线性方程组的几种迭代法.doc

线性方程组的几种迭代法专业班级：信计061 姓名： 袁金龙 学号：15206012 一 实验目的1. 熟悉Matlab编程。2. 学习线性方程组迭代解法的程序设计算法。二 实验题目 研究解线性方程组Ax=b迭代法收敛速度。三 实验原理与理论基础 (一) 雅克比（Jacobi）迭代法算法设计： 输入矩阵a与右端向量b及初值x(1,i); 按公式计算得 （二）高斯赛得尔迭代法算法设计：1. 输入矩阵a与右端向量b及初值x(1,i). 2. （i = 1, 2, n） （三）超松驰法算法设计：输入矩阵a与右端向量b及初值x(1,i)。 ,四 实验内容（一）问题重述： 研究解线性方程组Ax=b迭代法收敛速度。要求：(1)选取不同的初始向量x0 及右端向量b，给定迭代误差要求，用雅可比迭代和高斯-赛德尔迭代法求解，观察得到的序列是否收敛？若收敛，记录迭代次数，分析计算结果并得出你的结论。(2)用SOR迭代法求解上述方程组，松弛系数取1 =1 for i=1:n m=0; %此步也可以用ifj=i条件判定一下。 for j=1:(i-1) m=m+a(i,j)*x(k,j); end for j=(i+1):n m=m+a(i,j)*x(k,j); end x(k+1,i)=(b(i)-m)/a(i,i); end l=0; %判定满足条件使循环停止迭代。 for i=1:n l=l+abs(x(k+1,i)-x(k,i); end if l=1 for i=1:n p=0;q=0; for j=1:(i-1) p=p+a(i,j)*x(k+1,j); end for j=(i+1):n q=q+a(i,j)*x(k,j); end x(k+1,i)=(b(i)-q-p)/a(i,i); end l=0; %判定满足条件使循环停止迭代。 for i=1:n l=l+abs(x(k+1,i)-x(k,i); end if l=1 if w=2|w=1 请重新输入w的值，w在1与2之间; break end for i=1:n p=0;q=0; for j=1:(i-1) p=p+a(i,j)*x(k+1,j); end for j=i:n q=q+a(i,j)*x(k,j); end x(k+1,i)=x(k,i)+w*(b(i)-q-p)/a(i,i); end l=0; %判定满足条件使循环停止迭代。 for i=1:n l=l+abs(x(k+1,i)-x(k,i); end if l yakebi(0.00001)ans = Columns 1 through 4 0.97928690957815 0.97865527545592 0.99413112710292 0.99702522668084 Columns 5 through 8 0.99892294027653 0.99953623723806 0.99981798692368 0.99992456206301 Columns 9 through 12 0.99996827213618 0.99998318569782 0.99998318569782 0.99996827213618 Columns 13 through 16 0.99992456206301 0.99981798692368 0.99953623723806 0.99892294027653 Columns 17 through 20 0.99702522668084 0.99413112710292 0.97865527545592 0.97928690957815k = 12+此时初值全取1；b=2.5 1.9 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.9 2.5;yakebi(0.00001)ans = Columns 1 through 4 1.09693565579242 1.07066544555281 1.02189792894430 1.01031973400603 Columns 5 through 8 1.00387123915809 1.00163550420902 1.00064915592407 1.00026828691766 Columns 9 through 12 1.00011377674600 1.00006071270830 1.00006071270830 1.00011377674600 Columns 13 through 16 1.00026828691766 1.00064915592407 1.00163550420902 1.00387123915809 Columns 17 through 20 1.01031973400603 1.02189792894430 1.07066544555281 1.09693565579242k = 14+*高斯赛得尔迭代法*+此时初值全取1；b=2.2 1.7 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.7 2.2;gaoshisaideer(0.00001)ans = Columns 1 through 4 0.97928672785055 0.97865497724617 0.99413072858837 0.99702477289046 Columns 5 through 8 0.99892247916398 0.99953582092589 0.99981765831007 0.99992433911341 Columns 9 through 12 0.99996813689365 0.99998309087521 0.99998307528667 0.99996810066011 Columns 13 through 16 0.99992430481311 0.99981764236491 0.99953582378631 0.99892249072258 Columns 17 through 20 0.99702478323251 0.99413073321783 0.97865497639848 0.97928672383457k = 9+ 此时初值全取1；b=2.5 1.9 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.9 2.5;gaoshisaideer(0.00001)ans = Columns 1 through 4 1.09693579352931 1.07066567042013 1.02189822461772 1.01032005741270 Columns 5 through 8 1.00387154239400 1.00163574176580 1.00064930348687 1.00026835270258 Columns 9 through 12 1.00011379767484 1.00006073762614 1.00006078469333 1.00011392257747 Columns 13 through 16 1.00026851424618 1.00064945546375 1.00163585410523 1.00387160976025 Columns 17 through 20 1.01032009166096 1.02189824119170 1.07066567972025 1.09693580005268k =10+ *超松驰法*+ caosongci(0.00001,1.5)ans = Columns 1 through 4 0.97928701590089 0.97865490271041 0.99413026958549 0.99702487773165 Columns 5 through 8 0.99892287174441 0.99953541295914 0.99981725926558 0.99992443005231 Columns 9 through 12 0.99996769517344 0.99998266391833 0.99998297217233 0.99996765389656 Columns 13 through 16 0.99992414492584 0.99981754006743 0.99953561936811 0.99892253859029 Columns 17 through 20 0.99702466465668 0.99413076724435 0.97865494383646 0.97928672836829k = 25+caosongci(0.00001,1.4)ans = Columns 1 through 4 0.97928664567787 0.97865497276614 0.99413091195128 0.99702478812825 Columns 5 through 8 0.99892206829160 0.99953566779334 0.99981800024840 0.99992405385342 Columns 9 through 12 0.99996735159261 0.99998299267004 0.99998292835147 0.99996756954667 Columns 13 through 16 0.99992425828155 0.99981747914644 0.99953562290255 0.99892255444268 Columns 17 through 20 0.99702464632961 0.99413077876016 0.97865493945573 0.97928672953468k = 19+caosongci(0.00001,1.3)ans = Columns 1 through 4 0.97928673960074 0.97865499490251 0.99413068018505 0.99702463941122 Columns 5 through 8 0.99892245408798 0.99953597158762 0.99981750927177 0.99992371086985 Columns 9 through 12 0.99996781471043 0.99998316284797 0.99998265476437 0.99996760364789 Columns 13 through 16 0.99992428696880 0.99981744084318 0.99953566673332 0.99892252714720 Columns 17 through 20 0.99702466742315 0.99413076397455 0.97865494656202 0.97928672698708k = 15+caosongci(0.00001,1.6)ans = Columns 1 through 4 0.97928734202960 0.97865524817880 0.99413049485300 0.99702500134349 Columns 5 through 8 0.99892252381996 0.99953549297181 0.99981769782089 0.99992404686608 Columns 9 through 12 0.99996771015468 0.99998291173863 0.99998260667004 0.99996787506558 Columns 13 through 16 0.99992410875882 0.99981740832167 0.99953581362369 0.99892233973293 Columns 17 through 20 0.99702480955440 0.99413067446860 0.97865498570698 0.97928671447299k = 34+caosongci(0.00001,1.7)ans = Columns 1 through 4 0.97928742815626 0.97865496966424 0.99413080915957 0.99702520913033 Columns 5 through 8 0.99892224601752 0.99953595378623 0.99981762797535 0.99992394886168 Columns 9 through 12 0.99996811815082 0.99998261630139 0.99998287220233 0.99996788477464 Columns 13 through 16 0.99992400317099 0.99981765085329 0.99953559957412 0.99892254975631 Columns 17 through 20 0.99702465244167 0.99413077958034 0.97865495091184 0.97928669699158k = 47+caosongci(0.00001,1.9)ans = Columns 1 through 4 0.9792