迭代法求解线性方程组.ppt

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1、第六章线性方程组的迭代解法/*iterationmethodsforthesolutionoflinearsystems*/Linearsystems：Ax=bMatrixformAx=bAx*=bx(k+1)=f(x(k))x(k),k=0,1,2,…hopefully,limx(k)=x*Iterativemethod:givenalinearsystemAx=b,designaniterationformulax(k+1)=f(x(k))andchooseaninitialapproximatesolut

2、ionx(0).iterationresultsinaseriesapproximatesolutions{x(k)

3、k∈Z}whichapproachestotherealsolutionx*hopefully.x(0)Howtodesigntheiterationformula?Bisnotunique,sotheiterationformulaisnotunique!Ax=bx=Bx+fx(k+1)=Bx(k)+fEquavalentreformationIterationmatrix迭代法思想：第一步第

4、二步B与k无关，称为一阶定常迭代法收敛？发散？判断收敛的方法：计算中判断迭代终止条件的方法：LUD6.2基本迭代法Jacobiiteration取M为DMatrixformJacobi迭代法简单，迭代一次只需作一次矩阵与向量的乘法即可。ComponentformGauss-Seideliteration高斯－塞德尔迭代法取M为D+LGauss-SeideliterationComponentformJacobi分量形式comparisonJacobiiterationGauss-Seideliteration计

5、算x(k+1)时需要x(k)的所有分量，因此需开两组存储单元分别存放x(k)和x(k+1)计算xi(k+1)时只需要x(k)的i+1~n个分量，因此x(k+1)的前i个分量可存贮在x(k)的前i个分量所占的存储单元，无需开两组存储单元迭代收敛性定义3矩阵收敛性Errorvectorofiteration例2Howtocheckifacertainiterationsystemconvergesornot?定理3迭代收敛的等价条件NotflexibletouseactuallyPosteriorerror–est

6、imatedintheprocessofiterationPriorerror—estimatedbeforetheiterationBotherrorestimationcanbeusedtocontrolwhentohalttheiterationexampleJacobiiterationG-SiterationInthisexample,G-SiterationconvergesfasterthanJacobiiteration.It’snotalwaystrue.Sometimesthelaterco

7、nvergesfaster.Forsomelinearsystems,G-SconvergeswhileJdiverges,andviceversa.收敛速度G-SiterationconvergesexampleJacobiiterationmatrixB=-D-1(L+U)G-SiterationmatrixG=-(D+L)-1UJacobiiterationdivergesConvergencyfortwospecialmatrix1.A是对称正定矩阵G-S收敛.2.A是严格对角占优矩阵或A是弱对角占优

8、，且不可约矩阵，则J和G-S均收敛。Strictlydiagonallydominantweaklydiagonallydominantorderrordern-rAreducible,elseAnotreducible.Example3G-SiterationdivergesJacobiiterationdivergesStrictlydiagonallydominantJ,G-SiterationconvergeExample4StrictlydiagonallydominantJ,G-Siterat

9、ionconvergeEuavalentreformation:2*(1)+(2)(1)+(3)!note:ifthegivenlinearsystemdoesn’tsatisfytheconvergencecondition,wecanmodifytheorderoftheequationsordosomelinearcombinationstogetanequivalentsyst