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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
$0^0$ definition      59
$2 \times 2$ problems, reliable solution of      497—498
$\gamma_\mathrm{n}$ (error constant) definition      63
$\gamma_\mathrm{n}$ (error constant) properties      67
$\infty$ (IEEE arithmetic)      42 490 492
$\textrm{LDL}^\mathrm{T}$ factorization      197
$\tilde{\gamma}_\mathrm{n}$ (error constant)      68
3M method      437—438 447—448 502
3M method, error analysis      444—446
Aasen's method      222—225
Aasen's method, growth factor      224
Aasen, Jan Ole      214 222 224 227
Abdelmalek, Nabih N.      376 403
Abramowitz, Milton      30
Absolute error      3
Absolute norm      107
Accuracy versus precision      6 28
ACRITH      483
Acton, Forman S.      30 195 283 486
Adams, Duane A.      103
Adjugate matrix      282
Aelfric      xxi
Aggarwal, Vijay B.      76
Ahac, Alan A.      190
Ahlberg, J.H.      154
Aitken extrapolation      91
Aitken, Alexander Craig      91 374
Albers, Donald J.      511
Alefeld, Goeltz      483
Alexopolous, Aristides G.      49
Allen, Jr., Richard C.      76
Almacany, Montaha      415
Alt, H.      447
Alternating directions method      474—475
Aluru, Srinivas      446
Alvarado, Fernando L.      153
Amato, James J.      317
Amodio, Pierluigi      190
Anda, Andrew A.      376
Anderson, Edward      287 397 404
Anderson, Iain J.      88
Anderson, T.W.      524 525
Ando, T.      188
Antonov, A.G.      285
Arioli, Mario      130 301 324 337 402—404 413 414
Arnold, William F.      318
Ashcraft, Cleve      188 219 227
Ashenhurst, R.L.      486
Asplund, Edgar      302
Atanasoff, John V.      139
Atlas      579
Augment precompiler      483 501
Augmented system      383
Augmented system, iterative refinement on      389
Augmented system, scaling and conditioning      391
Automatic Computing Engine (ACE)      53 185 188 337
Automatic differentiation      485
Automatic error analysis      471—487; see also “Interval analysis” “Running
Automatic error analysis using direct search optimization      472—474
Automatic error analysis, condition estimation      477—478
Automatic error analysis, roots of a cubic      479—481
Automatic error analysis, Strassen's inversion method      478—479
Axelsson, Owe      337
Babuska, Ivo      90
Bachelis, Boris      50
Back substitution      140
Backward error      6—7
Backward error analysis in differential equations      29
Backward error analysis, development by Wilkinson      29—30 185—186
Backward error analysis, motivation      6
Backward error analysis, not a panacea      1
Backward error analysis, purpose      65 195
Backward error columnwise      122
Backward error linear system, Oettli — Prager theorem      122
Backward error linear system, Rigal — Gaches theorem      12 120
Backward error Lyapunov equation      311—312
Backward error, componentwise      122 128
Backward error, componentwise relative      122
Backward error, componentwise, evaluating      130
Backward error, definition      6
Backward error, least squares problem      392—395 406
Backward error, mixed forward-backward error      7 456
Backward error, normwise      120
Backward error, normwise relative      120
Backward error, preserving symmetric structure      394—395
Backward error, row-wise      122
Backward error, structured      129
Backward error, Sylvester equation      308—311
Backward error, symmetric structure, preserving      136
Backward error, underdetermined system      411
Backward stability, componentwise      129
Backward stability, definition      7
Backward stability, normwise      129
Backward stability, row-wise      130
Bai, Zhaojun      317 346 397 404 524
Bailey, David H.      436 446—448 457 478 489 501 502
Baksalary, J.K.      318
Balle, Susanne M.      448
Ballester, C.      429
Banded matrix, growth factor      173
Bank, Randolph E.      257
Bareiss, E.H.      49 189
Bargmann, V.      183
Barlow, Jesse L.      29 47 49 181 189 299 376 397
Barnett, S.      317
Barone, John L.      381 402
Barrett, Geoff      56
Barrlund, Anders      181 190 209 405
Bartels — Stewart method      307—308
Bartels, R.H.      307
Bartels, Sven G.      129 301 429
Base of floating point arithmetic      36
Bau III, David      169 337
Bauer's scaling theorem      127 133
Bauer, F.L.      53 76 107 114 127 133 135 177 281 547
Beam, Richard M.      525
Beaton, Albert E.      381 402
Bell, E.T.      279
Bellman, Richard      318
Benford, Frank      47
Benoit, Commandant      209
Benschop, N.F.      323
Berman, Abraham      133 152
Bhatia, Rajendra      317 374
Bilinear noncommutative matrix multiplication algorithm      436—437
Bilinear noncommutative matrix multiplication algorithm, error analysis      443—444
Binary-decimal conversion      57
Bini, Dario      429 443 444
Birkhoff, Garrett      279 479
Bischof, Christian H.      297—299 363 376
Bit      56
Bjoerck, Ake      76 241 353 371 375—377 379 380 386 388 389 391 391 392 399 402—405 413 414 423 423 425 429 565
Bjorstad, Petter      448
Blanch, G.      505
BLAS (Basic Linear Algebra Subprograms)      578
BLAS (Basic Linear Algebra Subprograms), Extended and Mixed Precision      64 241 462 503 579
BLAS (Basic Linear Algebra Subprograms), fast level      3 447
BLAS (Basic Linear Algebra Subprograms), Technical Forum Standard      503 579
BLAS (Basic Linear Algebra Subprograms), xnrm2 (2-norm)      499—500 507
Bliss, B.      485
Block algorithm, advantages of      245
Block algorithm, definition      246
Block diagonal dominance      251—255 257
Block diagonal dominance and block LU factorization      252—255
Block diagonal dominance, definition      251
Block LDLl factorization (of skew-symmetric matrix)      225—226
Block LDLl factorization (of skew-symmetric matrix), growth factor      226
Block LDLT factorization (of symmetric matrix)      214—222
Block LDLT factorization (of symmetric matrix) for tridiagonal matrix      221—222
Block LDLT factorization (of symmetric matrix), complete pivoting and its stability      215—216
Block LDLT factorization (of symmetric matrix), growth factor, complete pivoting      216
Block LDLT factorization (of symmetric matrix), growth factor, partial pivoting      218
Block LDLT factorization (of symmetric matrix), growth factor, rook pivoting      221
Block LDLT factorization (of symmetric matrix), partial pivoting and its stability      216—219
Block LDLT factorization (of symmetric matrix), rook pivoting and its stability      219—221
Block LU factorization      246 247
Block LU factorization computation      247
Block LU factorization error analysis      250—256
Block LU factorization existence and uniqueness      247
Block LU factorization stability for (point) diagonally dominant matrix      254—255
Block LU factorization stability for block diagonally dominant matrix      251—255
Block LU factorization stability for block tridiagonal matrix      257
Block LU factorization stability for symmetric positive definite matrix      255—256
Blue, James L.      499
Bodewig, E.      185
Bohlender, Gerd      88
Bohte, Z.      173
Boley, Daniel      242
Bollen, Jo A.M.      323
Bondeli, S.      522
Boros, T.      429
Borwein, J.M.      489
Borwein, P.B.      48
Bowden, B.V.      19
Bowdler, H.J.      188
Boyd, David W.      289 291 301
Boyle, Jeff      47
Brent, Richard P.      47 150 151 436 439 440 447 483 501 505
Brezinski, Claude      209
Briggs, William L.      456
Brightman, Tom      56
Brown, W.S.      495 498 49
Brunet, Marie-Christine      485 486
Buchan, John      57
Buchanan, James L.      152
Buchholz, W.      56
Bukhberger, B.      302
Bulirsch, R.      76 187
Bunch — Kaufman partial pivoting strategy      216—219
Bunch — Parlett complete pivoting strategy      215—216
Bunch, James R.      129 133 136 213 215 216 221 225—227 231
Buoni, John J.      190
Burdakov, Oleg      282
Burgmeier, James W.      76
Businger, Peter A.      133 180 403
Butcher, J.C.      90
Byers, Ralph      302 315 318 319 346 562
BYTE      56
Caffney, John      518
Calculator, displaying words on      32
Calvetti, D.      428—430
Calvin (and Hobbes)      471
Campbell, S.L.      332
Campbell-Kelly, Martin      245
Cancellation      9—10 27
Cancellation in summation      83 539
Cancellation not a bad thing      10
Cancellation of rounding errors      19—22
Cao, Wei-Lu      302
Caprani, Ole      90
Cardano, Geronimo      479
Carr III, John W.      53
Carter, Russell      493 494
Cauchy matrix      514—515
Cauchy matrix determinant      515
Cauchy matrix inverse      515
Cauchy matrix LU factors      515
Cauchy — Schwarz inequality      106
Cauchy, August in-Louis      515
Cayley, Arthur      434
CELEFUNT package      496
CESTAC      486
Chaitin-Chatelin, Francoise      351 468 485 486
Chan, N.N.      524
Chan, Raymond H.      457
Chan, Tony F.      11 29 133 134 189 377
Chandrasekaran, Shivkumar      123 129 377
Chang, Xiao-Wen      190 209 377
Chartres, Bruce A.      186
Chatelin, Francoise      see “Chaitin-Chatelin Frangoise”
Chebyshev spectral differentiation matrix      340 348
Cheng, Sheung Hun      219 221 224 227 295
Cho, Choong Yun      515
Choi, Man-Duen      511 523
Cholesky factorization      196
Cholesky factorization conditions for success in floating point      200
Cholesky factorization error analysis      197—200
Cholesky factorization existence and uniqueness      196
Cholesky factorization perturbation bounds      201
Cholesky factorization semidefinite matrix, complete pivoting      202
Cholesky factorization semidefinite matrix, computation of      202
Cholesky factorization semidefinite matrix, error analysis      205—208
Cholesky factorization semidefinite matrix, existence and uniqueness      201
Cholesky factorization semidefinite matrix, perturbation theory      203—205
Cholesky factorization semidefinite matrix, termination criteria      207—208
Cholesky factorization, computation of      197
Cholesky, Andre-Louis      209
Chopping      54
Christiansen, Soren      133
Chu, Eleanor      181
Chu, King-wah Eric      318
Chu, Moody T.      525
Circulant matrix      454
Circulant system, error analysis for solution by FFT      454—456
Clasen, B.-L.      281
Clenshaw, C.W.      49 102
Cline, Alan K.      287 295 297 404
Cline, R.E.      413
Clinger, William D.      57
Codenotti, B.      282
Cody, Jr., William J.      51 55 56 493 495 496 501
Cohen, A.M.      169 170 520
Colon notation      2
Commutation matrix      317
Companion matrix      522—523
Companion matrix, singular values      523
Comparison matrix      145
Compensated summation      83—88
Complete pivoting      158
Complete pivoting for symmetric indefinite matrices      215—216
Complete pivoting growth factor      169—170 189
Complete pivoting growth factor, conjecture proved false      170
Complete pivoting, early use of      188
Complete pivoting, fallacious criticism of      193
Complete pivoting, use of, in practice      188 562
Complex arithmetic, error analysis      71—73 77
Complex number division without overflow      500 506
Complex number, square root of      32
Componentwise relative error      4
Concus, P.      257
Condition number minimizing by scaling      123 125—127 133
Condition number of function      8
Condition number of linear system componentwise      123
Condition number of linear system normwise      121
Condition number of matrix (rectangular)      382
Condition number of matrix (square)      109 110 114
Condition number of nonlinear system      464—467
Condition number of summation      91
Condition number, distance to singularity and      111 114 127
Condition number, estimation      287—304
Condition number, estimation for tridiagonal matrices      299—301
Condition number, estimation, asymptotic cost      288
Condition number, estimation, block 1-norm estimator      294—295
Condition number, estimation, counterexamples      287 288 292—294 297 302
Condition number, estimation, counterexamples by direct search      477—478
Condition number, estimation, incremental      298
Condition number, estimation, LAPACK estimator      292—294 477—478
Condition number, estimation, LINPACK estimator      295—297
Condition number, estimation, probabilistic methods      297—298
Condition number, general theory      29
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