不动点迭代法非线性方程求解Word下载.docx
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不动点迭代法非线性方程求解Word下载.docx
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90
0.2588
0.5000
0.7071
0.8660
0.9659
1.0000
0.2679
0.5774
1.7320
3.7320
(2)按表6.5用3次多项式方法插值计算1~100之间整数的平方根。
表6.51~100内特殊值的平方根表
1
4
9
16
25
36
49
64
81
100
2
3
5
6
7
8
10
1、不动点迭代非线性方程求解
解:
算法说明:
在Matlab中编程实现不动点迭代法的函数为StablePoint
功能:
用不动点迭代法求函数的一个零点。
调用格式:
[root,n]=StablePoint(f,x0,eps)。
其中,f为函数名;
x0为初始迭代向量;
eps为根的精度;
root为求出的函数零点;
n为迭代步数。
流程图:
不动点迭代法的MATLAB程序代码:
function[root,n]=StablePoint(f,x0,eps)
%用不动点迭代法求函数f的一个零点
%初始迭代量:
x0
%根的精度:
eps
%求出的函数零点:
root
%迭代步数:
n
if(nargin==2)
eps=1.0e-4;
end
tol=1;
root=x0;
n=0;
while(tol>
eps)
n=n+1;
r1=root;
root=subs(sym(f),findsym(sym(f)),r1)+r1;
%迭代的核心公式
tol=abs(root-r1);
实例:
采用不动点迭代法求方程
的一个根。
在MATLAB命令窗口中输入程序代码:
>
[r,n]=StablePoint('
1/sqrt(x)+x-2'
0.5)
结果输出:
r=
0.3820
n=
4
从计算结果可以看出,经过四步迭代,得出方程的一个根为0.3820
2.编程解决以下科学计算问题
7、某工厂2005年度各季度产值(单位:
万元)分别为450.6,395.9,410.2,450.9,试绘制折线图和饼图,并说明图像的实际意义。
源程序代码:
%折线图
subplot(1,2,1)
plot([450.6,395.9,410.2,450.9])
title('
2005年度各季度产值-折线图'
);
%饼状图
subplot(1,2,2)
pie([450.6,395.9,410.2,450.9],1:
4,{'
第一季度'
'
第二季度'
第三季度'
第四季度'
})
2005年度各季度产值-饼图'
)
从折线图可以看出该工厂效益变化趋势,效益在第二季度最低随后逐渐提高,并在第四季度恢复到第一季度的水平;
从饼状图可以看出各个季度该工厂效益的比例关系。
从这两个图可以合理安排工厂的生产计划。
8.根据
symsaxy
eq=1/a^2*x^2+y^2/(25-a^2)-1;
aa=[0.5:
0.5:
3.5,5/sqrt
(2),3.6:
6.6];
[m,n]=size(aa);
fori=1:
eq1=subs(eq,a,aa(i));
ezplot(eq1,[-2020])
drawnow
axis([-2020-1010])
pause(0.5)
时,随着a增大曲线形状由长轴在y轴的椭圆逐渐转变为圆(此时
);
时a继续增大曲线形状由圆转变为长轴在x轴的椭圆;
a>
5时曲线变为双曲线。
2.按要求对指定函数进行插值和拟合。
A.正弦值算法:
x=0:
pi/12:
pi/2;
y=[00.25880.50000.70710.86600.96591.0000];
xi=0:
pi/180:
%三次样条差值
yi=interp1(x,y,xi,'
spline'
%五次多项式拟合
A=polyfit(x,y,5);
yj=polyval(A,xi)
运行结果:
yi=
Columns1through11
00.01750.03490.05240.06980.08720.10450.12190.13920.15640.1737
Columns12through22
0.19080.20790.22490.24190.25880.27560.29230.30900.32550.34200.3583
Columns23through33
0.37460.39070.40670.42260.43840.45400.46950.48480.50000.51500.5299
Columns34through44
0.54460.55920.57360.58780.60180.61570.62930.64280.65610.66910.6820
Columns45through55
0.69470.70710.71930.73130.74310.75470.76600.77710.78800.79860.8090
Columns56through66
0.81910.82900.83870.84800.85710.86600.87460.88290.89100.89870.9062
Columns67through77
0.91350.92040.92710.93350.93960.94540.95100.95630.96120.96590.9703
Columns78through88
0.97440.97820.98170.98490.98780.99040.99270.99460.99630.99770.9987
Columns89through91
0.99950.99991.0000
yj=
0.00000.01740.03490.05230.06970.08710.10450.12180.13910.15640.1736
0.19080.20790.22490.24190.25880.27560.29240.30900.32560.34200.3584
0.69460.70710.71930.73130.74310.75470.76600.77710.78800.79860.8090
0.81910.82900.83860.84800.85710.86600.87460.88290.89100.89880.9063
0.91350.92050.92720.93360.93970.94550.95100.95630.96120.96590.9703
0.97430.97810.98160.98480.98770.99020.99250.99450.99620.99750.9986
0.99940.99981.0000
通过比较,两种方法得到的结果近似相等。
B.正切值算法:
5*pi/12;
y=[00.26790.57741.00001.73203.7320];
00.01840.03650.05450.07240.09020.10790.12550.14310.16070.1784
0.19610.21380.23170.24970.26790.28630.30480.32360.34270.36200.3817
0.40170.42210.44290.46410.48580.50790.53050.55370.57740.60170.6266
0.65200.67800.70460.73170.75930.78760.81630.84560.87540.90580.9367
0.96811.00001.03251.06581.10031.13641.17431.21451.25721.30281.3516
1.40411.46041.52111.58631.65651.73201.81311.90021.99362.09372.2008
Columns67through76
2.31522.43742.56752.70602.85323.00953.17523.35063.53613.7320
-0.00000.02350.04540.06580.08500.10320.12060.13750.15400.17010.1862
0.20220.21830.23450.25110.26790.28510.30280.32080.33940.35850.3781
0.39820.41880.44000.46160.48380.50650.52970.55330.57740.60200.6270
0.65240.67830.70470.73150.75880.78670.81500.84400.87360.90390.9351
0.96701.00001.03411.06931.10601.14421.18411.22591.26991.31621.3652
1.41711.47231.53101.59351.66041.73201.80871.89101.97932.07422.1762
2.28602.40402.53102.66772.81472.97273.14273.32533.52143.7320
通过比较知,角度较小时五次多项式算得的值较大,角度增大则两种方法得到的结果近似相等。
x=[149162536496481100];
y=[12345678910];
xi=1:
100;
f=interp1(x,y,xi,'
cubic'
结果:
f=
1.00001.37291.71252.00002.24052.45512.64942.82923.00003.16363.3186
3.46613.60693.74223.87294.00004.12374.24354.35994.47304.58324.6907
4.79584.89885.00005.09935.19665.29215.38575.47775.56815.65705.7446
5.83095.91606.00006.08296.16476.24546.32496.40356.48106.55776.6334
6.70826.78236.85566.92817.00007.07127.14167.21137.28047.34877.4164
7.48357.55007.61597.68127.74597.81027.87397.93728.00008.06238.1242
8.18558.24648.30688.36688.42638.48548.54418.60248.66038.71788.7749
8.83178.88818.94429.00009.05559.11079.16559.22019.27449.32849.3821
Columns89through99
9.43549.48849.54129.59359.64569.69739.74869.79969.85029.90059.9505
Column100
10.0000
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