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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
LAPACK, linear system      191
LAPACK, LU factorization      178 191—192 257
LAPACK, matrix 1-norm estimator      292—294 477—478
LAPACK, matrix inversion      282—283
LAPACK, QR factorization      377—378
LAPACK, Sylvester equation      318
LAPACK, test matrix generation      525
LAPACK, triangular systems      153
LAPACK, underdetermined system      414
LAPACK, xlamch for determining machine parameters      495
Laratta, A.      413 414
Larson, John L.      485
Larsson, S.      29
Laszlo, Lajos      345
Latin, neoclassic, publishing papers in      456
Laub, Alan J.      298 315—318 523
Lawson, Charles L.      375 376 399 402 403 414 499
Least significant digit      36
Least squares problem      382
Least squares problem, augmented system      383
Least squares problem, augmented system, scaling and conditioning of      391
Least squares problem, backward error      392—395 406
Least squares problem, iterative refinement      388—391 403
Least squares problem, linear equality constrained      396—400
Least squares problem, linear equality constrained, backward error      405
Least squares problem, linear equality constrained, elimination method      399—400
Least squares problem, linear equality constrained, method of weighting      397
Least squares problem, linear equality constrained, null space method      397—399
Least squares problem, linear equality constrained, perturbation theory      396
Least squares problem, linear inequality constrained      405
Least squares problem, Longley test problem      402
Least squares problem, modified Gram — Schmidt, error analysis      386
Least squares problem, normal equations      382 405
Least squares problem, normal equations, error analysis      386—388
Least squares problem, normal equations, versus QR factorization      388
Least squares problem, perturbation theory      382—384
Least squares problem, QR factorization, error analysis      384—385
Least squares problem, quadratic constrained      405
Least squares problem, seminormal equations      391—392
Least squares problem, weighted      395
LeBlanc, E.      483
Lee, King      446 447
Lefevre, Vincent      50
Lehmer, D.H.      527
Lehoucq, Richard B.      563
Leja ordering      100 101 103 104 427
Lemeire, Prans      152
Leoncini, M.      282
Leuprecht, H.      88
Level index arithmetic      49
LeVeque, Randall J.      11 29
Lewis, John G.      29 188 219 227 299 302 524
Lewis, Robert Michael      472
Li, Kim-Hung      524
Li, T.Y.      282
Li, Xiaoye S.      232 492
Lickteig, Thomas      447
Linear system, large dense, in applications      191
Linear system, perturbation theory      119—137
Linear system, practical forward error bounds      131
Linear system, records for largest solved      191
Linear system, scaling before Gaussian elimination      177—179 190
Linear system, times for solution on early computers      185
Linnainmaa, Seppo      49 76 84 90 501
LINPACK      579
LINPACK, Cholesky factorization of semidefinite matrix      207
LINPACK, condition estimator      295—297 302
LINPACK, iterative refinement      241
LINPACK, LU factorization      178
LINPACK, matrix inversion      264 268 269 271
LINPACK, tridiagonal system solution      303
Linz, Peter      90
Linzer, Elliot      456
Liu, Joseph W.H.      209 227
log(1+x), accurate evaluation      32
Logarithmic distribution of numbers      47 49
Longley test problem      402
Longley, James W.      402
Lotti, Grazia      443 444
LU factorization      161; see also “Gaussian elimination”
LU factorization a posteriori stability tests      180—181
LU factorization complete pivoting      158; see also “Complete pivoting”
LU factorization Crout's method      163
LU factorization error analysis      163—166
LU factorization error analysis, history of      183—187
LU factorization existence and uniqueness      161
LU factorization for nonsymmetric positive definite matrix      208—209
LU factorization growth factor      165—173; see also “Growth factor”
LU factorization loop orderings      187
LU factorization of diagonally dominant matrix      170—172
LU factorization of Hessenberg matrix      24—25
LU factorization of tridiagonal matrix      174
LU factorization partial pivoting      158 162;
LU factorization perturbation bounds      181—182
LU factorization pivoting strategy, choice of      178—179
LU factorization rank-revealing      182—183
LU factorization rook pivoting      159; see also “Rook pivoting”
LU factorization scaling, row and column      177—179
LU factorization stability for M-matrix      190
LU factorization threshold pivoting      193
LU factorization versus Cramer's rule      13—14
LU factorization without pivoting, instability of      15
LU factorization, determinantal formulae for factors      161
LU factorization, Doolittle's method      162—163
LU factorization, parallel variants of      179—180
LU factorization, partitioned      246
LU factorization, partitioned, error analysis of      249—250
LU factorization, recursively partitioned      248
Lu, Hao      429
Luszczek, Piotr      232
Lyapunov equation      311
Lyapunov equation, backward error      311—312
Lyapunov equation, discrete-time      316—317
Lynch, Thomas W.      56
Lyness, J.N.      429
Lynn, M. Stuart      323
M-matrix      145 152
M-matrix, stability of LU factorization      190
M-matrix, triangular      145—147
Mac Lane, Saunders      479
Machar code      495
Machine epsilon      37
Macleod, Allan J.      189
Magic square matrix, p norm of      115
Makhoul, John      210
Malajovich, Gregorio      288
Malcolm, Michael A.      29 86 88 188 245 260 302 495 506 522
Malyshev, Alexander N.      152 282 404 405
Manne, F.      448
Manteuffel, Thomas A.      152
Mantissa      36
Maple      6 170 502
Markov chain, perturbation analysis for      133
Marovich, Scott B.      570
Martin, R.S.      188
Mascarenhas, Walter      189
Mathematica      6 170 502
Mathias, Roy      152 208 377 514
MATLAB      3 575
MATLAB gallery      512 513 583
MATLAB gallery (“chebspec”)      348 513
MATLAB gallery (“clement”)      239 513
MATLAB gallery (“frank”)      463
MATLAB gallery (“kalian”)      513
MATLAB gallery (“orthog”)      239 473 477 513
MATLAB gallery (“pei”)      513
MATLAB gallery (“randsvd”)      513 517 524 525
MATLAB gallery (“toeppen”)      513 525
MATLAB gallery (“tridiag”)      513
MATLAB, compan      513 523
MATLAB, condest      xxii 295
MATLAB, eig      464
MATLAB, eps      39
MATLAB, fft      454
MATLAB, frank      513
MATLAB, hadamard      513
MATLAB, hilb      51
MATLAB, ifft      454
MATLAB, inv      261 268
MATLAB, invhilb      51
MATLAB, magic      51
MATLAB, Matrix Computation Toolbox      583—585
MATLAB, normestl      295
MATLAB, pascal      513 524
MATLAB, pow2      493
MATLAB, rand      513 516
MATLAB, randn      513 516
MATLAB, rcond      294
MATLAB, realmax      39
MATLAB, realmin      39
MATLAB, roots      480
MATLAB, Symbolic Math Toolbox      3 6 463 514 519
MATLAB, toeplitz      513
MATLAB, vander      513
Matrix adjugate      282
Matrix block diagonally dominant      251
Matrix circulant      454
Matrix commutation      317
Matrix companion      522—523
Matrix comparison      145
Matrix condition number      109 110 114 382
Matrix confluent Vandermonde-like      419
Matrix convergent      332 340
Matrix correlation      525
Matrix diagonally dominant      172
Matrix distance to singularity      111 127
Matrix group inverse      331
Matrix inversion      259—285; see also “Inverse matrix”
Matrix involutary      519
Matrix irreducible      127
Matrix magic square      115
Matrix Market      524
Matrix moment      518
Matrix multiplication backward error      77
Matrix multiplication error analysis      69—71 78
Matrix multiplication fast methods      433—449
Matrix norm      see “Norm”
Matrix polynomials      102
Matrix second difference      522
Matrix test      511—525
Matrix, Cauchy      514—515
Matrix, Drazin inverse      331—332
Matrix, Fourier      168
Matrix, Frank      463
Matrix, Hadamard      116 168 170 193
Matrix, Hilbert      512—515 523—524
Matrix, Kahan      149 154 205
Matrix, M-matrix      145 152
Matrix, nonsymmetric positive definite      208
Matrix, Pascal      518—521
Matrix, Pei      284 304
Matrix, powers of      339—352; see also “Powers of a matrix”
Matrix, pseudo-inverse      382 402 405 408
Matrix, random      515—518
Matrix, randsvd      517—518 525
Matrix, semiconvergent      332
Matrix, skew-symmetric      214 225
Matrix, submatrices, number of      192
Matrix, Sylvester's introduction of term      305
Matrix, symmetric indefinite      214
Matrix, symmetric positive definite      196
Matrix, symmetric positive semidefinite      201
Matrix, symmetric quasidefinite      229
Matrix, totally nonnegative      164
Matrix, tridiagonal      174—176
Matrix, tridiagonal, Toeplitz      521—522
Matrix, Vandermonde      416
Matrix, Vandermonde-like      418
Matrix, vec-permutation      314 317
Matsui, Shouichi      49
Mattheij, R.M.M.      257
Matula, David W.      49 53 57
Mazzia, Francesca      190
McCarthy, Charles      133
McCracken, Daniel D.      76 90 102
McKeeman, William Marshall      188 240
McKenney, Alan      317 524
McKinnon, K.I.M.      476
Mead, J.L.      180
Meaningless answers, why you might get them      30—31
Meinguet, Jean      209 281
Mendelsohn, N.S.      283
Mer, Ole      83
Metropolis, N.      486
Meurant, Gerard      257 302
Meyer, Jr., Carl D.      133 332
Milenkovic, Victor J.      29
Miller, D.F.      317
Miller, Webb      76 104 438 441 471 484 485 504 515 570
Miranker, Willard L.      483
Mirsky, L.      554
Misconceptions, of floating point arithmetic      28
Mixed forward-backward error      7 456
Moler, Cleve B.      29 30 55 76 86 152 186 188 190 231 235 240 242 245 259 260 279 295 302 317 339 342 352 381 407 429 500 505 507 523
Moment matrix      518
Monotone norm      107
Monotonicity of floating point arithmetic      56
Monotonicity of rounding      38
Montgomery, D.      183
Moore — Penrose conditions      405
Moore, J. Strother      56
Moore, Ramon E.      483
Morrison, Donald      500 507
Most significant digit      36
Mueller, K.H.      102
Mukherjea, Kalyan      374
Muller, Jean-Michel      32 50
Multidirectional search method      475—477
Multiple precision arithmetic      501—503
Multiple precision arithmetic, Bailey's package MPFUN      501—502
Multiple precision arithmetic, Brent's package      483 501
Multiple precision arithmetic, GNU MP library      502
Murakami, H.      480
Murray, Walter      210 226 227 413 476
Mutation testing      515—516
NAG Library      575
NAG Library, LAPACK in      580
NAG Library, machine constants      497
Nan (not a number)      41—42 490 492
Nash, Stephen G.      308 319 381 562
Nashed, M. Zuhair      402
Neal, Larry      188
Nelder — Mead simplex method      476—477
NETLIB      574—575
Neumaier, A.      57 85
Neumann, M.      190
Neville elimination      180 189
Newbery, A.C.R.      102 103
Newcomb, Simon      47
Newman, Morris      512 518
Newton — Schulz iteration      183
Newton's method      460
Newton's method for eigenproblem      463—464
Newton's method for matrix inverse      278
Newton's method for reciprocation      46
Newton's method limiting accuracy      461
Newton's method limiting residual      462
Newton's method stopping criteria      467—468
Newton's method, inexact      468
Newton's method, sources of error in      460
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