| 词条 | 数值算法的精确性与稳定性 |
| 释义 | 图书信息出版社: 清华大学出版社; 第1版 (2011年2月1日) 外文书名: Accuracy and Stability of Numerical Algorithms 2nd edition 丛书名: 国际著名数学图书 平装: 680页 正文语种: 英语 开本: 16 ISBN: 9787302244936, 7302244936 条形码: 9787302244936 尺寸: 24.4 x 17.4 x 3.4 cm 重量: 939 g 作者简介作者：（美国）汉安（Nicholas J.Higham） Nicholas J. Higham is Richardson Professor of Applied Mathematics at the University of Manchester, England. He is the author of more than 80 publications and is a member of the editorial boards of Foundations of Computational Mathematics, the IMA Journal of Numerical Analysis, Linear Algebra and Its Applications, and the SIAM Journal on Matrix Analysis and Applications. 内容简介《数值算法的精确性与稳定性(第2版)(影印版)》内容简介：accuracy and stability of numerical algorithms 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, all enlivened by historical perspective and informative quotations. this second edition expands and updates the coverage of the first edition (1996) and includes numerous improvements to the original material. two new chapters treat symmetric indefinite systems and skew-symmetric systems, and nonlinear systems and newton's method. twelve new sections include coverage of additional error bounds for gaussian elimination, rank revealing lu factorizations, weighted and constrained least squares problems, and the fused multiply-add operation found on some modern computer architectures. although not designed specifically as a textbook, this new edition is a suitable reference for an advanced course. it can also be used by instructors at all levels as a supplementary text from which to draw examples, historical perspective, statements of results, and exercises. 媒体评论"This definitive source on the accuracy and stability of numerical algorithms is quite a bargain and a worthwhile addition to the library of any statistician heavily involved in computing." ——Robert L. Strawderman, Journal of the American Statistical Association, March 1999. "This text may become the new 'Bible' about accuracy and stability for the solution of system of linear equations. It covers 688 pages carefully collected, investigated, and written.. One will find that this book is a very suitable and comprehensive reference for research in numerical linear algebra, software usage and development, and for numerical linear algebra courses." —— N. Kockler, Zentrablatt for Mathematik, Band 847/96. "Nick Higham has assembled an enormous amount of important and useful material in a coherent, readable form. His book belongs on the shelf of anyone who has more than a casual interest in rounding error and matrix computation." —— G.W. Stewart, SIAM Review, March 1997. 目录list of figures list of tables preface to second edition preface to first edition about the dedication 1 principles of finite precision computation 1.1 notation and background 1.2 relative error and significant digits 1.3 sources of errors 1.4 precision versus accuracy 1.5 backward and forward errors 1.6 conditioning 1.7 cancellation 1.8 solving a quadratic equation 1.9 computing the sample variance 1.10 solving linear equations 1.10.1 gepp versus cramer's rule 1.11 accumulation of rounding errors 1.12 instability without cancellation 1.12.1 the need for pivoting 1.12.2 an innocuous calculation? 1.12.3 an infinite sum 1.13 increasing the precision 1.14 cancellation of rounding errors 1.14.1 computing (ex - 1)ix 1.14.2 qr factorization 1.15 rounding errors can be beneficial 1.16 stability of an algorithm depends on the problem 1.17 rounding errors are not random 1.18 designing stable algorithms 1.19 misconceptions 1.20 rounding errors in numerical analysis 1.21 notes and references problems 2 floating point arithmetic 2.1 floating point number system 2.2 model of arithmetic 2.3 ieee arithmetic 2.4 aberrant arithmetics 2.5 exact subtraction 2.6 fused multiply-add operation 2.7 choice of base and distribution of numbers 2.8 statistical distribution of rounding errors 2.9 alternative number systems 2.10 elementary functions 2.11 accuracy tests 2.12 notes and references problems 3 basics 3.1 inner and outer products 3.2 the purpose of rounding error analysis 3.3 running error analysis 3.4 notation for error analysis 3.5 matrix multiplication 3.6 complex arithmetic 3.7 miscellany 3.8 error analysis demystified 3.9 other approaches 3.10 notes and references problems 4 summation 4.1 summation methods 4.2 error analysis 4.3 compensated summation 4.4 other summation methods 4.5 statistical estimates of accuracy 4.6 choice of method 4.7 notes and references problems 5 polynomials 5.1 hornet's method 5.2 evaluating derivatives 5.3 the newton form and polynomial interpolation 5.4 matrix polynomials 5.5 notes and references problems 6 norms 6.1 vector norms 6.2 matrix norms 6.3 the matrix p-norm 6.4 singular value decomposition 6.5 notes and references problems 7 perturbation theory for linear systems 7.1 normwise analysis 7.2 componentwise analysis 7.3 scaling to minimize the condition number 7.4 the matrix inverse 7.5 extensions 7.6 numerical stability 7.7 practical error bounds 7.8 perturbation theory by calculus 7.9 notes and references problems 8 triangular systems 8.1 backward error analysis 8.2 forward error analysis 8.3 bounds for the inverse 8.4 a parallel fan-in algorithm 8.5 notes and references 8.5.1 lapack problems 9 lu factorization and linear equations 9.1 gaussian elimination and pivoting strategies 9.2 lu factorization 9.3 error analysis 9.4 the growth factor 9.5 diagonally dominant and banded matrices 9.6 tridiagonal matrices 9.7 more error bounds 9.8 scaling and choice of pivoting strategy 9.9 variants of ganssian elimination 9.10 a posteriori stability tests 9.11 sensitivity of the lu factorization 9.12 rank-revealing lu factorizations 9.13 historical perspective 9.14 notes and references 9.14.1 lapack problems 10 cholesky factorization 10.1 symmetric positive definite matrices 10.1.1 error analysis 10.2 sensitivity of the cholesky factorization 10.3 positive semidefinite matrices 10.3.1 perturbation theory 10.3.2 error analysis 10.4 matrices with positive definite symmetric part 10.5 notes and references 10.5.1 lapack problems 11 symmetric indefinite and skew-symmetric systems 11.1 block ldlt factorization for symmetric matrices 11.1.1 complete pivoting 11.1.2 partial pivoting 11.1.3 rook pivoting 11.1.4 tridiagonal matrices 11.2 aasen's method 11.2.1 aasen's method versus block ldlt factorization 11.3 block ldlt factorization for skew-symmetric matrices 11.4 notes and references 11.4.1 lapack problems 12 iterative refinement 12.1 behaviour of the forward error 12.2 iterative refinement implies stability 12.3 notes and references 12.3.1 lapack problems 13 block lu factorization 13.1 block versus partitioned lu factorization 13.2 error analysis of partitioned lu factorization 13.3 error analysis of block lu factorization 13.3.1 block diagonal dominance 13.3.2 symmetric positive definite matrices 13.4 notes and references 13.4.1 lapack problems 14 matrix inversion 14.1 use and abuse of the matrix inverse 14.2 inverting a triangular matrix 14.2.1 unblocked methods 14.2.2 block methods 14.3 inverting a full matrix by lu factorization 14.3.1 method a 14.3.2 method b 14.3.3 method c 14.3.4 method d 14.3.5 summary 14.4 gauss-jordan elimination 14.5 parallel inversion methods 14.6 the determinant 14.6.1 hyman's method 14.7 notes and references 14.7.1 lapack problems 15 condition number estimation 15.1 how to estimate componentwise condition numbers 15.2 the p-norm power method 15.3 lapack 1-norm estimator 15.4 block 1-norm estimator 15.5 other condition estimators 15.6 condition numbers of tridiagonal matrices 15.7 notes and references 15.7.1 lapack problems 16 the sylvester equation 16.1 solving the sylvester equation 16.2 backward error 16.2.1 the lyapunov equation 16.3 perturbation result 16.4 practical error bounds 16.5 extensions 16.6 notes and references 16.6.1 lapack problems 17 stationary iterative methods 17.1 survey of error analysis 17.2 forward error analysis 17.2.1 jacobi's method 17.2.2 successive overrelaxation 17.3 backward error analysis 17.4 singular systems 17.4.1 theoretical background 17.4.2 forward error analysis 17.5 stopping an iterative method 17.6 notes and references problems 18 matrix powers 18.1 matrix powers in exact arithmetic 18.2 bounds for finite precision arithmetic 18.3 application to stationary iteration 18.4 notes and references problems 19 qr factorization 19.1 householder transformations 19.2 qr factorization 19.3 error analysis of householder computations 19.4 pivoting and row-wise stability 19.5 aggregated householder transformations 19.6 givens rotations 19.7 iterative refinement 19.8 gram-schmidt orthogonalization 19.9 sensitivity of the qr factorization 19.10 notes and references 19.10.1 lapack problems 20 the least squares problem 20.1 perturbation theory 20.2 solution by qr factorization 20.3 solution by the modified gram-schmidt method 20.4 the normal equations 20.5 iterative refinement 20.6 the seminormal equations 20.7 backward error 20.8 weighted least squares problems 20.9 the equality constrained least squares problem 20.9.1 perturbation theory 20.9.2 methods 20.10 proof of wedin's theorem 20.11 notes and references 20.11.1 lapack problems 21 underdetermined systems 21.1 solution methods 21.2 perturbation theory and backward error 21.3 error analysis 21.4 notes and references 21.4.1 lapack problems 22 vandermonde systems 22.1 matrix inversion 22.2 primal and dual systems 22.3 stability 22.3.1 forward error 22.3.2 residual 22.3.3 dealing with instability 22.4 notes and references problems 23 fast matrix multiplication 23.1 methods 23.2 error analysis 23.2.1 winograd's method 23.2.2 strassen's method 23.2.3 bilinear noncommutative algorithms 23.2.4 the 3m method 23.3 notes and references problems 24 the fast fourier transform and applications 24.1 the fast fourier transform 24.2 circulant linear systems 24.3 notes and references problems 25 nonlinear systems and newton's method 25.1 newton's method 25.2 error analysis 25.3 special cases and experiments 25.4 conditioning 25.5 stopping an iterative method 25.6 notes and references problems 26 automatic error analysis 26.1 exploiting direct search optimization 26.2 direct search methods 26.3 examples of direct search 26.3.1 condition estimation 26.3.2 fast matrix inversion 26.3.3 roots of a cubic 26.4 interval analysis 26.5 other work 26.6 notes and references problems 27 software issues in floating point arithmetic 27.1 exploiting ieee arithmetic 27.2 subtleties of floating point arithmetic 27.3 cray peculiarities 27.4 compilers 27.5 determining properties of floating point arithme 27.6 testing a floating point arithmetic 27.7 portability 27.7.1 arithmetic parameters 27.7.2 2 x 2 problems in lapack 27.7.3 numerical constants 27.7.4 models of floating point arithmetic 27.8 avoiding underflow and overflow 27.9 multiple precision arithmetic 27.10 extended and mixed precision blas 27.11 patriot missile software problem 27.12 notes and references problems 28 a gallery of test matrices 28.1 the hilbert and cauchy matrices 28.2 random matrices 28.3 "randsvd" matrices 28.4 the pascal matrix 28.5 tridiagonal toeplitz matrices 28.6 companion matrices 28.7 notes and references 28.7.1 lapack problems a solutions to problems b acquiring software b.1 internet b.2 netlib b.3 matlab b.4 nag library and nagware f95 compiler c program libraries c.1 basic linear algebra subprograms c.2 eispack c.3 linpack c.4 lapack c.4.1 structure of lapack d the matrix computation toolbox bibliography 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