第五节松弛法.ppt

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1、§5松弛法/*RelaxationMethods*/换个角度看Jacobi方法：Jacobi第K次迭代结果§5RelaxationMethods§5RelaxationMethodsGauss-SeidelIterativeMethod…………写成矩阵形式：BGauss-Seidel迭代阵§5RelaxationMethodsGauss-Seidel…………§5RelaxationMethods§5RelaxationMethods定理设A可逆，且aii0，松弛法从任意出发对某个收敛(L)<1。§5

2、RelaxationMethods§5RelaxationMethods定理（Kahan必要条件）设A可逆，且aii0，松弛法从任意出发收敛0<<2。证明：，而且收敛

3、i

4、<1总成立已知收敛

5、det(L)

6、<1

7、det(L)

8、=

9、1

10、n<10<<2§5RelaxationMethods定理（Ostrowski-Reich充分条件）若A对称正定，且有0<<2，则松弛法从任意出发收敛。（证明以后给出）Q:Whatfactordeterminesthespeedofconvergence

11、?A:Thesmaller(B)is,thefastertheiterationswillconverge.对于SOR法，希望找使得(L)最小。§5RelaxationMethods定理若A为对称正定三对角阵，则且SOR的最佳松弛因子/*optimalchoiceofforSORmethod*/为，此时。例：，考虑迭代格式问：取何值可使迭代收敛？取何值时迭代收敛最快？解：考察B=I+A的特征根1=1+,2=1+3收敛要求(B)<1-2/3<<0(B)=max{

12、1+

13、,

14、

15、1+3

16、}当取何值时最小？-2/3-1/30=-1/2§5RelaxationMethodsLab08.SORMethodUsetheSORmethodtosolveagivenn×nlinearsystemwithaninitialapproximationandasetof’s.InputThereareseveralsetsofinputs.Foreachset:The1stlinecontainsaninteger100n0whichisthesizeofamatrix.n=1sig

17、nalstheendoffile.Thefollowingnlinescontaintheaugmentedmatrixinthefollowingformat:Thenumbersareseparatedbyspacesandnewlines.ThenextlinecontainsarealnumberTOL,whichisthetolerancefor

18、

19、·

20、

21、norm,andanintegerN0whichisthemaximumnumberofiterations.Thelastlineofeac

22、htestcasecontainsanintegerm>0,followedbymreal’s.§5RelaxationMethodsOutput(representsaspace)Foreach,theremustbeasetofoutputsinthefollowingformat:The1stlinecontainsanandthecorrespondingnumberofiterationstaken.IntheCprintf:fprintf(outfile,"%4.2f%d",omeg

23、a,iter_no);Thecorrespondingsolutionorerrormessagesaretobeprintedasthefollowing:EachentryofthesolutionistobeprintedasintheCfprintf:fprintf(outfile,"%12.8f",x);IfthematrixAhasazerocolumn,printthemessage“Matrixhasazerocolumn.Nouniquesolutionexists.

24、n”.IfthemethodfailstogiveasolutionafterNiterations,printthemessage“Maximumnumberofiterationsexceeded.”.Ifthereisanentryofthatisoutoftherange[2127,2127],printthemessage“Noconvergence.”.Theoutp