数值分析新课件教学作者教学专用 NA07b.ppt

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1、§2复合求积/*CompositeQuadrature*/Haven’twehadenoughformulae?What’supnow?Ohcomeon,youdon’tseriouslyconsiderh=(ba)/2acceptable,doyou?Whycan’tyousimplyrefinethepartitionifyouhavetobesopicky?Don’tyouforgettheoscillatorynatureofhigh-degreepolynomials!Uh-oh高次插值有Runge现象，故采用分段低次插值分段低次合成的Newton-Cotes复合求积公式。

2、复合梯形公式：在每个上用梯形公式：=Tn/*中值定理*/§2CompositeQuadrature复化Simpson公式：44444=Sn注：为方便编程，可采用另一记法：令n’=2n为偶数，这时，有§2CompositeQuadrature收敛速度与误差估计：定义若一个积分公式的误差满足且C0，则称该公式是p阶收敛的。~~~例：计算解：其中=3.138988494其中=3.141592502运算量基本相同例：用复化梯形公式、复化Simpson公式和复化Cotes公式计算要使精度要求达到问N各取多少？HW:p.2731，3，4，5，6§2CompositeQuadratureQ:给

3、定精度，如何取n?例如：要求，如何判断n=？?上例中若要求，则即：取n=409通常采取将区间不断对分的方法，即取n=2k上例中2k409k=9时，T512=3.14159202S4=3.141592502注意到区间再次对分时可用来判断迭代是否停止。§2CompositeQuadratureLab13.CompositeTrapezoidalRuleUsetheCompositeTrapezoidalrulewithagivenn>0toapproximateagivenintegral.YouaresupposedtowriteafunctiondoubleCTR(intn,dou

4、blea,doubleb,double(*f)())toapproximatetheintegralfromatobofthefunctionfusingthetrapezoidalruleonnequal-lengthsubintervals.InputThereisnoinputfile.Instead,youmusthandinyourfunctionina*.hfile.Theruleofnamingthe*.hfileisthesameasthatofnamingthe*.cor*.cppfiles.OutputForeachtestcase,youaresupposedtor

5、eturntheapproximationoftheintegral.§2CompositeQuadratureSampleJudgeProgram#include#include#include"98115001_13.h"doublef1(doublex){return(1.0/(1.0+sin(x)*sin(x)));}doublef2(doublex){return(x*log(x));}voidmain(){FILE*outfile=fopen("out.txt","w");intn;doublea,b;a=0.0;b=1.0;n=10;fpr

6、intf(outfile,"%lf",CTR(n,a,b,f1));a=1.0;b=2.0;n=4;fprintf(outfile,"%lf",CTR(n,a,b,f2));fclose(outfile);}SampleOutput0.8090930.639900§3龙贝格积分/*RombergIntegration*/例：计算已知对于=106须将区间对分9次，得到T512=3.14159202由来计算I效果是否好些？考察=3.141592502=S4一般有：Romberg序列Romberg算法：<?<?<?………………T1=)0(0TT8=)3(0TT4=)2

7、(0TT2=)1(0TS1=)0(1TR1=)0(3TS2=)1(1TC1=)0(2TC2=)1(2TS4=)2(1T§3RombergIntegrationLab14.RombergIntegrationUseRombergintegrationtoapproximateagivenintegral.YouaresupposedtowriteafunctiondoubleRomberg(doublea,doubleb