迭代法求解线性方程组.ppt

第六章线性方程组的迭代解法 iterationmethodsforthesolutionoflinearsystems Linearsystems Ax b Matrixform Ax bAx b Iterativemethod givenalinearsystemAx b designaniterationformulax k 1 f x k andchooseaninitialapproximatesolutionx 0 iterationresultsinaseriesapproximatesolutions x k k Z whichapproachestotherealsolutionx hopefully Howtodesigntheiterationformula Bisnotunique sotheiterationformulaisnotunique Ax b x Bx f Equavalentreformation Iterationmatrix 迭代法思想 第一步 第二步 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 L U D 6 2基本迭代法 Jacobiiteration 取M为D Matrixform Jacobi迭代法简单 迭代一次只需作一次矩阵与向量的乘法即可 Componentform Gauss Seideliteration高斯 塞德尔迭代法 取M为D L Gauss Seideliteration Componentform Jacobi分量形式 comparison 迭代收敛性 定义3矩阵收敛性 Errorvectorofiteration 例2 Howtocheckifacertainiterationsystemconvergesornot 定理3迭代收敛的等价条件 Notflexibletouseactually Posteriorerror estimatedintheprocessofiteration Priorerror estimatedbeforetheiteration example Jacobiiteration G Siteration 收敛速度 G Siterationconverges example JacobiiterationmatrixB D 1 L U G SiterationmatrixG D L 1U Jacobiiterationdiverges Example3 G Siterationdiverges Jacobiiterationdiverges Strictlydiagonallydominant J G Siterationconverge JacobiorG Siterationcanbeusedtosolvelinearsystemsbutsometimesitconvergesveryslowly howtoaccelerateit SOR successiveoverrelaxedmethod accelerationofG Siteration W1 overrelaxedW 1 G S SupposehasbeenfoundbyusingG S nowwehavetofind SOR successiveoverrelaxedmethod accelerationofG Siteration NOTE ThekeyprobleminSORishowtochoosesuchawthatSORconvergesfastest theproblemofhowtochoosethebestrelaxedfactorw Presently theproblemhasbeensolvedforafewspecialmatrices Forthegeneralcase successivesearchingmethodisused Atthestart chooseoneormoredifferentwtotrySOR Thenmodifywaccordingtothespeedofconvergenceandsuccessivelyfindthebestw Finallyfixwandcontinueiteration Intheory byiterationwecangetapproximatesolutiontoanyaccuracyexpected Actually however duetothelimitofcomputerwordlength wecan tarriveatanyaccuracybutthemachineaccuracyatmost Sow