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**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。

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