《数学数值分析》PPT课件

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1、第九章线性与非线性方程组的迭代解法/*iterationmethodsforthesolutionoflinearornonlinearsystems*/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))andchooseaninitialapproximatesolutionx(0).ite

2、rationresultsinaseriesapproximatesolutions{x(k)

3、kZ}whichapproachestotherealsolutionx*hopefully.x(0)Howtodesigntheiterationformula?LUDAx=bx=Bx+fx(k+1)=Bx(k)+fJacobiiterationMatrixformComponentformConvenientinprogrammingGauss-SeideliterationComponentformConvenientinprogrammingMatrixfo

4、rmcomparisonJacobiiterationGauss-Seideliteration计算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ConvergenceofiterationConvergenceofmatrixErrorvectorofiterationexampleJacobiiterationG-SiterationHowtocheckifacertaini

5、terationsystemconvergesornot?ConditionsofconvergenceNotflexibletouseactuallyPosteriorerror–estimatedintheprocessofiterationPriorerror—estimatedbeforetheiterationProof:fromJordanstandardformofB,weknowcolumnj+1Rowj+1Proof:Fromabove,wehaveSummary:Bp或(B)越小，迭代法的收敛速度越快；若事先给出误差精度，由

6、事先误差估计式可得到迭代次数的估计实际计算中，若Bp不太接近于1，利用事后误差估计作为控制迭代停止的条件，即当x(k)-x(k-1)p<时，迭代终止，并取x(k)作为近似解；迭代法是否收敛主要取决于迭代矩阵的性质，可用条件2，3判断迭代法是否收敛。G-SiterationdivergesexampleJacobiiterationmatrixB=D-1(L+U)G-SiterationmatrixG=-(D+L)-1UJacobiiterationdivergesConvergencyfortwospecialmatrix1.(Th4)Aiss

7、ymmetricandpositivedefiniteG-Sconverges.(seeproofofTh7later.)2.(Th5)AisstrictlydiagonallydominantorweaklydiagonallydominantandnotreducibleJandG-Sbothconverge.Strictlydiagonallydominantweaklydiagonallydominantorderrordern-rAreducible,elseAnotreducible.!note:1.Permutationmatrix2.Are

8、ducibleAx=bcanbereducedtotwolinearsystemsoflowerorderbypermutation(exchangetworows,e.g.rowiandjaswellastwocorrespondingcolumns,coliandj).3.Areduciblethereexistsanonemptysubsetoftheindex,J{1,2,,…,n},s.t.akj=0,kJ,jJ.r-dimExchangrowianj(n-r)-dimExchangcolianjLemma1:Aisstrictlydiago

9、nallydominan