课题一 迭代格式的比较.doc

课题一 迭代格式的比较 要求 :1、 编制一个程序进行运算，最后打印出每种迭代格式的敛散情况； 、建立迭代程序的M文件：function k,piancha,xk=diedail(x0,k) x(1)=x0for i = 1:k x(i+1)=funl(x(i)%用所用的格式迭代 piancha=abs(x(i+1)-x(i);%偏差 i=i+1; xk=x(i);%第k次迭代的结果 (i-1) piancha xkend p=(i-1) piancha xk;%输出迭代次数、偏差和第k次迭代的结果、对于不同的迭代式子建立不同的funl.m文件（1）建立迭代式子（1）的M文件：function y1 =funl(x) y1=(3*x+1)/x2;在matlab命令窗口中运行以下命令 k,piancha,xk=diedail(0.5,5) %初值是0.5，迭代5次x = 0.5000x = 0.5000 10.0000ans = 1.0000 9.5000 10.0000x = 0.5000 10.0000 0.3100ans = 2.0000 9.6900 0.3100x = 0.5000 10.0000 0.3100 20.0832ans = 3.0000 19.7732 20.0832x = 0.5000 10.0000 0.3100 20.0832 0.1519ans = 4.0000 19.9314 0.1519x = 0.5000 10.0000 0.3100 20.0832 0.1519 63.1191ans = 5.0000 62.9673 63.1191k = 5piancha = 62.9673xk = 63.1191由以上结果可知迭代式（1）是发散的。（2）建立迭代式子（2）的M文件：function y1 =funl(x) y1=(x3-1)/3;在matlab命令窗口中运行以下命令 k,piancha,xk=diedail(0.5,7)x = 0.5000x = 0.5000 -0.2917ans = 1.0000 0.7917 -0.2917x = 0.5000 -0.2917 -0.3416ans = 2.0000 0.0499 -0.3416x = 0.5000 -0.2917 -0.3416 -0.3466ans = 3.0000 0.0050 -0.3466x = 0.5000 -0.2917 -0.3416 -0.3466 -0.3472ans = 4.0000 0.0006 -0.3472x = 0.5000 -0.2917 -0.3416 -0.3466 -0.3472 -0.3473ans = 5.0000 0.0001 -0.3473x = 0.5000 -0.2917 -0.3416 -0.3466 -0.3472 -0.3473 -0.3473ans = 6.0000 0.0000 -0.3473x = 0.5000 -0.2917 -0.3416 -0.3466 -0.3472 -0.3473 -0.3473 -0.3473ans = 7.0000 0.0000 -0.3473k = 7piancha = 1.0399e-006xk = -0.3473由以上结果可知迭代式（2）是收敛的。（3）建立迭代式子（3）的M文件：function y1 =funl(x) y1=(3*x+1)(1/3);在matlab命令窗口中运行以下命令： k,piancha,xk=diedail(0.5,10)x = 0.5000x = 0.5000 1.3572ans = 1.0000 0.8572 1.3572x = 0.5000 1.3572 1.7181ans = 2.0000 0.3609 1.7181x = 0.5000 1.3572 1.7181 1.8326ans = 3.0000 0.1145 1.8326x = 0.5000 1.3572 1.7181 1.8326 1.8660ans = 4.0000 0.0335 1.8660x = 0.5000 1.3572 1.7181 1.8326 1.8660 1.8756ans = 5.0000 0.0096 1.8756x = 0.5000 1.3572 1.7181 1.8326 1.8660 1.8756 1.8783ans = 6.0000 0.0027 1.8783x = 0.5000 1.3572 1.7181 1.8326 1.8660 1.8756 1.8783 1.8791ans = 7.0000 0.0008 1.8791x = 0.5000 1.3572 1.7181 1.8326 1.8660 1.8756 1.8783 1.8791 1.8793ans = 8.0000 0.0002 1.8793x = 0.5000 1.3572 1.7181 1.8326 1.8660 1.8756 1.8783 1.8791 1.8793 1.8794ans = 9.0000 0.0001 1.8794x = 0.5000 1.3572 1.7181 1.8326 1.8660 1.8756 1.8783 1.8791 1.8793 1.8794 1.8794ans = 10.0000 0.0000 1.8794k = 10piancha = 1.7458e-005xk = 1.8794由以上结果可知迭代式（3）是收敛的。（4）建立迭代式子（4）的M文件：function y1 =funl(x) y1=1/(x2-3);在matlab命令窗口中运行以下命令： k,piancha,xk=diedail(0.5,5)x = 0.5000x = 0.5000 -0.3636ans = 1.0000 0.8636 -0.3636x = 0.5000 -0.3636 -0.3487ans = 2.0000 0.0149 -0.3487x = 0.5000 -0.3636 -0.3487 -0.3474ans = 3.0000 0.0013 -0.3474x = 0.5000 -0.3636 -0.3487 -0.3474 -0.3473ans = 4.0000 0.0001 -0.3473x = 0.5000 -0.3636 -0.3487 -0.3474 -0.3473 -0.3473ans = 5.0000 0.0000 -0.3473k = 5piancha = 9.0701e-006xk = -0.3473由以上结果可知迭代式（4）是收敛的。（5）建立迭代式子（5）的M文件：function y1 =funl(x) y1=(3+1/x)(1/2);在matlab命令窗口中运行以下命令： k,piancha,xk=diedail(0.5,8)x = 0.5000x = 0.5000 2.2361ans = 1.0000 1.7361 2.2361x = 0.5000 2.2361 1.8567ans = 2.0000 0.3794 1.8567x = 0.5000 2.2361 1.8567 1.8811ans = 3.0000 0.0244 1.8811x = 0.5000 2.2361 1.8567 1.8811 1.8793ans = 4.0000 0.0019 1.8793x = 0.5000 2.2361 1.8567 1.8811 1.8793 1.8794ans = 5.0000 0.0001 1.8794x = 0.5000 2.2361 1.8567 1.8811 1.8793 1.8794 1.8794ans = 6.0000 0.0000 1.8794x = 0.5000 2.2361 1.8567 1.8811 1.8793 1.8794 1.8794 1.8794ans = 7.0000 0.0000 1.8794x = 0.5000 2.2361 1.8567 1.8811 1.8793 1.8794 1.8794 1.8794 1.8794ans = 8.0000 0.0000 1.8794k = 8piancha = 5.9875e-008xk =1.8794由以上结果可知迭代式（5）是收敛的。（6）建立迭代式子（6）的M文件：function y1 =funl(x) y1=x-(1/3)*(x3-3*x-1)/(x2-1);在matlab命令窗口中运行以下命令： k,piancha,xk=diedail(0.5,5)x = 0.5000x = 0.5000 -0.5556ans = 1.0000 1.0556 -0.5556x = 0.5000 -0.5556 -0.3168ans = 2.0000 0.2388 -0.3168x = 0.5000 -0.5556 -0.3168 -0.3470ans = 3.0000 0.0302 -0.3470x = 0.5000 -0.5556 -0.3168 -0.3470 -0.3473ans = 4.0000 0.0003 -0.3473x = 0.5000 -0.5556 -0.3168 -0.3470 -0.3473 -0.3473ans = 5.0000 0.0000 -0.3473k = 5piancha = 4.5086e-008xk = -0.3473由以上结果可知迭代式（6）是收敛的。2、 用事后误差估计来控制迭代次数，并且打印出迭代的次数； 建立diedail.m文件程序如下：function k,piancha,xk=diedail(x0)x(1)=x0for i = 1:100x(i+1)=funl(x(i)piancha=abs(x(i+1)-x(i);(i-1) piancha x(i)if (piancha0.0000000005)%误差小于0.0000000005k=i,xk=x(i)breakendi=i+1;endk piancha xk对于不同的收敛迭代式运行命令k,piancha,xk=diedail(0.5)，有以下结果：（2）k = 11piancha = 2.2008e-010xk = -0.3473（3）k = 19piancha = 2.0403e-010xk = 1.8794（4）k = 9piancha = 4.4683e-010xk = -0.3473（5）k = 10piancha = 3.3969e-010xk = 1.8794（6）k = 6piancha = 8.3267e-016xk = -0.3473 3、 初始值的选取对迭代收敛有何影响； 对于f(x)，如果其迭代式满足不动点存在的两个条件，则在其要求的区间内存在不动点，那么对于其所要求的区间上的所有点此迭代式都收敛。也就是说，此时初值的选取对于迭代收敛没有影响。又称全局收敛性。以上四个收敛式都有全局收敛性。 初值的不同对于迭代式的敛散性和收敛阶没有影响，只是对于达到所要求误差的达到次数有一定的影响。4、分析迭代收敛和发散的原因。