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Higham N.J. — Accuracy and Stability of Numerical Algorithms
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Название: Accuracy and Stability of Numerical AlgorithmsАвтор: 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.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Издание: 2Год издания: 2002Количество страниц: 680Добавлена в каталог: 19.06.2009Операции: Положить на полку |
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Condition number, Hadamard 279 282 284 Condition number, Skeel's 123 Conjugate gradient method 324 336 Conjugate gradient method, circulant preconditioner 457 Conn, Andrew R. 297 Conte, Samuel D. 187 Continued fraction, algorithms and error analysis 505 Continued fraction, evaluating in IEEE arithmetic 490—491 Continued fraction, running error bound 77 Convergent matrix 332 340 Conversion, binary-decimal 57 Cooley, James W. 456 Coomes, Brian A. 29 Coonen, Jerome T. 55 493 Cope, J.E. 132 Coppersmith, Don 436 Correct significant digits 3—4 28 Correlation matrix 525 Cortes, Joaquin 189 Cottle, Richard W. 209 Cox, Anthony J. 362 375 395 396 398 400 404 405 Cramer's rule, (in)stability of 13—14 30 33 Cray computers 669 Cray computers, adoption of IEEE arithmetic 44 Cray computers, arithmetic on 35 493 Cray computers, puzzling results from Cray Y-MP and Cray 2 493—494 Cray computers, UNICOS library 436 438 Crout's method 163 Crout, Prescott D. 187 Cryer, Colin W. 169 170 188 189 CS decomposition 380 400 Csanky, L. 278 Cubic equation, Newton's method 486—487 Cubic equation, stability of explicit formulae for roots 479—481 Curtis, A.R. 190 Cuyt, Annie 496 Cybenko, George 210 Cyclic reduction 190 Dahlquist, Germund 76 147 Daniel, J.W. 376 Datta, Karabi 311 Davies, Philip L. 525 Davis, Philip J. 90 457 Dax, Achiya 227 332 Day, David 524 Day, Jane M. 189 de Boor, Carl 186—188 de Jong, Lieuwe Sytse 29 de Rijk, P.P.M. 402—404 Dekker, T.J. 53 84 282 501 del Ferro, Scipione 479 Demeure, Cedric J. 429 Demmel, James W. 42 49 56 57 77 90 115 128—130 136 152 182 188 191 198—200 209 232 242 249—253 256 257 282 288 301 317 337 346 409 413 414 483 492 496 501 515 524 534 Dennis, Jr., John E. 323 468 475 476 Denormalized numbers see “Subnormal numbers” Departure from normality (Henrici's) 344—345 Descloux, J. 469 Determinant 279—280 Determinant perturbation bound 285 Determinant, computation of 279—280 Determinant, of upper Hessenberg matrix 24—25 Determinant, testing sign of 282 Dhillon, Inderjit S. 56 301 304 534 Diagonal dominance 172; see also “Block diagonal dominance” Diagonally dominant matrix and block LU factorization 254—255 Diagonally dominant matrix, Gaussian elimination, error bounds for 177 Diagonally dominant matrix, growth factor, bound for 172 Diagonally dominant matrix, inverse by Gauss — Jordan elimination, error bound for 277 Diagonally dominant matrix, inverse, bound for 154 Diagonally dominant matrix, nonsingularity of 190 Diagonally dominant matrix, triangular, bound for cond 144 Diagonally dominant matrix, tridiagonal, bound for inverse 300 Diagonally dominant matrix, tridiagonal, bound for LU factors 175 Diament, Benjamin 288 Diamond, Harold G. 57 Differential equations see “Ordinary differential equations” Direct search optimization methods 474—477 Discretization error 5 Distance to singularity, componentwise 127 Distance to singularity, normwise 111 Divided differences 21 99—101 Divided differences, confluent 419—420 Dixon, John D. 298 Dongarra, Jack J. 185 187 226 231 232 256 397 404 496 524 573 Doolittle's method 162—163 187 Doolittle, Myrick Hascall 187 Dorn, William S. 76 90 102 Double rounding 43 58 Douglas, Craig C. 446 569 Douglas, Jr., Jim 185 Doyle, Sir Arthur Conan 287 Drake, J.B. 25 Drazin inverse 331—332 Drift in floating point arithmetic 54 Drmac, Zlatko 210 Du Croz, Jeremy J. 262 269 281 Dual norm 107 Dual vector 107 Dubrulle, Augustin A. 282 507 Duff, Iain S. 131 187 193 226 227 256 301 337 402—404 524 Dumitrescu, Bogdan 447 Duncan, Martin 87 Dunham, C.B. 415 Dwyer, Paul S. 15 Eckart, Carl 114 Edelman, Alan 59 150 169 170 189 191 284 377 480 516 516 524 532 Effective conditioning 133 Egecioglu, Oemer 103 Eijkhout, Victor 232 Eirola, Timo 29 EISPACK 579 Elden, Lars 396 404 Eldersveld, Samuel K. 257 ELEFUNT package 496 Elementary functions 50 Elfving, Tommy 429 Emel'yanenko, G.A. 302 Enright, Wayne H. 29 Equilibration 123 126 178 Erisman, A.M. 180 181 186 193 Error analysis see “Rounding error analysis” Error, absolute 3 Error, mixed forward-backward error 7 456 Error, relative 3 4 Error, sources of 5—6 Espelid, Terje O. 90 ESSL library (IBM) 436 438 Euler's method, with compensated summation 86 expml function 30 Faddeeva, V.N. 188 Fairgrieve, Thomas F. 50 528 Fallat, Shaun M. 189 Fan, Ky 377 Fan-in algorithm 152 Fan-in algorithm for summation 80 Fan-in algorithm for triangular system solution 149—151 Farebrother, R.W. 402 Farkas, I. 103 Farnum, Charles 494 494 Fast Fourier Transform 451—457 Fast Fourier transform for solving circulant systems 454—456 Fast Fourier transform, Cooley — Tukey factorization of DFT matrix 452 Fast Fourier transform, error bound 453 Fast Fourier transform, inverse transform 454 Fast matrix multiplication 433—449 Fast matrix multiplication in the level-3 BLAS 447 Fast matrix multiplication, 3M method for complex multiplication 437—438 Fast matrix multiplication, bilinear noncommutative algorithm 436—437 Fast matrix multiplication, deriving methods 446 Fast matrix multiplication, error analysis 438—446 Fast matrix multiplication, Miller's error results 438 Fast matrix multiplication, record exponent 436 Fast matrix multiplication, Strassen's method 434—436 Fast matrix multiplication, Strassen's method, Winograd's variant 435—436 442—443 Fast matrix multiplication, Winograd's method 434 Fateman, Richard J. 55 Feingold, David G. 257 Feldstein, A. 57 Ferguson, H.R.P. 448 478 Ferguson, Jr., Warren E. 45 56 Ferng, William R. 299 FFT see “Fast Fourier transform” FFTW 457 Fike, C.T. 93 104 Fischer, Patrick C 446 Fixed point arithmetic 53 Fl operator (rounding) 38 Flannery, Brian P. 476 487 505 Fletcher, Roger 133 136 188 210 227 Floating point arithmetic 35—60 Floating point arithmetic base 36 Floating point arithmetic base, choice of 47 56 Floating point arithmetic chopping 54 Floating point arithmetic guard digit see “Guard digit” Floating point arithmetic mantissa 36 Floating point arithmetic model 54 498—499 Floating point arithmetic model with underflow 56—57 Floating point arithmetic model without guard digit 44 Floating point arithmetic model, Brown's 495 498—499 Floating point arithmetic model, complex arithmetic 71 Floating point arithmetic model, standard 40 Floating point arithmetic multiple precision 501—503 Floating point arithmetic parameters for selected machines 3 Floating point arithmetic parameters in software, specifying 496—497 Floating point arithmetic representation error 47 Floating point arithmetic rounding see “Rounding” ”Double Floating point arithmetic significand 36 Floating point arithmetic software issues 489—509 Floating point arithmetic speed of operations (relative) 56 Floating point arithmetic subnormal numbers 37 42 492 Floating point arithmetic subtraction done exactly 45 Floating point arithmetic unit roundoff 3 38 Floating point arithmetic wobbling precision 39 47 Floating point arithmetic, alternatives to 49 Floating point arithmetic, banned from safety-critical systems 496 Floating point arithmetic, binary-decimal conversion 57 Floating point arithmetic, compiler optimization, dangers of 494 Floating point arithmetic, complex arithmetic, error analysis 71—73 Floating point arithmetic, determining properties of 494—495 Floating point arithmetic, drift in 54 Floating point arithmetic, earliest subroutines 35 Floating point arithmetic, elementary functions 50 Floating point arithmetic, formal op algebra 54—55 Floating point arithmetic, fused multiply-add operation 46—47 60 Floating point arithmetic, gradual underflow see “Gradual underflow” Floating point arithmetic, IEEE arithmetic see “IEEE arithmetic” Floating point arithmetic, Language Independent Arithmetic (LIA-1) 499 Floating point arithmetic, monotonic 56 Floating point arithmetic, testing accuracy of 51—52 Floating point arithmetic, testing correctness of 495—496 Floating point coprocessors 42 Floating point numbers characterization 36 Floating point numbers, normalized 36 Floating point numbers, spacing between 37 Floating point numbers, subnormal 37 42 Floating point numbers, testing for equality 493 FLOP 3 Forsgren, Anders 210 226 Forsythe, George E. 29 30 48 53 76 86 126 133 152 186 188 190 235 240 242 245 259 260 279 302 321 489 523 FORTRAN 95 Fortran, environmental inquiry functions 495 Fortran, matmul 447 Forward error 6—7 Forward error for linear system 12 Forward error, definition 6 Forward error, linearized expression for 484 Forward error, mixed forward-backward error 7 456 Forward stability componentwise 130 Forward stability normwise 130 Forward stability, definition 9 Forward substitution 141 Foster, Leslie V. 159 167 170 403 Foulser, David E. 133 134 Fourier matrix 168 Fox, L. xxix xxx 30 31 105 184 186 FPV (floating point verification) package 495—496 Francois, Philippe 32 Frank matrix 463 Fraysse, Valerie 351 468 486 Friedland, Shmuel 342 352 Frobenius norm 107 Frommer, Andreas 482 Funderlic, R.E. 190 Fused multiply-add operation 46—47 60 Gaches, J. 120 132 Gahinet, Pascal M. 318 Gal, Shmuel 50 Gallivan, K.A. 180 Gallopoulos, E. 103 485 Gander, Walter 522 Gantmacher, F.R. 161 Gardiner, Judith D. 317 318 Gardner, Martin 115 Garner, Harvey L. 57 Gasca, M. 180 189 Gastinel, Noel 111 114 Gauss — Jordan elimination 273—277 281—282 Gauss — Jordan elimination error analysis 273—277 Gauss — Jordan elimination, algorithm 273 Gauss — Seidel method 321 329 Gauss, Carl Friedrich 1 187 215 321 381 456 Gaussian elimination 158—163; see also “LU factorization” Gaussian elimination complete pivoting 158; see also “Complete pivoting” Gaussian elimination computer programs, first 188 Gaussian elimination computer programs, history of 188 Gaussian elimination connection with LU factorization 161 Gaussian elimination error analysis 163—166 Gaussian elimination error analysis, history of 183—187 Gaussian elimination in ancient China 187 Gaussian elimination loop orderings 187 Gaussian elimination need for pivoting 158 Gaussian elimination on diagonally dominant matrix 170—172 Gaussian elimination on Hessenberg matrix 24—25 Gaussian elimination on tridiagonal matrix 174 Gaussian elimination pairwise elimination 180 189 Gaussian elimination partial pivoting 158 162; Gaussian elimination pivoting strategy, choice of 178—179 Gaussian elimination rook pivoting 159; see also “Rook pivoting” Gaussian elimination row-wise error bounds 177 Gaussian elimination scaling, row and column 177—179 Gaussian elimination threshold pivoting 193 Gaussian elimination use by Gauss 187 Gaussian elimination versus Cramer's rule 13—14 Gaussian elimination, a posteriori stability tests 180—181 Gaussian elimination, growth factor 165—173; see also “Growth factor” Gaussian elimination, parallel variants of 179—180 Gaussian elimination, pessimism of its accuracy in 1940 183 Gaussian elimination, without pivoting, instability of 15 Gautschi, Walter 415 419 Gautschi, Werner 344 Gay, David M. 57 ge see “Gaussian elimination” Geist, G.A. 25 Gelfand's problem 48 Geman, Stuart 516 Generalized QR factorization 397—398 Gentle, James E. 29 Gentleman, W. Morven 103 375 376 456 570 Geometric computation, accuracy of algorithms in 29 George, Alan 181 209
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