Numerical Analysis

1. Numerical Analysis    

2. Chen Huaitang 陈 怀 堂

3. Preliminaries

4. Introduction to Computional Methods

5. The main part of Computional Methods is numerical analysis. It involves the study, development, and analysis of algorithms for obtaining numerical solutions to various mathematical problems. Frequently, numerical analysis is called the mathematics of scientific computing.

6. The algorithms that we study invariably are destined for use on high-speed computers, and therefore another crucial step intervenes before the solution to a problem can be obtained: a computer program or code must be written to communicate the algorithm to the computer.

7. This is, of course, a nontrivial matter, but there are so many choices of computers and computer languages that it is a topic best left out of the science of numerical analysis per se.

8. There are certainly many other purposes to which computers can be put besides the numerical solution of mathematical problems: providing basic communications, keeping large data bases,

9. playing games, “net surfing,” writing novels, accounting, and so on. Solving mathematical problems numerically on the computer is scientific computing.

10. The development of the associated algorithms (procedures) and the study of their behavior are the mathematics of scientific computing.

11. Often the development of an algorithm is stimulated by a constructive proof in mathematics. In classical analysis, nonconstructive methods are frequently used, but generally they do not lead to algorithms.

12. For example, existence and uniqueness theorems might be established by assuming that they are not true and then following the trail of a logical argument until arriving at a contradiction.

13. Not every constructive proof will lead to a successful algorithm, however. A difficulty that may arise is that an analytical solution to a given problem may be several steps away from a numerical solution.

14. Or it might be completely impractical because of slow convergence or the need for lengthy computation. As an example of the gap between an existence theorem and a numerical solution of a problem, consider the ubiquitous matrix equation Ax = b.

15. We know that it has a unique solution whenever A is nonsingular. But this fact may be of little solace when we are faced with a large linear system containing empirical data and we wish to compute an approximate numerical solution.

16. In general, in this course, we will begin each topic with a basic mathematical problem that arises frequently in practical applications. Then a certain amount of analysis will be presented in order to arrive at an algorithm for solving the problem.

17. Algorithms are usually given in the form of a pseudocode. Finally, additional analysis of the algorithm may be given to help in understanding its behavior, such as its convergence or its resistance to corruption by roundoff error.

18. Such analysis may take the form of either forward or backward error analysis. We have developed this material for a sequence of courses on the theory and application of numerical approximation techniques.

19. The text is designed primarily for junior-level mathematics, science, and engineering majors who have completed at least the first year of the standard college calculus sequence.

20. Familiarity with the fundamentals of matrix algebra and differential equations is also useful, but adequate introductory material on these topics is presented in the text so that those courses need not be prerequisites.

21. We have tried to adapt the text to fit these diverse users without compromising our original purpose: To give an introduction to modern approximation techniques; to explain how, why, and when they can be expected to work;

22. and to provide a firm basis for future study of numerical analysis and scientific computing. The text contains sufficient material for a full year of study, but we expect you to use the text only for a single-term course.

23. In such a course, students learn to identify the types of problems that require numerical techniques for their solution and see examples of the error propagation that can occur when numerical methods are applied.

24. They accurately approximate the solutions of problems that cannot be solved exactly and learn techniques for estimating error bounds for the approximations. The remainder of the text serves as a reference for methods that are not considered in the course.

25. A number of software packages have been developed to produce symbolic mathematical computations. Predominant among them in the academic environment are Derive, Maple, and Mathematica.

26. Student versions of these software packages are available at reasonable prices for most common computer systems. Although there are significant differences among the packages, both in performance and price, all can perform standard algebra and calculus operations.

27. Having a symbolic computation package available can be very useful in the study of approximation techniques. The results in most of our examples and exercises have been generated using problems for which exact values can be determined, since this permits the performance of the approximation method to be monitored.

28. Exact solutions can often be obtained quite easily using symbolic computation. In addition, for many numerical techniques the error analysis requires bounding a higher ordinary or partial derivative of a function,

29. which can be a tedious task and one that is not particularly instructive once the techniques of calculus have been mastered. Derivatives can be quickly obtained symbolically, and a little insight often permits a symbolic computation to aid in the bounding process as well.

30. We have chosen Maple as our standard package because of its wide distribution, but Derive or Mathematica can be substituted with only minor modifications. Examples and exercises have been added whenever we felt that a computer algebra system would be of significant benefit,

31. and we have discussed the approximation methods that Maple employs when it is unable to solve a problem exactly.

32. The error that results from replacing a number with its floating-point form is called roundoff error(regardless of whether the rounding or chopping method is used). The following definition describes two methods for measuring approximation errors.

33. Definition1If p* is anapproximation to p, the absolute error is |p-p*|, and the relative |p-p*| error is — , provided that p0. |p|

34. Definition2 The number p* is said to approximate p to t significant digits(or figures) if t is the largest nonnegative integer for which |p-p*| -t —— < 5  10 |p|

35. Table 1 illustrates the continuous nature of significant digits by listing, for the various values of p, the least upper bound |p-p*|, denoted max |p-p*| , when p* agrees with p to four significant digits.

36. Table 1 P 0.1 0.5 100 1000 5000 9990 10000 ————————————————————————————— max |p-p*| 0.00005 0.00025 0.05 0.5 2.5 4.995 5.

37. Contents of the text • Chapter 1 Direct Methods for Solving Linear Systems • Chapter 2 Iterative Techniques in Matrix Algebra • Chapter 3 Interpolation and Polynomial Approximation • Chapter 4 Numerical Integration • Chapter 5 Initial-Value Problems for Ordinary Differential Equations

38. Text Book and References • Computional Methods, Chen Gongning and Shen Jiaji,Higher Education Press. • Computional Methods, Deng Jianzhong, Ge Renjie and Cheng Zhengxing, Xi’an Jiao Tong University Press. • Numerical Analysis,Richard L.Burden and J.Douglas Faires,Higher Education Press.

39. Numerical Analysis,David Kincaid Ward Cheney, China Machine Press. • Numerical Analysis,Li Qingyang,Wang Nengchao and Yi Dayi, Huazhong University of Science and Technology Press. • Computional Methods—Examples and Solutions,Gao Peiwang and Lei Yongjun, National University of Defense Technology Press.