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Higham N. — Accuracy and stability of numerical algorithms
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Название: Accuracy and stability of numerical algorithmsАвтор: Higham N. Аннотация: 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.
Язык: Рубрика: Математика /Численные методы /Численный анализ /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1996Количество страниц: 688Добавлена в каталог: 23.02.2005Операции: Положить на полку |
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Lyapunov equation, discrete-time 322 Lynch, Robert E. 32 Lyness, J. N. 32 441 Lynn, M. Stuart 327 M-matrix 580 M-matrix, stability of LU factorization 198 M-matrix, triangular 157 159 160 Mac Lane, Saunders 483 Machar code 497—498 Machine epsilon 41 Macleod, Allan J. 197 Magic square matrix, p norm of 127 Makhoul, John 225 Malcolm, Michael A. 33 95 97 196 245q 262 305 497 509 525 Manteuffel, Thomas A. 165 Mantissa 40 Maple 6 181 506 Marine, F. 461 Markov chain, perturbation analysis for 146 Marovich, Scott B. 576 Marsaglia, George 527 Martin, R, S. 196 Mascarenhas, Walter 197 Mathematical 6 181 506 Mathias, Roy 165 223 386 515 MATLAB 3 43 583 MATLAB, condest 304 480 MATLAB, fft 468 MATLAB, inv 271 MATLAB, rand 518 MATLAB, randn 518 MATLAB, rcond 305 480 489 MATLAB, roots 483 MATLAB, special matrices 514 MATLAB, Symbolic Math Toolbox 3 521 MATLAB, Test Matrix Toolbox 591—594 Matrix multiplication, backward error 85 Matrix multiplication, error analysis 76—78 85 Matrix multiplication, fast methods 445—463 Matrix norm see “Norm” Matrix, block diagonally dominant 251 Matrix, Cauchy 516—517 Matrix, circulant 468 Matrix, companion 525—526 Matrix, comparison 157 Matrix, condition number 121—123 126 392 Matrix, confluent Vandermonde-like 429 Matrix, convergent 348 Matrix, diagonally dominant 181 Matrix, distance to singularity 123 140 Matrix, Drazin inverse 336—337 Matrix, Fourier 179 Matrix, Hadamard 128 179 181 201 Matrix, Hilbert 514—517 Matrix, inversion 261—287 (see also “Inverse matrix”) Matrix, involuntary 521 Matrix, Kahan 161 167 214 Matrix, M-matrix 580 Matrix, magic square 127 Matrix, moment 520 Matrix, nonsymmetric positive definite 223 Matrix, Pascal 520—524 Matrix, powers of 345—359 (see also “Powers of a matrix”) Matrix, pseudo-inverse 392 409 412 416 Matrix, random 517—520 Matrix, randsvd 519—520 Matrix, second difference 525 Matrix, semiconvergent 337 Matrix, Sylvester’s introduction of term 309 Matrix, symmetric indefinite 218 Matrix, symmetric positive definite 204 Matrix, symmetric positive semidefinite 210 Matrix, symmetric quasidefinite 229 Matrix, test 513—528 Matrix, totally nonnegative 176 Matrix, tridiagonal 183—186 Matrix, tridiagonal, Toeplitz 524—525 Matrix, Vandermonde 426 Matrix, Vandermonde-like 429 Matrix, vet-permutation 319 323 Matsui, Shouichi 53 Mattheij, R. M. M. 257 Matula, David W. 54 57 62 Mazzia, Francesca 197 McCarthy, Charles 146 McCracken, Daniel D. 83 100 113 McKeeman, William Marshall 196 241 McKenney, Alan 322 527 Meaningless answers, why you might get them 35 Meinguet, Jean 224 284 Mendelssohn, N. S. 286 Metcalf, Michael 3 Metropolis, N. 489 Meurant, Gerard 257 305 Meyer, Jr., Carl D. 147 337 Miiller, K. H. 113 Milenkovic, Victor J. 34 Miller, D. R. 322 Miller, Webb 83 115 451 454 473q 487 488 507 517 576 Miranker, Willard L. 486 Mirsky, L. 558 Misconceptions, of floating point arithmetic 31 Mixed forward-backward error 8 Modified Gram — Schmidt method see “Gram — Schmidt method modified” Moler, Cleve B. 33 34 59 84 95 164 190 196 197 231q 235 241 243 245q 261q 262 282 297 304 305 323 345q 348 359 391q 415q 441 503 508 511 526 579q Moller, Ole 92 Moment matrix 520 Monotone norm 119 Montgomery, D. 186 Moore — Penrose conditions 412 Moore, Ramon E. 486 Morrison, Donald 503 511 579 Morton, K. W. 32 Most significant digit 40 Mukherjea, Kalyan 382 Muller-Merbach, H. 32 Multi-directional search method 477—479 Multiple precision arithmetic 504—506 Multiple precision arithmetic, Bailey’s package MPFUN 505 Multiple precision arithmetic, Brent’s package 486 504 Murakami, H. 483n Murray, Walter 32 225 226 422 479 Mutation testing 517 NAG Library 583—584 NAG Library, LAPACK in 588 NAG Library, machine constants 500 Nagy, James G. 471 Nan (not a number) 46 492 495 Nash, Stephen G. 312 324 391q 567 Nashed, M. Zuhair 409 Nelder — Mead simplex method 479 NETLIB 582—583 Neumaier, A. 61 94 Neumann, M. 198 Newbery, A. C. R 113 Newcomb, Simon 51 Newman, Morris 514 520 Nickel, Karl 94 486 Nilson, E. N. 167 Nonlinear equations, references for rounding error analysis 32 Nonsymmetric positive definite matrix 223 Nonsymmetric positive definite matrix, LU factorization, stability of 223—224 Nordio, Marcelo 197 Norm 117—129 Norm equivalence constants 122t Norm, 128 Norm, 2-norm, evaluation without overflow 502—503 Norm, absolute 119 Norm, consistent 120 Norm, dual 119 Norm, Frobenius 120 Norm, Holder inequality 118 Norm, matrix 120—124 Norm, matrix p-norm 124—126 Norm, matrix p-norm of magic square matrix 127 Norm, monotone 119 Norm, subordinate matrix 120 121 Norm, unitarily invariant 121 Norm, vector 118—119 Norm, vector, norm equivalence constants 121t Normal equations, error analysis 397—399 Notation, explanation of 2—3 73—76 Notay, Yvan 329 NPSOL 197 Numerical analysis, definition 6 Numerical radius 350 Numerical stability for linear equation solvers 141—142 Numerical stability, definition 8 33 Oberaigner, W. 98 Oettli — Prager backward error theorem 135 Oettli, W. 135 145 Ofman, Yu. 461 Olesky, D. D. 198 Oliver, J. 113 440 Olkin, Ingram 527 Olshevsky, V. 440 441 Olver, F. W. J. 53 76 84 113 190 Omladic, Matjaz 225 Opfer, Gerhard 441 Optimization, references for rounding error analysis 32 Ordinary differential equations, accuracy of mesh point formation 101 Ordinary differential equations, backward error in 33 Ordinary differential equations, Euler’s method with compensated, summation 95 Ordinary differential equations, references for rounding error analysis 32 Ortega, James M. 384 Osborne, M. R. 421 Ostrowski, A. M. 350 546 Outer product, error analysis 71 Overflow 18 42 Overflow, avoiding 502—504 O’Cinneide, Colm Art 147 O’Leary, Dianne Prost 305 P-norm power method 291—294 304 Paige, C. C. 225 379 385 388 397 411 415q 421 422 570 Palmer, John 525 Palmer, Kenneth J. 32 Pan, Victor 441 448 450 459 Papadimitriou, Pythagoras 386 Papadopoulos, Philip M. 323 Parallel prefix operation 165 Paranoia code 497—498 Park, Haesun 385 Parlett, Beresford N. xxiv xxviii 27 34 58 87q 187 219 225 361q 383—385 Partial differential equations, references for rounding error analysis 32 Partial pivoting 170 173 Partial pivoting, early use of 196 Partial pivoting, growth factor 177—183 Partial pivoting, growth factor, large growth in practical problems 178 Partial pivoting, threshold pivoting 200 Partitioned algorithm, definition 246 Partitioned LU factorization, error analysis 248—250 Pascal matrix 520—524 Pascal matrix, Cholesky factor 521 Pascal matrix, inverse 522 Pascal matrix, total positivity 523 Pasternak, Mary E. 488 Patashnik, Oren 87q 520 Patrick, Merrell L. 226 Patriot missile software problem 506—507 Patterson, David A. 39n 57 Paxson, Vern 61 Pelz, Richard 505 Pena, J. M. 192 196 Pentium chip, division bug 59 Pereyra, Victor 434 436 441 Performance profile, for LAPACK norm estimator 296—297 Perturbation theory by calculus 144—145 Perturbation theory, linear systems 131—150 Perturbation theory, statistical 147 149—150 Perturbation theory, Sylvester equation 318—320 Peters, G. 113 196 279 284 287 411 Peterson, Brian 197 Philippe, Bernard 385 pi 49lq Pichat, M. 98 Pierce, Daniel J. 301 Piessens, Robert 32 Pinkus, Allan 32 190 196 Plemmons, Robert J. 147 198 301 337 411 422 471 579q 580 Polar decomposition 386 389 Poljak, Svatopluk 140 Polman, Ben 257 Polynomials 103—115 (see also “Horner’s method”) Polynomials, divided differences 109—112 Polynomials, fast evaluation schemes 114 115 Polynomials, Newton form 109—112 Ponceleon, Dulce B. 226 Poromaa, Peter 305 320 324 PORT library, machine constants 500 Portability of software 499—502 Positive definite matrix see “Nonsymmetric positive definite matrix; symmetric positive definite matrix” Pothen, Alex 165 Powell, M. J. D. 225 411 474 477 Power method 26 Power method for matrix 1-norm estimation 294—297 Power method for matrix p-norm estimation 291—294 304 Power, Stephen 224 Powers of a matrix 345—359 Powers of a matrix in exact arithmetic 346—353 Powers of a matrix in finite precision arithmetic 353—358 Powers of a matrix, behaviour of stationary iteration 358 Powers of a matrix, departure from normality 351—352 Powers of a matrix, hump 348—349 Powers of a matrix, pseudospectrum 352 356—358 Powers of a matrix, role of spectral radius 346—348 Prager, W. 135 145 Precision, effect of increasing 19—21 Precision, versus accuracy 7 33 Press, William H. 479 490 507 Priest, Douglas M. 33 34 58 96—99 102 502 504 531 Program verification, applied to error analysis 488 Pryce, J. D. 76 242 Pseudo-inverse 392 409 412 416 Pseudospectrum 352 356—358 Pseudospectrum of companion matrix 526 Puglisi, Chiara 371 Pukelsheim, Friedrich 323 Puschmann, Heinrich 197 Pythagorean sum 503 511—512 QR factorization 361—389 QR factorization, column pivoting 387 QR factorization, Givens 371—373 QR factorization, Givens, cancellation of errors in 24—26 QR factorization, Givens, error analysis 373—375 QR factorization, Householder 363—364 QR factorization, Householder, error analysis 368—369 QR factorization, Householder, error analysis for application to LS problem 395—396 QR factorization, Householder, error analysis for partitioned (WY representation) 370—371 QR factorization, iterative refinement for linear system 375—376 QR factorization, perturbation theory 381—383 QR factorization, rank-revealing 386 Quadratic equation, solving 11—12 33 Quadrature, accuracy of grid formation 101 Quadrature, error bound for evaluation of rule 86 Quadrature, references for rounding error analysis 32 Quarteroni, Alfio 32 Quinlan, Gerald D. 96 Quinn, Thomas R. 96 Rabinowitz, Philip 32 100 Rail, Louis B. 409 486 488
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