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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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Curtis, A. R 197 Cybenko, George 225 Cyclic reduction 197 Dahlquist, Germund 83 160 Daniel, J. W. 385 Datta, Karabi 315 Davis, Philip J. 32 100 471 Dax, Achiya 226 337 Day, Jane M. 197 de Boor, Carl 32 190 196 de Jong, Lieuwe Sytse 33 de Rijk, P. P. M. 410 411 Dekker, T. J. 57 93 284 504 del Ferro, Scipione 483 Demeure, Cedric J. 441 Demmel, James W. 47 53 60 61 84 126 140 141 143 149 165 198 207 208 224 243 248 250—253 257 304 322 352 417 422 486 495 499 504 527 536 Dennis, Jr., J. E. 32 328 477 478 Denormalized numbers see “Subnormal numbers” Departure from normality (Henrici’s) 351—352 Descloux, J. 343 Determinant 281—283 Determinant of upper Hessenberg matrix 27—28 Determinant, computation of 282—283 Determinant, condition number of 287 Dhillon, Inderjit 60 536 Diagonal dominance 181 Diagonal dominance and block LU factorization 255 Diagonal dominance, block 251—255 257 Diagonal dominance, bound for LU factors of tridiagonal matrix 185 Diagonal dominance, growth factor 181 Diagonal dominance, matrix inverse bound 167 Diagonal dominance, matrix inverse bound for tridiagonal matrix 303 Diagonal pivoting method 218—223 Diagonal pivoting method, complete pivoting and its stability 219—220 Diagonal pivoting method, growth factor, complete pivoting 220 Diagonal pivoting method, growth factor, partial pivoting 222 Diagonal pivoting method, partial pivoting and its stability 221—223 Diagonally dominant matrix, bound for inverse 167 Diagonally dominant matrix, growth factor for 181 Diamond, Harold G. 62 Differential equations see “Ordinary differential equations; partial differential equations references Direct search optimization methods 477—479 Discretization error 6 Distance to singularity, componentwise 140 Distance to singularity, normwise 123 Divided differences 109—112 Divided differences, confluent 430 Dixon, John D. 300 Dongarra, Jack J. 188 195 225 231q 257 499 581 Doolittle, Myrick Hascall 195 Doolittle’s method 173—174 195 Dorn, William S. 83 100 113 Double rounding 48 63 541 Douglas, Craig C 460 575 Douglas, Jr., Jim 32 188 Doyle, Sir Arthur Conan 289q Drake, J. B. 261q Drazin inverse 336—337 Drift, in floating point arithmetic 58 Drmac, Zlatko 225 Du Croz, Jeremy J. 265 272 273 284 Dual norm 119 Dual vector 119 Dubrulle, Augustin A. 285 511 Duff, Iain S. 143 195 200 225 226 257 304 343 410 411 527 Duncan, Martin 96 Dunham, C. B. 425q Dwyer, Paul S. 169 q Dynamical systems, references for rounding error analysis 32 Eckart, Carl 126 Edelman, Alan 63 162 180 181 197 198 287 386 483n 518 518q 527 534 Effective conditioning 146 Eirola, Time 33 EISPACK 587 Elden, Lars 412 Eldersveld, Samuel K. 257 ELEFUNT package 499 Elfving, Tommy 441 Emel’yanenko, G. A. 305 Enright, Wayne H. 33 Equilibration 136 138 192 Erisman, A. M. 190 193 200 Error analysis see “Rounding error analysis” Error, absolute 4 Error, backward see “Backward error” Error, forward see “Forward error” Error, mixed forward-backward error 8 Error, relative 4 5 Error, sources of 5—6 Espelid, Terje O. 100 ESSL library (IBM) 448 450 Expm 1 function 34 Faddeeva, V. N. 195 Fairgrieve, Thomas F. 61 530 Fan, Ky 386 Fan-in algorithm 165 Fan-in algorithm for summation 88 Fan-in algorithm for triangular system solution 162—164 Farebrother, R. W. 409 Farkas, I. 113 Farnum, Charles 497 Fast Fourier Transform 465—471 Fast Fourier transform for solving circulant systems 468—470 Fast Fourier transform, Cooley — Tukey factorization of DFT matrix 466 Fast Fourier transform, error bound 467 Fast matrix multiplication 445—463 Fast matrix multiplication in the level 3 BLAS 460 Fast matrix multiplication, , 3M method for complex multiplication 450 Fast matrix multiplication, bilinear noncommutative algorithm 449—450 Fast matrix multiplication, deriving methods 459—460 Fast matrix multiplication, error analysis 450—459 Fast matrix multiplication, Miller’s error results 451 Fast matrix multiplication, record exponent 448 Fast matrix multiplication, Strassen’s method 446—448 Fast matrix multiplication, Strassen’s method, Winograd’s variant 448 455—456 Fast matrix multiplication, Winograd’s method 446 Fateman, Richard J. 59 Feingold, David G. 257 Feldstein, A. 62 Ferguson, H. R. P. 461 482 Ferguson, Jr., Warren E. 49 60 Ferng, William R. 301 Fike, C. T. 114 Fischer, Patrick C. 460 Fixed point arithmetic 56 Fl operator (rounding) 42 Flannery, Brian P. 479 490 507 Fletcher, R. 32 147 149 225 226 Floating point arithmetic 39—65 Floating point arithmetic, alternatives to 53—54 Floating point arithmetic, banned from safety-critical systems 499 Floating point arithmetic, binary-decimal conversion 61—62 Floating point arithmetic, choice of base 51 60—61 Floating point arithmetic, compiler optimization, dangers of 497 Floating point arithmetic, determining properties of 497—498 Floating point arithmetic, drift in 58 Floating point arithmetic, earliest subroutines 39q Floating point arithmetic, formal op algebra 58 Floating point arithmetic, fused multiply-add operation 60 65 Floating point arithmetic, IEEE arithmetic see “IEEE arithmetic” Floating point arithmetic, Language Independent Arithmetic (LIA-1) 502 Floating point arithmetic, model 58 501—502 Floating point arithmetic, model with underflow 61 Floating point arithmetic, model without guard digit 49 Floating point arithmetic, model, Brown’s 498 501—502 Floating point arithmetic, model, standard 44 Floating point arithmetic, multiple precision 504—506 Floating point arithmetic, parameters for selected machines 41t Floating point arithmetic, parameters in software, specifying 499—500 Floating point arithmetic, representation error 51 Floating point arithmetic, rounding see “Rounding” Floating point arithmetic, software issues 491—512 Floating point arithmetic, speed of operations (relative) 60 Floating point arithmetic, subnormal numbers 41 47 495 Floating point arithmetic, subtraction done exactly 49—50 Floating point arithmetic, testing accuracy of 54—56 Floating point arithmetic, testing correctness of 498—499 Floating point arithmetic, unit roundoff 3 42 Floating point arithmetic, wobbling precision 43 51 Floating point coprocessor 47 Floating point numbers, characterization 40 Floating point numbers, normalized 40 Floating point numbers, spacing between 41 Floating point numbers, subnormal 41 47 Floating point numbers, testing for equality 495 FLOP 3 Forsgren, Anders 225 Forsythe, George E. 32 33 35 52 57 84 95 138 146 164 190 196 197 235 241 242 245q 261q 262 282 305 325q 491q 526 Fortran 90 3 Fortran, environmental inquiry functions 498 Fortran, matmul 460 Forward error 7—8 Forward error for linear system 13 Forward error, definition 7 Forward error, mixed forward-backward error 8 Forward stability, componentwise 142 Forward stability, definition 10 Forward stability, normwise 142 Foster, Leslie V. 178 411 Foulser, David E. 146 147 Fourier matrix 179 Fox, L. xxviiq xxviii 35 117n 188—190 FPV (floating point verification) package 498—499 Fraysse, Valerie 53 358 Friedland, Shmuel 348 359 Frobenius norm 120 Funderlic, R. E. 198 Fused multiply-add operation 60 65 Gaches, J. 132 145 Gahinet, Pascal M. 323 Gal, Shmuel 61 Gallopoulos, E. 488 Gander, Walter 525 Gantmacher, F. R. 172 Gardiner, Judith D. 323 Gardner, Martin 127q Garner, Harvey L. 62 Gasca, M. 196 Gastinel, Noel 123 126 Gauss — Jordan elimination 275—281 Gauss — Jordan elimination, algorithm 276 Gauss — Jordan elimination, error analysis 277—281 Gauss — Seidel method 325q 334 Gauss, Carl Friedrich 1q 195 219 325q 391 Gaussian elimination 170—174 (see also “LU factorization”) Gaussian elimination in ancient China 195 Gaussian elimination without pivoting, instability of 17 Gaussian elimination, a posteriori stability tests 192—194 Gaussian elimination, complete pivoting 170 Gaussian elimination, computer programs, first 195—196 Gaussian elimination, computer programs, history of 196 Gaussian elimination, connection with LU factorization 171 Gaussian elimination, error analysis 174—177 Gaussian elimination, error analysis, history of 186—191 Gaussian elimination, growth factor 177—183 (see also “Growth factor”) Gaussian elimination, loop orderings 195 Gaussian elimination, need for pivoting 170 Gaussian elimination, on Hessenberg matrix 27—28 Gaussian elimination, partial pivoting 170 173 Gaussian elimination, pessimism of its accuracy in 1940s 186—187 Gaussian elimination, row and column scaling 191—192 Gaussian elimination, threshold pivoting 200 Gaussian elimination, use by Gauss 195 Gaussian elimination, versus Cramer’s rule 14—15 Gautschi, Walter 425q 428 429 Gautschi, Werner 351 Gay, David M. 61 Geist, G. A. 261q Gelfand’s problem 51 Geman, Stuart 518 Gentle, James E. 34 Gentleman, W. Morven 113 384 385 470 576 Geometric computation, accuracy of algorithms in 34 George, Alan 193 224 226 Geuder, James C 190 Ghavimi, Ali R. 320 322 Gill, Philip E. 32 225 226 229 422 479 Gill, S. 92 Givens rotation 371 Givens rotation, disjoint rotations 373—375 387 Givens rotation, fast 385 Givens, Wallace J. 33 67q Gluchowska, J. 411 Gohberg, I. 141 440 516 Goldberg, David 39q 57 93 534 Goldberg, I. Bennett 62 Goldstine, Herman H. 1n 33 187 196 261q 263 517 Golub, Gene H. xxiv 13 27 33 146 182 190 195 223 231q 257 285 301 312 327 352 384 386 388 391 392 400 409—412 441 580 Goodman, R. 62 Goodnight, James H. 285 Gould, Nicholas I. M. 181 197 Govaerts, W. 242 Gradual underflow 47 61 Gragg, W. B. 297 385 Graham, Ronald L. 87q 520q Gram — Schmidt method 376—381 Gram — Schmidt method, classical algorithm 377 Gram — Schmidt method, classical error analysis 378—379 381 Gram — Schmidt method, modified algorithm 377 Gram — Schmidt method, modified connection with Householder QR factorization 36lq 379 385 Gram — Schmidt method, modified error analysis 378—381 Gram — Schmidt method, modified error analysis for application to LS problem 396—397 Gram — Schmidt method, modified stability 27 Gram — Schmidt method, reorthogonalization 385 Grebogi, Celso 32 Greenbaum, A. 328 329 Gregory, Robert T. 514 525 Grimes, Roger G. 305 527 Grosse, Eric 581q Growth factor 177—183 Growth factor for banded matrix 183 Growth factor for complete pivoting 180—181 Growth factor for diagonal pivoting method 220 222 Growth factor for diagonally dominant matrix 181 Growth factor for partial pivoting 177—183 Growth factor for random matrices 196—197 Growth factor for tridiagonal matrix 183 Growth factor for upper Hessenberg matrix 182 Growth factor, a posteriori estimates for 193 Growth factor, define using exact or computed quantities? 177 196 Growth factor, large growth in practical problems 178 Growth factor, lower bound for 179 Growth factor, maximization by direct search 475—476 Growth factor, numerical maximization for complete pivoting 181 197 Growth factor, statistical model of 180 Gu, Ming 352 536 Guard digit 48 Guard digit, test for 56 Gudmundsson, Thorkell 301 Guggenheimer, Heinrich W. 287 Gulliksson, Marten 412 Gurwitz, Chaya 32 Gustafson, John L. 459 Gustavson, F. G. 195 Haar distribution, random orthogonal matrix from 519—520 Hadamard condition number 281 287 Hadamard matrix 128 179 181 201 Hadamard’s inequality 287 Hager, William W. 294 304 Hall, Jr., Marshall 179
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