數(shù)值分析

內(nèi)容概要

　　本書介紹了現(xiàn)代數(shù)值近似技術(shù)的理論及實(shí)用知識(shí)，解釋了它們的工作原理。同它的前幾個(gè)版本一樣，該書仍將重點(diǎn)放在近似技術(shù)的數(shù)值分析上，以便為讀者今后的學(xué)習(xí)打下堅(jiān)實(shí)的數(shù)值分析與科學(xué)計(jì)算基礎(chǔ)。本書內(nèi)容豐富、翔實(shí)，可以根據(jù)不同的學(xué)習(xí)對(duì)象和學(xué)習(xí)目的，選擇、組織、串聯(lián)相應(yīng)的章節(jié)，形成側(cè)重于理論或是側(cè)重于實(shí)用的兩種學(xué)習(xí)策略。書中的每個(gè)概念均以大量的例子說明，同時(shí)書中還包含2000多個(gè)習(xí)題，范圍從方法、算法的基本應(yīng)用到理論的歸納與擴(kuò)展，涉及物理、計(jì)算機(jī)、生物、社會(huì)科學(xué)等多個(gè)不同的領(lǐng)域。通過這些實(shí)例，進(jìn)一步說明在現(xiàn)實(shí)世界中，數(shù)值方法是如何被應(yīng)用的。第七版新增了兩個(gè)突出的部分，一是前承條件共軛梯度方法，為線性方程系統(tǒng)提供了更完備的解決方法；另一部分是同倫與連續(xù)方法，為非線性方程系統(tǒng)的近似求解提供了不同的方法。

作者簡介

作者：（美國）伯登（Richard L.Burden） J.Douglas Faires

書籍目錄

1. MATHEMATICAL PRELIMINARIES 1.1 Review of Calculus 　　1.2 Round-off Errors and Computer Arithmetic 　1.3 Algorithms and Convergence 　1.4 Numerical Software 2. SOLUTIONS OF EQUATIONS IN ONE VARIABLE 　2.1 The Bisection Method 　2.2 Fixed-Point Iteration 　2.3 The Newton's Method 　2.4 Error Analysis for Iterative Methods 　2.5 Accelerating Convergence 　2.6 Zeros of Polynomials and Muller's Method 　2.7 Survey of Methods and Software 3. INTERPOLATION AND POLYNOMIAL APPROXIMATION 　3.1 Interpolation and the LaGrange Polynomial 　3.2 Divided Differences 　3.3 Hermite Interpolation 　3.4 Cubic Spline Interpolation 　3.5 Parametric Curves 　3.6 Survey of Methods and Software 4. NUMERICAL DIFFERENTIATION AND INTEGRATION 　4.1 Numerical Differentiation 　4.2 Richardson's Extrapolation 　4.3 Elements of Numerical Integration 　4.4 Composite Numerical Integration 　4.5 Romberg Integration 　4.6 Adaptive Quadrature Methods 　4.7 Gaussian Quadrature 　4.8 Multiple Integrals 　4.9 Improper Integrals 　4.10 Survey of Methods and Software 5. INITIAL-VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS 　5.1 The Elementary Theory of Initial-Value Problems 　5.2 Euler's Method 　5.3 Higher-Order Taylor Methods 　5.4 Runge-Kutta Methods 　5.5 Error Control and the Runge-Kutta-Fehlberg Method 　5.6 Multi-Step Methods 　5.7 Variable Step-Size Multi-Step Methods 　5.8 Extrapolation Methods 　5.9 Higher-Order Equations and Systems of Differential Equations 　5.10 Stability 　5.11 Stiff Differential Equations 　5.12 Survey of Methods and Software 6. DIRECT METHODS FOR SOLVING LINEAR SYSTEMS 7. ITERATIVE TECHNIQUES IN MATRIX ALGEBRA 8. APPROXIMATION THEORY 9. APPROXIMATING EIGENVALUES 10. NUMERICAL SOLUTIONS OF NONLINEAR SYSTEMS OF EQUATIONS 11. BOUNDARY-VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS 12. NUMERICAL SOLUTIONS TO PARTIAL DIFFERENTIAL EQUATIONS BIBLIOGRAPHY ANSWERS TO SELECTED EXERCISES INDEX

章節(jié)摘錄

版權(quán)頁：   插圖：   Factorizations take a simpler form when the matrix A is positive definite.For example,the Choleski factorization has the form A=LLt,where L is lower triangular.A symmetric matrix that has an LU factorization can also be factored in the form A =LDLt,where D is diagonal and L is lower triangular with 1's on the diagonal.With these factorizations,marupulations involving A can be simplified.If A is tridiagonal,the LU factorization takes a particularly simple form,with U having 1's on the main diagonal and 0s elsewhere,except on the diagonal immediately above the main diagonal.In addition,L has its only nonzero entries on the main diagonal and one diagonal below. The direct methods are the methods of choice for most linear systems.For tridiagonal,banded,and positive definite matrices,the special methods are recommended.For the general case,Gaussian elimination or LU factorization methods,which allow pivoting,are recommended.In these cases,the effects of roundoff error should be monitored.In Section 7.4 we discuss estimating errors in direct methods. Large linear systems with primarily 0 entries occurring in regular patterns can be solved efficiently using an iterative procedure such as those discussed in Chapter 7.Systems of this type arise naturally,for example,when finite-difference techniques are used to solve boundary-value problems,a common application in the numerical solution of partial-differential equations. It can be very difficult to solve a large linear system that has primarily nonzero entries or one where the 0 entries are not in a predictable pattern.The matrix associated with the system can be placed in secondary storage in partitioned form and portions read into main memory only as needed for calculation.Methods that require secondary storage can be either iterative or direct,but they generally require techniques from the fields of data structures and graph theory.The reader is referred to(BuR)and(RW)for a discussion of the current techniques. The software for matrix operations and the direct solution of linear systems implemented in IMSL and NAG is based on LAPACK,a subroutine package in the public domain.There is excellent documentation available with it and from the books written about it.We will focus on several of the subroutines that are available in all three sources. Accompanying LAPACK is a set of lower-level operations called Basic Linear Algebra Subprograms(BLAS).Level 1 of BLAS generally consists of vector-vector operations with input data and operation counts of O(n).Level 2 consists of the matrix-vector operations with input data and operation counts of O(n2).Level 3 consists of the matrix-matrix operations with input data and operation counts of O(n3).For example,in Level 1,the subroutine SCOPY overwrites a vector y with a vector x; SSCAL computes a scalar a times a vector x; SAXPY adds a scalar a times a vector x to a vector y(y = a?x+y); SDOT computes the inner,or scalar,product of two vectors; SNRM2 computes the Euclidean norm of a vector by a method similar to that discussed in Section 1.4; and ISAMAX computes the index of the vector component that gives the maximum absolute value of all the components.In Level 2,SGEMV computes the product of a matrix and a vector and in Level 3,SGEMM computes the product of a matrix and a matrix.

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《教育部高等教育司推薦?國外優(yōu)秀信息科學(xué)與技術(shù)系列教學(xué)用書:數(shù)值分析(第7版)(影印版)》在大力宣傳、推薦我國專家編著的面向21世紀(jì)和“九五”重點(diǎn)的信息科學(xué)和技術(shù)課程教材的基礎(chǔ)上，在有條件的高等學(xué)校的某些信息科學(xué)和技術(shù)課程中推動(dòng)使用國外優(yōu)秀教材的影印版進(jìn)行英語或雙語教學(xué)，以縮短我國在計(jì)算機(jī)教學(xué)上與國際先進(jìn)水平的差距，同時(shí)也有助于強(qiáng)化我國大學(xué)生的英語水平。

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