数值分析作业.docx

1、数值分析作业数值分析作业四 刘涌一、作业要求1. 算法的设计方案。2. 全部源程序（要求注明主程序和每个子程序的功能）。3. 用各种积分方法求出的数值解及节点的值。4. 在同一坐标系内画出积分方程精确解的曲线和用各种积分方法求出的数值解的曲线。5. 讨论不同积分法的优劣二、算法设计方案1. 复化梯形积分法：设 在区间a,b上有二阶连续导数，取等距节点复化梯形公式其求积系数总满足对于问题中所给积分方程中的积分项，分别用复化梯形积分法的数值积分算法公式展开，得到含有未知数u(x)和x的方程，有如下形式：再将各个方程积分节点带入各个方程，得到的(n+1)阶方程组。由于对角元素绝对占优，用Gauss消

2、去法解该方程组即可。由于节点数目未知，需要进行迭代确定，每迭代一次，需要判断是否满足精度要求，节点数目增加，然后再进行。2. 复化Simpon积分法：设在区间a,b上有二阶连续导数，取2m+1个等距节点复化Simpson公式为其求积系数总满足复化simpson积分法计算量较小，可以直接每次增加2个节点（区间必需取偶数）。3. Guass积分法：本题中采用下列公式若记，积分区间在a,b，其相应的正交多项式是Ledendre多项式，其形式为如上求积公式，查表得n=NG时的积分节点及求积系数求解得NG个差值节点的值。三、程序源代码：#include#include#include stdlib.h#

3、define N 2601#define M 39#define NG 9double a1N+1N+1,g1N+1,x1N+1,u1N+1;FILE *fp;/N:复化梯形计算的节点数：N+1/M:复化Simpson算法节点数：2*M+1/NG:gauss算法节点数：NG void gsxy(n)/列主元素的gauss消去法求解线性方程组，用于梯形法中方程组的求解 int n; double sum,max,tem; int i,j,r,im; for(r=0;rn;r+) im=r; max=fabs(a1rr); for(i=r;i=max) max=fabs(a1(i+1)r); im

4、=i+1; for(j=r;jn;j+) tem=a1imj; a1imj=a1rj; a1imj=tem; for(i=r+1;in;i+) tem=a1ir/a1rr; for(j=r+1;j=0;i-) sum=0.0; for(j=i+1;jn;j+) sum=sum+a1ij*u1j; u1i=(g1i-sum)/a1ii; double fhtix()/复合梯形法求解，因为计算量大，数组采用全局变量的形式存储，返回误差值，结果存入F:rezult4.txt double h,er; int i,j,n; n=N+1; h=2/(double)N; for(i=0;in;i+) x1

5、i=-1.0+h*(double)i; for(i=0;in;i+) g1i=exp(4*x1i)+(exp(x1i+4)-exp(-1.0*(x1i+4)/(x1i+4); for(i=0;in;i+) for(j=0;jn;j+) if(j=0)|(j=N) a1ij=h*exp(x1i*x1j)/2; else a1ij=h*exp(x1i*x1j); for(i=0;in;i+) a1ii=1.0+a1ii;/ lu(a1,g1,N+1,u1); gsxy(n); er=0.0; for(i=1;iN;i+) er=er+(exp(4*x1i)-u1i)*(exp(4*x1i)-u1i

6、); er=er*h; er=er+h*(exp(4*x10)-u10)*(exp(4*x10)-u10)/2.0; er=er+h*(exp(4*x1N)-u1N)*(exp(4*x1N)-u1N)/2.0; fprintf(fp,梯形解法： error is：%.8en,er); fputs(差值点，计算值，真实值n,fp); for(i=0;in;i=i+20) fprintf(fp, %f,%f,%fn,x1i,u1i,exp(4*x1i); return(er); void lu(a,b,n,x) double a,b,x; int n;/*Doolittle法解线性方程组，用于si

7、mpson和gauss方法中线性方程组的求解*/ int k,t,j,i; double *l,*u,*y; double sum; l=malloc(n*n*sizeof(double); u=malloc(n*n*sizeof(double); y=malloc(n*sizeof(double); for(k=0;kn;k+) lk*n+k=1.0; for(j=0;j0) lj*n=aj*n/u0; for(k=1;kn;k+) for(j=k;jn;j+) sum=0.0; for(t=0;tk;t+) sum=sum+lk*n+t*ut*n+j; uk*n+j=ak*n+j-sum;

8、 if(k(n-1) for(i=(k+1);in;i+) sum=0.0; for(t=0;tk;t+) sum=sum+li*n+t*ut*n+k; li*n+k=(ai*n+k-sum)/uk*n+k; y0=b0; for(i=1;in;i+) sum=0.0; for(t=0;t=0;i-) sum=0.0; for(t=i+1;tn;t+) sum=sum+ui*n+t*xt; xi=(yi-sum)/ui*n+i; free(l); free(u); free(y); return; double fhSimpson(double u2*M+1)/复合Simpson求解，返回误差

9、值，结果存入F:rezult4.txt double x2*M+1,a2*M+12*M+1,g2*M+1,h,er; int i,j; h=1/(double)M; for(i=0;i(2*M+1);i+) xi=-1.0+h*(double)i; for(i=0;i(2*M+1);i+) gi=exp(4*xi)+(exp(xi+4)-exp(-1.0*(xi+4)/(xi+4); for(i=0;i(2*M+1);i+) for(j=0;j(2*M+1);j+) if(j=0)|(j=2*M) aij=h*exp(xi*xj)/3; else if(j%2=0) aij=2*h*exp(x

10、i*xj)/3; else aij=4*h*exp(xi*xj)/3; for(i=0;i(2*M+1);i+) aii=1.0+aii; lu(a,g,2*M+1,u); er=0.0; for(i=2;i2*M;i=i+2) er=er+(exp(4*xi)-ui)*(exp(4*xi)-ui); er=er*h*2/3; for(i=1;i2*M;i=i+2) er=er+(exp(4*xi)-ui)*(exp(4*xi)-ui); er=er*h*4/3; er=er+h*(exp(4*x0)-u0)*(exp(4*x0)-u0)/3; er=er+h*(exp(4*x2*M)-u2*

11、M)*(exp(4*x2*M)-u2*M)/3; fprintf(fp,Simpson解法： error is：%.8en,er); for(i=0;i(2*M+1);i+) fprintf(fp,%f, %f,%fn,xi,ui,exp(4*xi); return(er); double gauss(double uNG)/Gauss积分法求解，返回误差值，结果存入F:rezult4.txt double xNG=-0.9681602395,-0.8360311073,-0.6133714327,-0.3242534234,0,0.3242534234,0.6133714327,0.8360

12、311073,0.9681602395; double a1NG=0.0812743884,0.1806481607,0.2606106964,0.3123470770,0.3302393550,0.3123470770,0.2606106964,0.1806481607,0.0812743884;/NG=9时节点值及系数/ double xNG=-0.9602898565,-0.7966664774,-0.5255324099,-0.1834346425,0.1834346425,0.5255324099,0.7966664774,0.9602898565;/ double a1NG=0.1

13、012285363,0.2223810345,0.3137066459,0.3626837834,0.3626837834,0.3137066459,0.2223810345,0.1012285363;/NG=8时节点值及系数/ double xNG=-0.9324695142,-0.6612093865,-0.2386191861,0.2386191861,0.6612093865,0.9324695142;/ double a1NG=0.1713244924,0.3607615730,0.4679139346,0.4679139346,0.3607615730,0.1713244924;/

14、NG=6时节点值及系数/ double xNG=-0.9491079123,-0.7415311856,-0.4058451514,0,0.4058451514,0.7415311856,0.9491079123;/ double a1NG=0.1294849662,0.2797053915,0.3818300505,0.4179591837,0.3818300505,0.2797053915,0.1294849662;/NG=7时节点值及系数 double aNGNG,gNG,er; int i,j; for(i=0;iNG;i+) gi=exp(4*xi)+(exp(xi+4)-exp(-

15、1.0*(xi+4)/(xi+4); for(i=0;iNG;i+) for(j=0;jNG;j+) aij=a1j*exp(xi*xj); for(i=0;iNG;i+) aii=1.0+aii; lu(a,g,NG,u); er=0; for(i=0;iNG;i+) er=er+a1i*(exp(4*xi)-ui)*(exp(4*xi)-ui); fprintf(fp,Gauss解法： error is：%.8en,er); for(i=0;iNG;i+) fprintf(fp,%f,%f,%fn,xi,ui,exp(4*xi); return(er); void main()/主函数，调

16、用各子函数求解方程，结果存入F:rezult4.txt double u22*M+1,u3NG; double r1,r2,r3; fp=fopen(F:rezult4.txt,w); r1=fhtix(u1); printf(er1 is:%.8en,r1); r2=fhSimpson(u2); printf(er2 is:%.8en,r2); r3=gauss(u3); printf(er3 is:%.8en,r3); fclose(fp); 四、计算结果复化梯形积分法复化Simpon积分法Guass积分法节点值数值解节点值数值解节点值数值解-10.018317-10.018317-0.9

17、68160.020803-0.992310.018889-0.975610.020194-0.836030.035291-0.984620.019479-0.951220.022263-0.613370.085993-0.976930.020087-0.926830.024544-0.324250.273347-0.969240.020715-0.902440.0270601-0.961550.021362-0.878050.0298320.3242533.658356-0.953860.022029-0.853660.032890.61337111.62881-0.946180.022717

18、-0.829270.036260.83603128.33576-0.938490.023426-0.804880.0399760.9681648.06917-0.93080.024158-0.780490.044072-0.923110.024913-0.75610.048589-0.915420.025691-0.731710.053568-0.907730.026493-0.707320.059057-0.900040.027321-0.682930.065109-0.892350.028174-0.658540.071781-0.884660.029054-0.634150.079137

19、-0.876970.029961-0.609760.087247-0.869280.030897-0.585370.096188-0.861590.031862-0.560980.106045-0.85390.032857-0.536590.116912-0.846210.033884-0.51220.128893-0.838520.034942-0.487810.142102-0.830830.036033-0.463420.156664-0.823150.037159-0.439020.172719-0.815460.038319-0.414630.190419-0.807770.0395

20、16-0.390240.209932-0.800080.040751-0.365850.231446-0.792390.042023-0.341460.255164-0.78470.043336-0.317070.281312-0.777010.04469-0.292680.310141-0.769320.046085-0.268290.341924-0.761630.047525-0.24390.376963-0.753940.049009-0.219510.415594-0.746250.05054-0.195120.458183-0.738560.052119-0.170730.5051

21、37-0.730870.053747-0.146340.556903-0.723180.055425-0.121950.613973-0.715490.057157-0.097560.676892-0.707810.058942-0.073170.746259-0.700120.060783-0.048780.822734-0.692430.062681-0.024390.907047-0.684740.06463901-0.677050.0666580.024391.102478-0.669360.068740.048781.215459-0.661670.0708870.0731711.3

22、40017-0.653980.0731010.0975611.47734-0.646290.0753850.1219511.628736-0.63860.0777390.1463411.795646-0.630910.0801680.1707321.979662-0.623220.0826720.1951222.182535-0.615530.0852540.2195122.406198-0.607840.0879170.2439022.652782-0.600150.0906630.2682932.924635-0.592460.0934950.2926833.224348-0.584780

23、.0964150.3170733.554775-0.577090.0994260.3414633.919064-0.56940.1025320.3658544.320684-0.561710.1057350.3902444.763462-0.554020.1090370.4146345.251615-0.546330.1124430.4390245.789794-0.538640.1159550.4634156.383124-0.530950.1195770.4878057.037259-0.523260.1233120.5121957.758428-0.515570.1271640.5365

24、858.553501-0.507880.1311360.5609769.430053-0.500190.1352320.58536610.39643-0.49250.1394560.60975611.46185-0.484810.1438110.63414612.63644-0.477120.1483030.65853713.93141-0.469440.1529360.68292715.35908-0.461750.1577130.70731716.93306-0.454060.1626390.73170718.66833-0.446370.1677190.75609820.58144-0.438680.1729580.78048822.6906-0.430990.178360.80487825.0159-0.42330.1839310.82926827.5795-0.415610.1896760.85365930.40581-0.407920.1956010.87804933.52176-0.400230.201710.90243936.95702-0.392540.2080110.92682940.74433-0.384850.2145080.9512244.91975-0.377160.2212080.9756149.52307