计算方法实验报告

1、 中北大学信息商务学院计算方法实验报告学生姓名： 刘昊文 学号： 1603042130 学 院： 中北大学信息商务学院 专 业： 电气工程及其自动化 指导教师： 薛晓健 2017 年 04 月 19 日推荐精选实验一：非线性方程的近似解法1 实验目的 1.掌握二分法和牛顿迭代法的原理 2.根据实验内容编写二分法和牛顿迭代法的算法实现 注：（可以用C语言或者matlab语言）2实验设备matlab3实验内容及步骤解方程f(x)=x5-3x3-2x2+2=04 实验结果及分析二分法：数据：f =x5-3*x3-2*x2+2 n xa xb xc fc 推荐精选 1 -3 3 0 2 2.0000

2、-3.0000 0 -1.5000 0.0313 3.0000 -3.0000 -1.5000 -2.2500 -31.6182 4.0000 -2.2500 -1.5000 -1.8750 -8.4301 5.0000 -1.8750 -1.5000 -1.6875 -2.9632 6.0000 -1.6875 -1.5000 -1.5938 -1.2181 7.0000 -1.5938 -1.5000 -1.5469 -0.5382 8.0000 -1.5469 -1.5000 -1.5234 -0.2405 9.0000 -1.5234 -1.5000 -1.5117 -0.1015 1

3、0.0000 -1.5117 -1.5000 -1.5059 -0.0343 11.0000 -1.5059 -1.5000 -1.5029 -0.0014 12.0000 -1.5029 -1.5000 -1.5015 0.0150 13.0000 -1.5029 -1.5015 -1.5022 0.0068 14.0000 -1.5029 -1.5022 -1.5026 0.0027 15.0000 -1.5029 -1.5026 -1.5027 0.0007 16.0000 -1.5029 -1.5027 -1.5028 -0.0003 17.0000 -1.5028 -1.5027 -

4、1.5028 0.0002 18.0000 -1.5028 -1.5028 -1.5028 -0.0001 19.0000 -1.5028 -1.5028 -1.5028 0.0001 20.0000 -1.5028 -1.5028 -1.5028 -0.0000推荐精选牛顿迭代法 syms x;f=(x5-3*x3-2*x2+2)x,k=Newtondd(f,0,1e-12) f = x5 - 3*x3 - 2*x2 + 2 x = NaN k =2实验二：解线性方程组的迭代法1实验目的 1.掌握雅克比迭代法和高斯-塞德尔迭代法的原理 2.根据实验内容编写雅克比迭代法和高斯-塞德尔迭代法的算

5、法实现 注：（可以用C语言或者matlab语言）2实验设备Matlab推荐精选3实验内容及步骤1、分别用雅克比迭代法和高斯-塞德尔迭代法解方程Ax=b 其中A=4 -1 0 -1 0 0 -1 4 -1 0 -1 0 0 -1 4 -1 0 -1 -1 0 -1 4 -1 0 0 -1 0 -1 4 -1 0 0 -1 0 -1 4 b=0 ;5;-2;5;-2;6 4实验结果及分析（雅克比迭代法）a=4 -1 0 -1 0 0;-1 4 -1 0 -1 0;0 -1 4 -1 0 -1;-1 0 -1 4 -1 0;0 -1 0 -1 4 -1;0 0 -1 0 -1 4 b=0;5;-2;

6、5;-2;6x=agui_jacobi(a,b)a = 4 -1 0 -1 0 0 -1 4 -1 0 -1 0 0 -1 4 -1 0 -1 -1 0 -1 4 -1 0 0 -1 0 -1 4 -1 0 0 -1 0 -1 4b = 0 5 -2 5 -2 6推荐精选k = 1 0 1.2500 -0.5000 1.2500 -0.5000 1.5000k = 2 0.6250 1.0000 0.5000 1.0000 0.5000 1.2500k = 3 0.5000 1.6563 0.3125 1.6563 0.3125 1.7500k = 4 0.8281 1.5313 0.7656

7、 1.5313 0.7656 1.6563k = 5 0.7656 1.8398 0.6797 1.8398 0.6797 1.8828k = 6 0.9199 1.7813 0.8906 1.7813 0.8906 1.8398k = 7 0.8906 1.9253 0.8506 1.9253 0.8506 1.9453k = 8 0.9626 1.8979 0.9490 1.8979 0.9490 1.9253k = 9 0.9490 1.9651 0.9303 1.9651 0.9303 1.9745k = 10 0.9826 1.9524 0.9762 1.9524 0.9762 1.

8、9651k = 11 0.9762 1.9837 0.9675 1.9837 0.9675 1.9881k = 12 0.9919 1.9778 0.9889 1.9778 0.9889 1.9837k = 13 0.9889 1.9924 0.9848 1.9924 0.9848 1.9944k = 14 0.9962 1.9896 0.9948 1.9896 0.9948 1.9924k = 15 0.9948 1.9965 0.9929 1.9965 0.9929 1.9974k = 16 0.9982 1.9952 0.9976 1.9952 0.9976 1.9965k = 17 0

9、.9976 1.9983 0.9967 1.9983 0.9967 1.9988k = 18 0.9992 1.9977 0.9989 1.9977 0.9989 1.9983k = 19 0.9989 1.9992 0.9985 1.9992 0.9985 1.9994k = 20 0.9996 1.9989 0.9995 1.9989 0.9995 1.9992k = 21 0.9995 1.9996 0.9993 1.9996 0.9993 1.9997k = 22 0.9998 1.9995 0.9998 1.9995 0.9998 1.9996推荐精选k = 23 0.9998 1.

10、9998 0.9997 1.9998 0.9997 1.9999k = 24 0.9999 1.9998 0.9999 1.9998 0.9999 1.9998k = 25 0.9999 1.9999 0.9998 1.9999 0.9998 1.9999k = 26 1.0000 1.9999 0.9999 1.9999 0.9999 1.9999k = 270.9999 2.0000 0.9999 2.0000 0.9999 2.0000x = 0.9999 2.0000 0.9999 2.0000 0.9999 2.0000（高斯-赛德尔迭代法迭代法）a=4 -1 0 -1 0 0;-1

11、 4 -1 0 -1 0;0 -1 4 -1 0 -1;-1 0 -1 4 -1 0;0 -1 0 -1 4 -1;0 0 -1 0 -1 4 b=0;5;-2;5;-2;6x= agui_GS(a,b)a = 4 -1 0 -1 0 0 -1 4 -1 0 -1 0 0 -1 4 -1 0 -1 -1 0 -1 4 -1 0 0 -1 0 -1 4 -1 0 0 -1 0 -1 4b = 0 5 -2 5 -2 6k = 1推荐精选 Columns 1 through 5 0 1.250000000000000 -0.187500000000000 1.203125000000000 0.1

12、13281250000000 Column 6 1.481445312500000k = 2 Columns 1 through 5 0.613281250000000 1.384765625000000 0.517333984375000 1.560974121093750 0.606796264648438 Column 6 1.781032562255859k = 3 Columns 1 through 5 0.736434936523438 1.715141296386719 0.764286994934082 1.776879549026489 0.818263351917267 C

13、olumn 6 1.895637586712837k = 4 Columns 1 through 5 0.873005211353302 1.863888889551163 0.884101506322622 1.893842517398298 0.913342248415574 Column 6 1.949360938684549k = 5 Columns 1 through 5 0.939432851737365 1.934219151618891 0.944355651925434 1.949282688019594 0.958215694580758 Column 6 1.975642

14、836626548k = 6 Columns 1 through 5 0.970875459909621 1.968361701603953 0.973321806562524 1.975603240263226 0.979901944623432 Column 6 1.988305937796489k = 7 Columns 1 through 5 0.985991235466795 1.984803746663188 0.987178231180726 1.988267852817738 0.990344384319354 Column 6推荐精选 1.994380653875020k =

15、 8 Columns 1 through 5 0.993267899870231 1.992697628842578 0.993836533883834 1.994362204518355 0.995360121808988 Column 6 1.997299163923206k = 9 Columns 1 through 5 0.996764958340233 1.996490403508264 0.997037942987456 1.997290755784169 0.997770080803910 Column 6 1.998702005947842k = 10 Columns 1 th

16、rough 5 0.998445289823108 1.998313328403619 0.998576522533907 1.998697973290231 0.998928326910423 Column 6 1.999376212361083k = 11 Columns 1 through 5 0.999252825423463 1.999189418716948 0.999315901092066 1.999374263356488 0.999484973608630 Column 6 1.999700218675174k = 12 Columns 1 through 5 0.9996

17、40920518359 1.999610448804764 0.999671232709106 1.999699281709024 0.999752487297240 Column 6 1.999855930001587k = 13 Columns 1 through 5 0.999827432628447 1.999812788158698 0.999841999967327 1.999855479973254 0.999881049533385 Column 6 1.999930762375178推荐精选k = 14 Columns 1 through 5 0.99991706703298

18、8 1.999910029133425 0.999924067870464 1.999930546109209 0.999942834404453 Column 6 1.999966725568729x = 0.999917067032988 1.999910029133425 0.999924067870464 1.999930546109209 0.999942834404453 1.999966725568729实验三：插值与拟合1实验目的1、掌握线性插值、抛物线插值、拉格朗日插值，三次样条插值与拟合2、根据实验内容，编写三次样条插值(一阶导数)的算法实现。3、根据实验内容，编写最小二乘

19、法的算法实现。2实验设备Matlab3实验内容及步骤 (1)、给定的插值条件如下：i01234567Xi8.1258.49.09.4859.69.95910.1710.2Yi0.07740.0990.280.600.7081.2001.8002.177 端点边界条件为第一类边界条件（给定一阶导数）： (2)、给出如下的数据，使用最小二乘法求一次和二次拟合多项式（取小数点后3位）x1.361.491.731.811.952.162.282.48y14.09415.06916.84417.37818.43519.94920.96322.495推荐精选4实验结果及分析结果b1 c1 d1 0.010

20、87 0.14489 0.368 0.17405 0.4485 -0.393 0.2878 -0.25891 2.1153 1.5294 2.8188 -69.141 -0.56548 -21.035 73.614 12.794 58.247 -512.32 -28.949 -259.9 42279推荐精选最小二乘法拟合代码：function p=funLSM(x,y,m)% x,y为序列长度相等的数据向量，m为拟合多项式次数format short;A=zeros(m+1,m+1);for i=0:m for j=0:m A(i+1,j+1)=sum(x.i.*y); end b(i+1)=

21、sum(x.i.*y)enda=Ab;p=fliplr(a)f=polyval(p,x)plot(x,y,b*,x,f,r-);disp(拟合方程系数按照降幂排列如下)一次拟合结果：b = 145.2270b = 145.2270 284.8363Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 9.978262e-17. In funLSM at 11 p = 1 0f = 1.3600 1.4900 1.7300 1.8100 1.9500 2.1600 2.28

22、00 2.4800拟合方程系数按照降幂排列如下p = 1 0推荐精选二次拟合结果b = 145.2270b = 145.2270 284.8363b = 145.2270 284.8363 577.3679Warning: Matrix is singular to working precision. In funLSM at 11 p = NaN NaN NaNf = NaN NaN NaN NaN NaN NaN NaN NaN拟合方程系数按照降幂排列如下p = NaN NaN NaN推荐精选实验四：数值积分1实验目的1、掌握牛顿-柯特斯求积公式、梯形公式、辛普生公式及误差（余项）。2、

23、根据实验内容，编写复合梯形公式、复合辛普生公式的算法实现。2实验设备Matlab3实验内容及步骤1、用复合梯形公式和辛普生公式求下面两式的积分（误差要求5e-8） （1） （2）推荐精选4实验结果及分析复化梯形积分代码：functiont=agui_trapz(fname,d2fname,a,b,e)%fname为北极函数，d2fname为函数fname的二阶导函数，a,b分别为上下界，e为精度y=abs(feval(d2fname,a:1e-5:b);m=max(y);h=abs(sqrt(12*e/(b-a)./m);n=ceil(b-a)/h)h=(b-a)/n;fa=feval(fname,a);fb=feval(fname,b);f=feval(fname,a+h:h:b-h+0.001*h);t=h*(0.5*(fa+fb)+s