511—525; see also “MATLAB” “Matrix” “Random
Test matrices, Harwell — Boeing collection      524
Test matrices, Matrix Computation Toolbox      583—585
Test matrices, Matrix Market      524
Teukolsky, Saul A.      476 487 505
Thacher, Jr., Henry C.      188
Thisted, Ronald A.      29
Thompson, Sir D'arcy Wentworth      1
Threshold pivoting      193
Thron, W.J.      505
Tienari, Martti      49
Tisserand, Arnaud      50
Tisseur, Franchise      294 461 462 468 523
Todd, John      114 115 512 518 523 524
Toeplitz matrix, pseudospectra      525 526
Toeplitz matrix, tridiagonal      521—522
Toh, Kim-Chuan      480 523
Toledo, Sivan      248
Torczon, Virginia J.      472 475 476
Tornheim, L.      169 170
Totally nonnegative matrix      164 520
Totally nonnegative matrix, LU factorization      164—165 188
Totally nonnegative matrix, row scaling in      179
Totally nonnegative matrix, test for      188
Transformations, well conditioned      27
Traub, J.F.      428
Trefethen, Lloyd N.      5 157 168 169 180 323 337 339 340 342 345 346 348 350 351 480 516 523 525
Tremaine, Scott      87
Triangular matrix, bounds for inverse      147—149
Triangular matrix, condition numbers      142—143
Triangular matrix, inversion methods, blocked      265—267
Triangular matrix, inversion methods, unblocked      262—264
Triangular matrix, M-matrix      145—147
Triangular systems      139—155
Triangular systems conditioning      144—145
Triangular systems fan-in algorithm      149—151
Triangular systems partitioned inverse method      153
Triangular systems substitution, backward error analysis      140—142
Triangular systems substitution, forward error analysis      142—147
Triangular systems, accurate solution of      139 143 144 147
Tridiagonal matrix      174—176
Tridiagonal matrix, condition number estimation      299—301
Tridiagonal matrix, growth factor      173
Tridiagonal matrix, LU factorization      174
Tridiagonal matrix, LU factorization, error analysis of      174—176
Tridiagonal matrix, structure of inverse      300—302
Tridiagonal matrix, Toeplitz      521—522
Tropp, Henry S.      489
Trossett, Michael W.      472
Trummer, Manfred R.      340 348 350
Truncation error      5
Tsao, Nai-kuan      76 375
Tucker, Warwick      482
Tukey, John W.      56 456
Turing Award of the ACM      xxix 55
Turing programming language      503
Turing programming language, Numerical Turing      503
Turing, Alan Mathison      xxix xxx 29 114 119 184 185 281 481
Turing, Alan Mathison, contributions in 1948 paper “Rounding-off errors”      184—185 281
Turnbull, H.W.      374 521
Turner, Peter R.      47 49 152
Twisted factorization      303
Tyrtyshnikov, Evgenij E.      523
Ulp (unit in last place)      39
Uncertainty, in data      5
Under determined system      408
Under determined system backward error, normwise      411
Under determined system backward error, row-wise      411
Under determined system, modified Gram — Schmidt      413
Under determined system, perturbation theory      409—411
Under determined system, Q method (QR factorization)      408
Under determined system, Q method (QR factorization), error analysis      411—413
Under determined system, seminormal equations      408
Under determined system, seminormal equations, error analysis      412
Underflow      16 38
Underflow, avoiding      499—501
Underflow, effects on software      501
Underflow, gradual      38 42 45 56
Underflow, model for error analysis      56—57
Underhill, L.G.      524 525
Ungar, Peter      447
UNICOS library (Cray)      436 438
Unit roundoff      3 38
Update formula, involving small correction      27
Urabe, Minoru      57
Uspensky, J.V.      487
Vajtersic, M.      448
van de Geijn, Robert      282
van der Sluis's theorem      125
van der Sluis, A.      125 126 190 199 381 402
Van der Vorst, Henk A.      187 226 256 324
van Leeuwen, J.      447
Van Loan, Charles F      xxviii 24 102 115 129 132 136 172 187 208 210 228 231 256 297 307 308 308 339 342 345 352 363 365 375 377 380 382 397 402 405 451 452 456
van Veldhuizen, M.      189
van Wijngaarden, A.      498
Vancouver Stock Exchange, inaccurate index      54
Vanderbei, Robert J.      229
Vandermonde matrix bounds and estimates for condition number      417—418
Vandermonde matrix definition      416
Vandermonde matrix inverse      416—418
Vandermonde matrix inversion algorithm      417
Vandermonde matrix LU factorization in factored form      422
Vandermonde matrix QR factorization      429
Vandermonde matrix structured condition number      428—429
Vandermonde system      415—431
Vandermonde system accuracy independent of condition number      425
Vandermonde system algorithm for dual      421
Vandermonde system algorithm for primal      422—423
Vandermonde system algorithm for residual of confluent system      427
Vandermonde system backward error analysis      425—426
Vandermonde system complexity results      429
Vandermonde system curing instability      427—428
Vandermonde system forward error analysis      424—425
Vandermonde system history of solution methods      429
Vandermonde system preventing instability      426—427
Vandermonde system structured backward error      429
Vandermonde-like matrix, confluent, definition      419
Vandermonde-like matrix, definition      418
Vandermonde-like matrix, determinant      430
Varah, James M.      133 154 257
Varga, Richard S.      147 154 257 505
Variance, algorithms for computing      11—12 29
Variance, condition numbers for      32
Variance, error bound for two-pass formula      33
Vavasis, Stephen A.      278 395
Vec operator      306
Vec-permutation matrix      314 317 584
Vemulapati, Udaya B.      299 397
Venus probe, loss due to program bug      489
Verdonk, Brigitte      496
Verschaeren, Dennis      496
Veselic, Kresimir      210
Vetterling, William T.      476 487 505
Vieta, Franciscus      479
Vignes, J.      486
Viswanath, D.      516
Vitasek, Emil      90
Viten'ko, I.V.      86
von Neumann, John      29 48 183 184 188 260 515
Waite, William      495 496
Walden, Bertil      392 404 406
Walker, Homer F.      323 468
Wallis, Jennifer Seberry      168
Wallis, W.D.      168
Ward, Robert C.      282
Warming, Robert F.      525
Wasilkowski, G.W.      133
Wasniewski, Jerzy      302
Watson, G.A.      133
Watterson, Bill      471
Wedderburn, J.H.M.      10
Wedin's least squares perturbation theorem      382
Wedin's least squares perturbation theorem, proof      400—402
Wedin, Per-Ake      382 400 402 405 413
Wegert, Elias      346
Wei, Musheng      380 402
1 2 3 4 5 6 7
blank
Реклама
blank
blank
HR
@Mail.ru
       © Электронная библиотека попечительского совета мехмата МГУ, 2004-2024
Электронная библиотека мехмата МГУ | Valid HTML 4.01! | Valid CSS! О проекте