数值分析

出版时间：2001-8  出版社：高等教育出版社  作者：Richard L.Bursen  页数：841  字数：1177000  
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内容概要

　　本书介绍了现代数值近似技术的理论及实用知识，解释了它们的工作原理。同它的前几个版本一样，该书仍将重点放在近似技术的数值分析上，以便为读者今后的学习打下坚实的数值分析与科学计算基础。本书内容丰富、翔实，可以根据不同的学习对象和学习目的，选择、组织、串联相应的章节，形成侧重于理论或是侧重于实用的两种学习策略。书中的每个概念均以大量的例子说明，同时书中还包含2000多个习题，范围从方法、算法的基本应用到理论的归纳与扩展，涉及物理、计算机、生物、社会科学等多个不同的领域。通过这些实例，进一步说明在现实世界中，数值方法是如何被应用的。第七版新增了两个突出的部分，一是前承条件共轭梯度方法，为线性方程系统提供了更完备的解决方法；另一部分是同伦与连续方法，为非线性方程系统的近似求解提供了不同的方法。

作者简介

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

书籍目录

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

章节摘录

版权页：   插图：   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荐•国外优秀信息科学与技术系列教学用书:数值分析(第7版)(影印版)》在大力宣传、推荐我国专家编著的面向21世纪和“九五”重点的信息科学和技术课程教材的基础上，在有条件的高等学校的某些信息科学和技术课程中推动使用国外优秀教材的影印版进行英语或双语教学，以缩短我国在计算机教学上与国际先进水平的差距，同时也有助于强化我国大学生的英语水平。

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•   一本很好的学习数值分析的书