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实验六 迭代（方程求解）

一.实验目的：认识迭代数列，考察迭代数列的收敛性.并学会用Mathematica系统对线性和非线性的方程组进行迭代求解.

二.实验环境：计算机，Mathematica 数学软件,Word文档，课本。 三.实验的基本理论和方法：

给定迭代函数f(x)以及一个初值x0利用xn?1?f(xn),n?0,1,???迭代得到数列xn，n?0,1,???.如果数列xn收敛与某个x，则有x?f(x).即x是方程

****x?f(x)的解.由此用如下的方法求方程g(x)?0的近似解。

将方程g(x)?0改写为等价的方程x?f(x)，然后选取一初值利用

xn?1?f(xn),n?0,1,???做迭代.迭代数列xn收敛的极限就是g(x)?0的解.线

性方程组以及非线性方程组的求解与单变量的方程求解方法类似.实验内容和步骤

四.实验内容与结果 1.线性方程组

⑴编写给定初值x0及迭代函数f(x)，迭代n次产生相应的序列.

⑵给函数f(x)?(x?2/x)初值为0进行迭代80次所产生的迭代序列并显示. 输入程序：

Iteratef_,x0_,n_Integer:ModuleForit1,i,i,tempn,itfx_:x2x2;x0,ftemp;;AppendTot,temp;,tempAppendTot,tempIteratef,1.,80

运行结果得：

1.,1.5,1.41667,1.41422,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421,1.41421

输入程序：

NTIterateg_,x0_,n_Integer:Modulei,varhx_Fori1,ix0,t,h,Dtgx,x;n,i,AppendTot,var;Nvargvarhvar,20,0,varIfhvarPrint\byZeroafter\i,\iterations.\;Breaktgx_:x^32;; NTIterateg,1,40

运行结果得：

1,1.3333333333333333333,1.2638888888888888889,1.2599334934499769665,1.259921050017769774,1.259921049894873165,1.259921049894873165,1.259921049894873165,1.25992104989487316,1.25992104989487316,1.25992104989487316,1.2599210498948732,1.2599210498948732,1.2599210498948732,1.259921049894873,1.259921049894873,1.259921049894873,1.259921049894873,1.25992104989487,1.25992104989487,1.25992104989487,1.2599210498949,1.2599210498949,1.2599210498949,1.259921049895,1.259921049895,1.259921049895,1.259921049895,1.25992104989,1.25992104989,1.25992104989,1.2599210499,1.2599210499,1.2599210499,1.259921050,1.259921050,1.259921050,1.259921050,1.25992105,1.25992105

2. 非线性方程组

⑴对于给定的矩阵M，数组f和初始向量x，由迭代x代程序，并选择初值分别迭代20和50次所产生的序列. 迭代40次运行结果： 输入程序：

LSIteratem_,f_List,f0_List,n_Integer:Modulei,varForitm1,0.4,0.5,1;f1,1;f00,0;1,if0,tn,iTable,ti,i,n,m.varf;var;var0n?1?Mxn?f编写迭

LSIteratem,f,f0,40

运行结果得：

0,0,1.,1.,2.4,2.5,4.4,4.7,7.28,7.9,11.44,12.54,17.456,19.26,26.16,28.988,38.7552,43.068,56.9824,63.4456,83.3606,92.9368,121.535,135.617,176.782,197.385,256.736,286.776,372.446,416.144,539.904,603.367,782.251,874.319,1132.98,1266.44,1640.56,1833.93,2375.13,2655.21,3438.22,3843.78,4976.73,5563.88,7203.28,8053.25,10425.6,11655.9,15088.9,16869.7,21837.8,24415.1,31604.9,35335.,45739.9,51138.5,66196.3,74009.4,95801.,107109.,138645.,155010.,200650.,224334.,290385.,324660.,420250.,469854.,608192.,679980.,880185.,984077.,1.2738210,1.4241710,1.8434910,2.0610810,2.6679210,2.9828210,3.8610510,4.316781066666666

迭代60次运行结果 输入程序：

LSIteratem_,f_List,f0_List,n_Integer:Modulei,varForitm1,0.4,0.5,1;f1,1;f00,0;1,if0,tn,iTable,ti,i,n,m.varf;var;varLSIteratem,f,f0,60

运行结果得：

1,1.3333333333333333333,1.2638888888888888889,1.2599334934499769665,1.259921050017769774,1.259921049894873165,1.259921049894873165,1.259921049894873165,1.25992104989487316,1.25992104989487316,1.25992104989487316,1.2599210498948732,1.2599210498948732,1.2599210498948732,1.259921049894873,1.259921049894873,1.259921049894873,1.259921049894873,1.25992104989487,1.25992104989487,1.25992104989487,1.2599210498949,1.2599210498949,1.2599210498949,1.259921049895,1.259921049895,1.259921049895,1.259921049895,1.25992104989,1.25992104989,1.25992104989,1.2599210499,1.2599210499,1.2599210499,1.259921050,1.259921050,1.259921050,1.259921050,1.25992105,1.25992105

⑵改写矩阵的等价形式，给定数组f和初始向量x，运用迭代格式

0x?(I?D?1A)x?D?1b编写迭代程序。

输入程序：

JacobIteratea_,b_List,x0_List,n_Integer:ModuleForiForiForjvar1Nb;var;ab3,1,3,21,1;x0;var1;jSumak,adj,kaj,jvark,j,20varad1,i1,i1,jjLengtha,i,j,k,varad,in,iad,j,Ifai,ivar10,;x0,Print\\i,\i,\,0.\;Abort,Printvar;0,0;JacobIteratea,b,x0,10

运行结果得：

0,0

0.33333333333333333333,0.50000000000000000000 0.16666666666666666667,0.1020

0.3333333333333333333,0.2500000000000000000 0.2500000000000000000,0.1020

0.3333333333333333333,0.1250000000000000000 0.291666666666666667,0.1019

0.333333333333333333,0.062500000000000000 0.312500000000000000,0.1018

0.33333333333333333,0.03125000000000000

输入程序：