计算方法上机练习题.docx

计算方法上机练习题信息44 肖卓昊 2140502094习题2-13：程序：A=0.4721 0.2352 -0.2613 0.8421; 0.2352 0.7411 -0.0463 0.1569; -0.2613 -0.0463 -0.8955 0.1748; 0.8421 0.1569 0.1748 0.9841;b=-0.2317;0.3219;0.6217;0.9838;% 不选主元的高斯消去法x=Gauss(A,b);x=x;disp(不选主元的解x=)x% 选主元的高斯消去法x=Gauss_pivot(A,b);disp(选主元的解x=)xfunction x=Gauss_pivot(A,b)n,n=size(A);x=zeros(n,1);C=zeros(1,n);B=A b;for i=1:n-1 max1,j=max(abs(B(i:n,i); C=B(i,:); B(i,:)=B(j+i-1,:); B(j+i-1,:)=C; if A(i,i)=0 break end for k=i+1:n; temp=B(k,i)/B(i,i); B(k,i+1:n+1)=B(k,i+1:n+1)-temp*B(i,i+1:n+1); end endx(n)=B(n,n+1)/B(n,n);for k=n-1:-1:1 x(k)=(B(k,n+1)-B(k,k+1:n)*x(k+1:n)/B(k,k);endfor s=1:1:n-1; for i=s+1:1:n c=B(i,s)/B(s,s); B(i,:)=B(i,:)-c*B(s,:); endenddisp(选主元的高斯消去后的增广矩阵A_b=)Bfunction x=Gauss(A,b)A_b=A,b;m,n=size(A_b);for k=1:1:m-1; for i=k+1:1:m c=A_b(i,k)/A_b(k,k); A_b(i,:)=A_b(i,:)-c*A_b(k,:); endenddisp(不选主元的高斯消去后的增广矩阵A_b=)A_bx(m)=A_b(m,n)/A_b(m,m);for k=m-1:-1:1 x(k)=(A_b(k,n)-A_b(k,k+1:m)*x(k+1:m)/A_b(k,k);end运行结果：（1）不选主元A_b = 0.4721 0.2352 -0.2613 0.8421 -0.2317 0 0.6239 0.0839 -0.2626 0.4373 0 0 -1.0514 0.6762 0.4347 0 0 0 -0.1936 1.8607解x = 14.2241 -2.4575 -6.5933 -9.6090选主元A_b= 0.8421 0.1569 0.1748 0.9841 0.9838 0.0000 0.6535 -0.1439 -0.3928 -0.2277 -0.0000 0 -0.7754 0.8166 1.2499 0.0000 0 0.0000 -0.8413 -2.3816解x= 14.2241 -2.4575 -6.5933 -9.6090（2）不选主元A_b= 0.5398 0.7161 -0.5554 -0.2982 0.2058 0 -0.0050 0.8974 -0.3351 -0.2507 0 0 -300.6974 112.9461 84.0065 0 0 0 -0.0835 -0.0866解x= 0.5162 0.4152 0.1100 1.0365选主元A_b= 0.6465 -0.8187 -0.1872 0.1291 0.1070 0 2.4125 -0.4412 -0.6364 -0.0066 0 0 1.5249 0.1200 -0.2915 0 0 0 0.4532 0.1046解x= 0.5162 0.4152 0.1100 1.0365（3）不选主元A_b= 0.7290 0.8100 0.9000 0.6867 0 -0.1111 -0.2346 -0.1037 0 0 0.0244 -0.0029解x= -0.2255 1.1814 -0.1176选主元A_b= 1.3310 1.2100 1.1000 1.0000 0 -0.5155 -0.3050 -0.4087 0 0 -0.3525 -0.1014解x= -0.2255 1.1814 -0.1176（4）不选主元A_b= 1.0e+03 * 4.1970 0.3050 -0.2060 -0.8400 0.1360 0 0.0708 -0.0471 0.0534 0.0113 0 0 0.0401 0.7670 0.0117 0 0 0 -0.1601 0.0027解x= 0.0170 0.5804 0.6144 -0.0168选主元A_b= 1.0e+03 * 4.1970 0.3050 -0.2060 -0.8400 0.1360 0 0.0703 -0.0467 0.0547 0.0111 0 0 0.0866 0.7353 -0.0012 0 0 0 -0.2823 0.0019解x= 0.0170 0.5804 0.6144 -0.0168习题3-27:程序：x=0.2 0.24 0.28 0.32 0.36 0.40;y=0.19867 0.2377 0.2764 0.3146 0.3523 0.3894;m=length(x);DQT=x,y;for j=2:m for i=m:-1:1 if (i=j) y(i)=(y(i)-y(i-1)/(x(i)-x(i-j+1); else y(i)=0; end end DQT=DQT,y;enddisp(牛顿差商表：) DQTfor x0=0.2:0.01:0.4 ff=DQT(1,2); ff=ff+DQT(2,3)*(x0-DQT(1,1); ff=ff+DQT(3,4)*(x0-DQT(1,1)*(x0-DQT(2,1); ff=ff+DQT(4,5)*(x0-DQT(1,1)*(x0-DQT(2,1)*(x0-DQT(3,1); ff=ff+DQT(5,6)*(x0-DQT(1,1)*(x0-DQT(2,1)*(x0-DQT(3,1)*(x0-DQT(4,1); ff=ff+DQT(6,7)*(x0-DQT(1,1)*(x0-DQT(2,1)*(x0-DQT(3,1)*(x0-DQT(4,1)*(x0-DQT(5,1); FF(round(100*x0-19)=double(ff);enddisp(牛顿插值函数值：) FFx=0.2 0.24 0.28 0.32 0.36 0.40;y=0.19867 0.2377 0.2764 0.3146 0.3523 0.3894;for x0=0.2:0.01:0.4 f=Lagrange(x,y,x0); F(round(100*x0-19)=double(f);endFfunction f=Lagrange(x,y,x0)syms t;n=length(x);f=0;for(i=1:n) l=y(i); for(j=1:i-1) l=l*(t-x(j)/(x(i)-x(j); end; for(j=i+1:n) l=l*(t-x(j)/(x(i)-x(j); end; f=f+l; simplify(f); if(i=n) if(nargin=3) f=subs(f,t,x0); else f=collect(f); end endend运行结果：（1）牛顿差商表： Columns 1 through 60.2000 0.1987 0 0 0 0 00.2400 0.2377 0.9758 0 0 0 00.2800 0.2764 0.9675 -0.1031 0 0 00.3200 0.3146 0.9550 -0.1562 -0.4427 0 00.3600 0.3523 0.9425 -0.1562 -0.0000 2.7669 0 0.4000 0.3894 0.9275 -0.1875 -0.2604 -1.6276 -21.9727（2）插值结果：拉格朗日插值结果： Columns 1 through 120.1987 0.2084 0.2182 0.2280 0.2377 0.2474 0.2571 0.2668 0.2764 0.2860 0.2956 0.3051 Columns 13 through 210.3146 0.3241 0.3335 0.3429 0.3523 0.3616 0.3709 0.3802 0.3894牛顿插值结果： Columns 1 through 120.1987 0.2084 0.2182 0.2280 0.2377 0.2474 0.2571 0.2668 0.2764 0.2860 0.2956 0.3051 Columns 13 through 210.3146 0.3241 0.3335 0.3429 0.3523 0.3616 0.3709 0.3802 0.3894习题5-22：程序：disp(简单迭代法)y,n=Jiandan(1.5)disp(牛顿法)r,n=Newtond(1.5)disp(改进牛顿法)r,n=VNewton(1.5)disp(弦割法)r,n=mulVNewton(1.5,1)function y,n=Jiandan(x