第五節(jié)松弛法演示文稿

第五節(jié)松弛法演示文稿當(dāng)前第1頁(yè)\共有12頁(yè)\編于星期三\9點(diǎn)優(yōu)選第五節(jié)松弛法當(dāng)前第2頁(yè)\共有12頁(yè)\編于星期三\9點(diǎn)§5RelaxationMethods當(dāng)前第3頁(yè)\共有12頁(yè)\編于星期三\9點(diǎn)Gauss-SeidelIterativeMethod…………寫成矩陣形式：BGauss-Seidel

迭代陣§5RelaxationMethods當(dāng)前第4頁(yè)\共有12頁(yè)\編于星期三\9點(diǎn)Gauss-Seidel…………當(dāng)前第5頁(yè)\共有12頁(yè)\編于星期三\9點(diǎn)§5RelaxationMethods當(dāng)前第6頁(yè)\共有12頁(yè)\編于星期三\9點(diǎn)§5RelaxationMethods當(dāng)前第7頁(yè)\共有12頁(yè)\編于星期三\9點(diǎn)定理設(shè)A

可逆，且aii0，松弛法從任意出發(fā)對(duì)某個(gè)收斂

(L)<1?！?RelaxationMethods當(dāng)前第8頁(yè)\共有12頁(yè)\編于星期三\9點(diǎn)§5RelaxationMethods定理

（Kahan必要條件）設(shè)A

可逆，且aii0，松弛法從任意出發(fā)收斂0<<2

。證明：，而且收斂|i|<1總成立已知收斂|det(L)|<1|det(L)|=|1|n<10<<2

當(dāng)前第9頁(yè)\共有12頁(yè)\編于星期三\9點(diǎn)§5RelaxationMethods定理

（Ostrowski-Reich充分條件）若A對(duì)稱正定，且有0<<2，則松弛法從任意出發(fā)收斂。（證明以后給出）Q:Whatfactordeterminesthespeedofconvergence?A:

Thesmaller

(B)is,thefastertheiterationswillconverge.對(duì)于SOR法，希望找使得(L)

最小。當(dāng)前第10頁(yè)\共有12頁(yè)\編于星期三\9點(diǎn)§5RelaxationMethods定理若A

為對(duì)稱正定三對(duì)角陣，則且SOR的最佳松弛因子

/*optimalchoiceofforSORmethod*/為，此時(shí)。例：，考慮迭代格式問(wèn)：取何值可使迭代收斂？

取何值時(shí)迭代收斂最快？解：考察B=I+A

的特征根1=1+,2=1+3

收斂要求

(B)<1-2/3<<0

(B)=

max{|1+

|,|1+3

|}

當(dāng)取何值時(shí)最?。?2/3-1/30=-1/2當(dāng)前第11頁(yè)\共有12頁(yè)\編于星期三\9點(diǎn)§5RelaxationMethodsLab08.SORMethod UsetheSORmethodtosolveagivenn×nlinearsystemwithaninitialapproximationandasetof’s.Input Thereareseveralsetsofinputs.Foreachset: The1stlinecontainsaninteger100

n

0whichisthesizeofamatrix.n=1signalstheendoffile. Thefollowingnlinescontaintheaugmentedmatrixinthefollowingformat:Thenumbersareseparatedbyspacesandnewlines.ThenextlinecontainsarealnumberTOL,whichisthetolerancefor||·||norm,andanintegerN

0whichisthemaximumnumberofiterations.