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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
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 $(e^x-1)$      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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