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Higham N. — Accuracy and stability of numerical algorithms
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Íàçâàíèå: Accuracy and stability of numerical algorithmsÀâòîð: Higham N. Àííîòàöèÿ: Treats the behavior of numerical algorithms in finite precision arithmetic, combining algorithmic derivations, perturbation theory, and rounding error analysis and emphasizing software practicalities, with particular reference to LAPACK and MATLAB. Includes historical perspectives, especially on the work of Wilkinson and Turing, with quotations introducing chapters on subjects such as floating point summation, condition number estimation, and the Sylvester equation. Although designed as a reference rather than a text, it includes problems and solutions. Annotation c. by Book News, Inc., Portland, Or.
ßçûê: Ðóáðèêà: Ìàòåìàòèêà /×èñëåííûå ìåòîäû /×èñëåííûé àíàëèç /Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö ed2k: ed2k stats Ãîä èçäàíèÿ: 1996Êîëè÷åñòâî ñòðàíèö: 688Äîáàâëåíà â êàòàëîã: 23.02.2005Îïåðàöèè: Ïîëîæèòü íà ïîëêó |
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Raimi, Ralph A. 52 Ramos, George U. 470 Random matrices 517—520 Random matrices with given singular values 519—520 Random matrices, 2-norm of 518 Random matrices, condition number of 518 Random matrices, expected number of real eigenvalues 518 Random matrices, orthogonal 519 527 Random matrices, spectral radius of 518 Random matrices, tend to be well conditioned 518 Randsvd matrix 519—520 527 Range reduction 54 Rath, Wolfgang 385 Ratz, Deitmar 487 Ratz, H. C 327 Razaz, M. 113 RCOND condition estimator (UNPACK, MATLAB) 305 480—481 489 Reichel, Lothar 113 358 440—442 527 Reichelt, Mark W. 527 Reid, J. K 3 190 193 197 200 226 411 Reinsch, C. xxviii 284 585 Reiser, John F. 58 Relative error 4 5 Relative error counter, <k> 75 Relative error, componentwise 5 Relative precision 76 Relative residual 14 Ren, Huan 60 Research problems 102 115 201 229 244 287 308 324 359 423 442 463 490 527 528 Residual, relative 14 Rew, R. K. 289q 299 480 Riccati equation, algebraic 322 Rice, John R. 33 385 Richtmyer, Robert D. 32 Rigal — Gaches backward error theorem 132 Rigal, J. L. 132 145 Robertazzi, T. G. 98 Roberts, J. D. 322 Rohn, Jiri 128 140 141 149 Romani, Francesco 328 Rose, Donald J. 257 Rosenthal, Peter 322 Ross, D. R. 97 Rounding 4 42 Rounding error analysis, automatic 473—490 Rounding error analysis, demystified 82—83 Rounding error analysis, graphs in 83 Rounding error analysis, model with underflow 61 Rounding error analysis, model without guard digit 49 Rounding error analysis, model, standard 44 Rounding error analysis, notation 73—76 Rounding error analysis, ordering of operations, effect of 77 154 Rounding error analysis, purpose of 71—72 203q Rounding error analysis, statistical approach 52—53 Rounding errors are not random 29 52 Rounding errors in subtraction 49—50 Rounding errors, accumulation of 16 Rounding errors, beneficial effects of 26—27 Rounding errors, cancellation of 21—26 Rounding errors, statistical assumptions on 52—53 Rounding to even versus to odd 58 Rounding, dealing with ties 42 58 Rowan, Thomas Harvey 488 489 Rubin, Donald B. 391q 410 Ruhe, Axel 385 Ruiz, Daniel 343 Rules of thumb, condition for computed powers of matrix to converge to zero 358 Rules of thumb, forward error related to backward error and condition number 10 Rules of thumb, relative speed of floating point operations 60 Rules of thumb, square root of constants in error bound 52 Runge — Kutta method 92 100 Running error analysis 73 489 Running error analysis for continued fraction 85 Running error analysis for Homer’s method 105—106 113 Running error analysis for inner product 72—73 Rust, B. W. 145 Rutishauser, Heinz 514 520 Sameh, Ahmed H. 162 164 488 Samelson, Klaus 57 Sample variance see “Variance” Sande, G. 470 Sanz-Serna, J. M. 32 33 Saunders, Michael A. 225 226 229 257 422 Sautter, Werner 190 Scalapack 588 Scaling a linear system before Gaussian elimination 191—192 197—198 Scaling to minimize the condition number 136—139 191 Scarborough, James B. 33 58 Schaumburg, Kjeld 305 Schelin, Charles W. 61 Scherer, R. 84 Schnabel, Robert B. 32 Schneider, Hans 124 359 Schonfelder, J. L. 113 Schreiber, Robert S. 165 180 245q 250—253 257 371 384 576 Schryer, N. L. 498 Schur complement 219 224 247 252 Schur complement, perturbation bounds for symmetric positive semidefinite matrix 212—218 Schwartz, S. C 98 Schwetlick, Hubert 84 406 411 418 422 Searle, Shayle R. 323 490 Second difference matrix 525 Semiconvergent matrix 337 Seminormal equations for least squares problem 403—404 Seminormal equations for underdetermined system 417 Separation (sep), of two matrices 318 Sha, Xuan-He 450 459 Shampine, Lawrence F. 32 33 84 95 527 Shannon, Claude E. 60 Shapiro, Alexander 146 Shepherd, David 59 Sherman — Morrison formula 197 490 Shinnerl, Joseph R. 225 229 Shroff, Gautam M. 301 Shub, Michael 147 518 Significance arithmetic 489 Significant digits, correct 4—5 32 Significant digits, least and most significant 40 Simon, Horst D. 460 Singular value decomposition (SVD) 580 Skeel, Robert D. 36 135 146 190 192 198 235 240 241 486 Slishman, Gordon 460 575 Smale, Steve 2n Smith, David M. 505 Smith, Francis J. 113 439 Smith, Jon M. 37 Smith, Robert L. 503 509 Smith, Roger M. 460 575 Smoktunowicz, Alicja 94 141 149 242 Snyder, James N. 231q 241 Software, avoiding underflow and overflow 502—504 Software, effects of underflow 504 Software, issues in floating point arithmetic 491—512 Software, portability 499—502 Software, specifying arithmetic parameters 499—500 Software, specifying numerical constants 501 Sokolnicka, Jolanta 242 SOR method, forward error analysis 334 Sorensen, Danny C 195 225 229 257 Sorevik, T. 461 Spellucci, P. 32 Spencer, Herbert 665 Spieb, J. 460 Spooner, David 488 Square root, of complex number 36 Stable algorithms, designing 30—31 Stationary iterative methods 325—343 Stationary iterative methods and powers of a matrix 358 Stationary iterative methods, backward error analysis 334—336 Stationary iterative methods, forward error analysis 329—334 Stationary iterative methods, forward error analysis, singular systems 338—341 Stationary iterative methods, Jacobi method 332—334 Stationary iterative methods, scale independence 331 Stationary iterative methods, singular systems, theory for 336—338 Stationary iterative methods, SOR 334 Stationary iterative methods, stopping criteria 341—342 Statistics 34 (see also “Variance computational references”) Steele, Jr., Guy L. 61 Steer, J. 83 84 119 126 127 190 Stegun, Irene A. 35 Sterbenz, Pat H. 33 34 50 57 60 489 Stewart, G W. (Pete) xxiv 75 113 119 126 131q 139 146 147 151q 164 174 195 197 198 224 231q 241 242 297 305 312 324 359 381 382 385 386 392 394 407 504 519 527 580 Stewart, William J. 305 Sticky bit 46 Stone, Betty Jane 126 Storey, C 323 Stoutemyer, David R. 489 Strakos, Zdenek 329 Strang, Gilbert 15 124 146 471 Strassen, Volker 446 461 462 Strassen’s method 446—448 Strassen’s method for inversion 461—462 481—482 Strassen’s method, accuracy compared with conventional multiplication 454—455 Strassen’s method, error analysis 452—456 Strassen’s method, error versus operation count 16 Strassen’s method, implementation issues 460 Strassen’s method, Winograd’s variant 448 455—456 Straus, E. G 138 146 Street, Anne Penfold 179 Stummel, Friedrich 34 83 191 Subdifferential, of a vector norm 291 Subgradient 292 Subnormal numbers 41 47 495 Summation 87—102 Summation, choice of method: summary 98—100 Summation, compensated and applications 92—97 Summation, condition number 100 Summation, criterion for minimizing error 90 Summation, distillation algorithms 98 Summation, doubly compensated 96—97 Summation, error analysis 89—92 Summation, insertion method 88 Summation, pairwise (fan-in) 88 Summation, recursive 88 Summation, recursive, ordering in 19 90—91 Summation, statistical error estimates 98 Sun, Ji-guang 119 126 131q 139 146 194 198 209 224 382 386 392 394 404 407 409 411 413 580 Sun, Xiaobai 384 SVD (singular value decomposition) 580 Swarm, W. H. 474 Swartzlander, Jr., Earl E. 54 Sweeney, D. W. 60 Swenson, J. R. 52 Sylvester equation 309—324 Sylvester equation, backward error 313—316 Sylvester equation, Bartels — Stewart method 311—313 Sylvester equation, generalizations 321—324 Sylvester equation, perturbation theory 318—320 Sylvester equation, practical error bounds 320—321 Sylvester equation, solution methods 311—313 Sylvester, James Joseph 309q 322 446 Symbolic manipulation package 6 Symmetric indefinite factorization 218 (see also “Diagonal pivoting method”) Symmetric indefinite matrix 218 Symmetric positive definite matrix 204 Symmetric positive definite matrix and block LU factorization 255—257 Symmetric positive definite matrix, practical test for 225 Symmetric positive semidefinite matrix 210 Symmetric positive semidefinite matrix, determinantal conditions for 228 Symmetric quasidefinite matrix 229 Synthetic division 107 Tablemaker’ s dilemma 5 Tang, Ping Tak Peterj 61 499 530 Tang, W. P. 441 Tao, Pham Dinh 128 304 Tartaglia, Niccolo 483 Taussky, Olga 513q 515 Test for accuracy of floating point arithmetic 54—56 Test for guard digit 56 Test matrices 513—528 Test matrices, Harwell — Boeing collection 527 Test matrices, Test Matrix Toolbox 591—594 Teukolsky, Saul A. 479 490 507 Thacher, Jr., Henry C 196 Thisted, Ronald A. 34 Thompson, Sir D’arcy Wentworth 1q Thron, W. J. 507 Tienari, Martti 53 Todd, John 126 514 520 526 527 Toeplitz matrix, pseudospectra 527 528 Toeplitz matrix, tridiagonal 524—525 Toh, Kim-Chuan 483n 526 Torczon, Virginia J. 477—479 Tornheim, L. 180 181 Totally nonnegative matrix 176 523 Totally nonnegative matrix, LU factorization 176 196 Totally nonnegative matrix, LU factorization, row scaling in 192 Totally nonnegative matrix, test for 196 Trammer, Manfred R. 32 346 355 358 Transformations, well conditioned 30 Transputer (Inmos), proof of arithmetic’s correctness 59 Traub, J. F. 440 Trefethen, Lloyd N. 6q 32 169q 180 328 345q 346 348 352 353 355 358 483n 526 527 Tremaine, Scott 96 Triangular matrix, bounds for inverse 159—161 Triangular matrix, condition numbers 155 Triangular matrix, inversion 265—270 Triangular matrix, inversion methods, blocked 267—270 Triangular matrix, inversion methods, unblocked 265—267 Triangular matrix, M-matrix 157 159 160 Triangular systems 151—168 Triangular systems, accurate solution of 151q 155 156 159 Triangular systems, conditioning 156—157 Triangular systems, fan-in algorithm 162—164 Triangular systems, partitioned inverse method 165 Triangular systems, substitution, backward error analysis 152—154 Triangular systems, substitution, forward error analysis 155—159 Tridiagonal matrix 183—186 Tridiagonal matrix, condition number estimation 301—303 Tridiagonal matrix, growth factor 183 Tridiagonal matrix, LU factorization 184 Tridiagonal matrix, LU factorization, error analysis of 184—186 Tridiagonal matrix, structure of inverse 303 305 Tridiagonal matrix, Toeplitz 524—525 Tropp, Henry S. 491n Truncation error 6 Tsao, Nai-kuan 84 384 Tukey, John W. 60 470 Turing Award of the ACM xxvii 59 Turing programming language 506 Turing programming language, Numerical Turing 506 Turing, Alan Mathison xxvii xxviii 33 126 131q 188 284 485 Turing, Alan Mathison, contributions in 1948 paper "Rounding-off errors..." 188 284 Turnbull, H. W. 383 523 Turner, Kathryn 225 Turner, Peter R. 51 53 165 Tyrtyshnikov, Evgenij E. 526 Ulp (unit in last place) 43 Uncertainty, in data 5 Underdetermined system 415—423 Underdetermined system, backward error 419 423 Underdetermined system, backward stability, definition 419 Underdetermined system, error analysis 421 Underdetermined system, modified Gram — Schmidt 421—422 Underdetermined system, perturbation theory 417—419 Underdetermined system, Q method (QR factorization) 416 Underdetermined system, Q method (QR factorization), error analysis 419—422 Underdetermined system, seminormal equations 417 Underflow 18 42 Underflow, avoiding 502—504
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