1、References forNumerical Analysis 李慶揚，數(shù)值分析基礎教程，高教出版社，2001，北京 李慶揚 等，現(xiàn)代數(shù)值分析，高教出版社，1995，北京 王仁宏，數(shù)值逼近，高教出版社，1999，北京 裘宗燕，Mathematica 數(shù)學軟件系統(tǒng)的應用及其程序設計，北大出版社，1994，北京 J.Thomas King, 數(shù)值計算引論（程序），南大出版社，1988，寧 James,S. Vandergraft, Introduction to Numerical Computation , (2nd edition), Academic press,1983 J.Stoer,

2、R. Bulirsch, Introduction to Numerical Analysis, Spring-Verlag,1980(中譯本) JH Mathews,數(shù)值方法(Matlab版)第3版,北京:電子工業(yè)出版社,2002 Http://mathews numerical.html,0 Preface,Purpose: To give an introduction to modern approximation techniques Mathematica,1 Mathematics Preliminaries,1.1 Review of Cal

3、culus 1.2 Round off Errors and Computer Arithmetic 1.3 Algorithms and Convergence 1.4 Numerical Soft ware,(Intermediate Value Theorem),1.1 Review of Calculus,Limit of a sequence; Continuity; Differentiable, Derivative; Integral; Rolles Theorem.,1.1 Review of Calculus,1.1 Review of Calculus,(Taylors

4、Theorem),n-degree Taylor Polynomial, Taylor Series; remainder term;,1.1 Review of Calculus,(Mean Value Theorem for Integrals),1.2 Resources Underflow, Overflow,Definition1.15 If p* is an approximation to p ,the absolute error is | p- p* |, the relative error is | p- p* | / | p| provided that p 0. Th

5、e absolute error can be misleading and the relative error is more meaningful since the relative error takes into consideration the size of the value.,Definition1.16 The number p* is said to approximate p to n significant digits(or figures) if n is the largest nonnegative integer for which | p- p* |

6、/ | p| 5x10-n (in form of relative error) or more strictly, | p- p* | 1/2x10k-n, (in form of absolute error) where k is in scientific form of p=0.a1a2 an an+1. x10k.,then the bounds of two cases can be ：,Remark It means that an rounding approximation : p*=a1a2.ak. ak+1.an approximates to p=a1a2.ak.

7、ak+1.anan+1., the absolute error is less than half of the remaining unit. Consequently, the Def.1.6 in form of relative error is also satisfied as exploited below, so we usually obtain significant digit by using rounding method.,For the two versions of definition of figure, there is a description :,

8、Notation The relations among the definitions with respect to :(I) relative error , (II) absolute error, (III) rounding error: If a number possess n figures according to (III), then to (II), then to (I). But it doesnt follows vice verse. -non-equivalent !,Example: p=0.51414, p*=0.5142, then p* posses

9、s 4 figures according to (I), but 3 to (II) and (III); -critical cases p=0.15,p*=0.1,0.2, then p* possess 1 figure according to (II) and (III) .,1.3 Algorithms and Convergence,An algorithm is called stable when small changes in the initial data produce correspondingly small changes in the final resu

10、lts. Some algorithm are stable only for certain choices of initial data, those are called conditionally stable .,Suppose that E00 denotes an initial error and En represents the magnitude of an error after n subsequent operations. If EnC n E0, where C is a constant independent of n, then the growth of error is said to be linear; (unavoidable; stable) If EnCn E0, for some C1, exponential. (should be avoid; ),1.4 Numerical Software,Matlab; Mathematica; Maple etc.,1.5 函數(shù)計算的誤差估計,誤差危害及其防止,避免兩個相近的數(shù)相減；絕對值小的數(shù)作除數(shù) 辦法：改變計算公式；增加字長。 注意運算次序，防止