matlab数学软件实验.doc

实验课程：科学计算与数学软件专 业：信息与计算科学班 级：09070242学 号：姓 名： 中北大学理学院实验一 Matlab入门【实验目的】1、MATLAB的基本操作； 2、MATLAB编程；【实验内容】 【实验所使用的仪器设备与软件平台】1PIV2.8/256M计算机；2Matlab6.0。【实验方法与步骤】1.1 a=1 2 3;4 5 6b=9;7;5;3;0c=b(2:4)d=b(4:-1:1)e=sort(b)f=3,b1.2（a）x=7 4 3y=-1 -2 -3u=y x（b） x=7 4 3y=-1 -2 -3u=x y1.3 x=7 4 3y=-1 -2 -3b=x;y【实验结果】1.1 a = 1 2 3 4 5 6b = 9 7 5 3 0c = 7 5 3d = 3 5 7 9e = 0 3 5 7 9f = 3 9 7 5 3 01.2 （a）x = 7 4 3y = -1 -2 -3u = -1 -2 -3 7 4 3（b）x = 7 4 3y = -1 -2 -3u = 7 4 3 -1 -2 -31.3 x = 7 4 3y = -1 -2 -3b = 7 4 3 -1 -2 -3实验二 Matlab绘图【实验目的】1、熟练掌握MATLAB二维曲线的绘制； 2、掌握图形的修饰； 3、掌握三维图形的绘制。【实验内容】 【实验所使用的仪器设备与软件平台】1PIV2.8/256M计算机；2Matlab6.0。【实验方法与步骤】2.1 x=(0:0.4:20); y=sin(x).*exp(-0.4*x); plot(x,y,*); xlabel(x);ylabel(y)2.2 x=(0:0.4:20); y=sin(x).*exp(-0.4*x); plot(x,y,r); xlabel(x);ylabel(y)2.3 x=(0:0.4:20); y=sin(x).*exp(-0.4*x);plot(x,y,+,x,y,r)xlabel(x);ylabel(y)【实验结果】2.1 2.2 2.3 实验三 线性代数【实验目的】1、熟练掌握MATLAB里的矩阵运算； 2、掌握利用MATLAB解线性方程组；【实验内容】 【实验所使用的仪器设备与软件平台】1PIV2.8/256M计算机；2Matlab6.0。【实验方法与步骤】3.1 A=1 2 3;0 1 4;3 0 2 B=4 1 2;3 2 1;0 1 2 C=A+B D=A-B E=A*B3.3 A=1 2 3;0 1 4;3 0 2 B=3;5;1 E=A*B3.5 B=3 2 1;0 4 3;0 0 6 C=1 0 2;-1 1 0;0 3 2 D=1 0 0;-2 1 0;5 2 7 E=B+C*D【实验结果】3.1A = 1 2 3 0 1 4 3 0 2B = 4 1 2 3 2 1 0 1 2C = 5 3 5 3 3 5 3 1 4D = -3 1 1 -3 -1 3 3 -1 0E = 10 8 10 3 6 9 12 5 103.3 A = 1 2 3 0 1 4 3 0 2B = 3 5 1E = 16 9 11 3.5 B = 3 2 1 0 4 3 0 0 6C = 1 0 2 -1 1 0 0 3 2D = 1 0 0 -2 1 0 5 2 7E = 14 6 15 -3 5 3 4 7 20实验四 多项式与插值【实验目的】1、掌握Lagrange插值的基本原理； 2、掌握Matlab对多项式的各种计算； 3、编写Matlab实现Lagrange插值法的程序；4、通过Matlab观察龙格现象。【实验内容】 【实验所使用的仪器设备与软件平台】1PIV2.8/256M计算机；2Matlab6.0。【实验方法与步骤】4.3 (1) r=3 4 -1 -3 poly(r)(2) r=0 2 1 -3 -5 poly(r)4.6 a=(5-1)*(5-2.5)*(5-4)*(5-6.1)*(5-7.2)*(5-10)r=1 2.5 4 6.1 7.2 10 poly(r)4.7 r=-2 1 2 poly(r)【实验结果】4.3 r = 3 4 -1 -3ans = 1 -3 -13 27 36 差5倍(2) r = 0 2 1 -3 -5ans = 1 5 -7 -29 30 04.6 a = -121.0000r = 1.0000 2.5000 4.0000 6.1000 7.2000 10.0000ans = 1.0e+003 * 0.0010 -0.0308 0.3682 -2.1606 6.4462 -9.0160 4.3920 差-(1/121)倍4.7 r = -2 1 2ans = 1 -1 -4 4 差4倍实验五 数值积分【实验目的】1、掌握数值积分的基本原理； 2、编写Matlab对数值积分的实现程序； 3、掌握Matlab实现数值积分的命令。【实验内容】【实验所使用的仪器设备与软件平台】1PIV2.8/256M计算机；2Matlab6.0。【实验方法与步骤】5.1（1）n=2x=0:pi/8:pi/4; y=tan(x); z=trapz(x,y)n=4x=0:pi/16:pi/4; y=tan(x); z=trapz(x,y)n=8x=0:pi/32:pi/4; y=tan(x); z=trapz(x,y)n=16x=0:pi/64:pi/4; y=tan(x); z=trapz(x,y)(2)n=2x=0:0.5:1; y=exp(x); z=trapz(x,y)n=4 x=0:0.25:1; y=exp(x); z=trapz(x,y)n=8 x=0:0.125:1; y=exp(x); z=trapz(x,y)n=16 x=0:0.0625:1; y=exp(x); z=trapz(x,y)(3)n=2x=0:0.5:1; y=1./(2+x); z=trapz(x,y)n=4 x=0:0.25:1; y=1./(2+x); z=trapz(x,y)n=8 x=0:0.125:1; y=1./(2+x); z=trapz(x,y)n=16 x=0:0.0625:1; y=1./(2+x); z=trapz(x,y)5.2n=2x=0:pi/4:pi/2; y=sin(x); z=trapz(x,y)n=4 x=0:pi/8:pi/2; y=sin(x); z=trapz(x,y)n=8 x=0:pi/16:pi/2; y=sin(x); z=trapz(x,y)n=25 x=0:pi/50:pi/2; y=sin(x); z=trapz(x,y)n=100 x=0:pi/200:pi/2; y=sin(x); z=trapz(x,y)5.11(a)clear a=0;b=pi;n=2; fprintf(n I Errorn);n I Error for k=1:4n=2*n; h=(b-a)/n; i=1:n+1;x=a+(i-1)*h; f=1./(2+cos(x);I= (h/3)*(f(1)+4*sum(f(2:2:n)+f(n+1);if n2,I=I+(h/3)*2*sum(f(3:2:n);endfprintf(%3.0f%10.5ffn,n,I);end（c）clear a=0;b=pi;n=2; fprintf(n I Errorn);n I Error for k=1:4n=2*n; h=(b-a)/n; i=1:n+1;x=a+(i-1)*h;f=1./(1+sin(x).2);I= (h/3)*(f(1)+4*sum(f(2:2:n)+f(n+1);if n2,I=I+(h/3)*2*sum(f(3:2:n);endfprintf(%3.0f%10.5f n,n,I);end【实验结果】5.1（1）n=2 z =0.3590n=4z = 0.3498n=8z = 0.3474n=16z = 0.3468(2)n=2z = 1.7539n=4z = 1.7272n=8z = 1.7205n=16z = 1.7188(3)n=2z = 0.4083n=4z = 0.4062n=8z = 0.4056n=16z = 0.40555.2n=2z = 0.9481n=4z = 0.9871n=8z = 0.9968n=25z = 0.9997n=100z = 1.00005.11(a) 4 1.80766 8 1.81377 16 1.81380 32 1.81380 （c） 4 24.35227 8 24.35227 16 24.35227 32 24.35227 实验六 数值微分【实验目的】1、掌握差分近似的定义以及高阶导数的数值计算的基本原理；2、编写相应的Matlab的实现程序。【实验内容】【实验所使用的仪器设备与软件平台】1PIV2.8/256M计算机；2Matlab6.0。【实验方法与步骤】6.1 x=0 xd=0 0.2000 0.4000 0.6000 0.8000 1.0000;yd=0.0000 8.6964 10.6800 8.7440 4.7727 0.0000;a=0; L=length(xd);d=polyfit(xd-a,yd,L-1); fact=1;for k=1:L-1 fact=factorial(k),fact;endderiv=d.*fact x=0.5 xd=0 0.2000 0.4000 0.6000 0.8000 1.0000;yd=0.0000 8.6964 10.6800 8.7440 4.7727 0.0000;a=0.5; L=length(xd);d=polyfit(xd-a,yd,L-1); fact=1;for k=1:L-1 fact=factorial(k),fact;endderiv=d.*fact 6.2 yd=0 2.000 4.0000 6.0000 8.0000;ud=0.0000 9.8853 15.4917 18.2075 19.0210;a=0; L=length(yd);d=polyfit(yd-a,ud,L-1); fact=1;for k=1:L-1 fact=factorial(k),fact;endderiv=d.*fact 【实验结果】6.1 x=0deriv = 807.8125 -891.1875 576.1156 -263.8644 66.3140 -0.0000X=0.5deriv = 807.8125 -487.2813 231.4984 -70.3756 -10.0665 10.06596.2 deriv = -0.0250 0.2485 -1.5085 6.2938 -0.0000实验七 非线性方程求根【实验目的】1、掌握非线性方程求根的几种算法：二分法、牛顿法、割线法等； 2、编写Matlab对几种算法的实现及比较程序。【实验内容】【实验所使用的仪器设备与软件平台】1PIV2.8/256M计算机；2Matlab6.0。【实验方法与步骤】7.1(a) x=0:0.1:20; y=0.5*exp(x/3)-sin(x); plot(x,y,x,zeros(size(x);(b) x=0:0.1:20; y=log(1+x)-x.*x; plot(x,y,x,zeros(size(x)7.4 bisec_n(fun_a,1,2)f_name =fun_aBisection SchemeIt.a b c fa=f(a) fc=f(c) abs(fc-fa)bisec_n(fun_a,4,5)f_name =fun_aBisection SchemeIt.a b c fa=f(a) fc=f(c) abs(fc-fa)bisec_n(fun_a,7,8)f_name =fun_aBisection SchemeIt.a b c fa=f(a) fc=f(c) abs(fc-fa)7.5（a） bisec_n(fun_c,0,0.4)f_name =fun_cBisection SchemeIt.a b c fa=f(a) fc=f(c) abs(fc-fa)（b）x=0.299828/2y=0.2969*sqrt(x)-0.126*x-0.3516*x2+0.2843*x3-0.1015*x4【实验结果】7.1(a)(b)7.4 1, 1.000000, 1.500000 2.000000, 1.833730, -0.565626 2.399e+000 2, 1.500000, 1.750000 2.000000, 1.166403, -0.565626 1.732e+000 3, 1.750000, 1.875000 2.000000, 0.471645, -0.565626 1.037e+000 4, 1.875000, 1.937500 2.000000, 0.000431, -0.565626 5.661e-001 5, 1.875000, 1.906250 1.937500, 0.000431, -0.270208 2.706e-001 6, 1.875000, 1.890625 1.906250, 0.000431, -0.131855 1.323e-001 7, 1.875000, 1.882813 1.890625, 0.000431, -0.064962 6.539e-002 8, 1.875000, 1.878906 1.882813, 0.000431, -0.032079 3.251e-002 9, 1.875000, 1.876953 1.878906, 0.000431, -0.015778 1.621e-002 10, 1.875000, 1.875977 1.876953, 0.000431, -0.007662 8.093e-003 11, 1.875000, 1.875488 1.875977, 0.000431, -0.003613 4.043e-003 12, 1.875000, 1.875244 1.875488, 0.000431, -0.001590 2.021e-003 13, 1.875000, 1.875122 1.875244, 0.000431, -0.000580 1.010e-003 14, 1.875000, 1.875061 1.875122, 0.000431, -0.000074 5.051e-004 15, 1.875061, 1.875092 1.875122, 0.000178, -0.000074 2.526e-004 16, 1.875092, 1.875107 1.875122, 0.000052, -0.000074 1.263e-004 17, 1.875092, 1.875099 1.875107, 0.000052, -0.000011 6.314e-005 18, 1.875099, 1.875103 1.875107, 0.000020, -0.000011 3.157e-005 19, 1.875103, 1.875105 1.875107, 0.000004, -0.000011 1.579e-005 20, 1.875103, 1.875104 1.875105, 0.000004, -0.000003 7.893e-006 21, 1.875104, 1.875104 1.875105, 0.000000, -0.000003 3.946e-006Tolerance is satisfied.Final result:Root= 1.875104 1, 4.000000, 4.500000 5.000000,-16.849852, 22.050556 3.890e+001 2, 4.500000, 4.750000 5.000000, -8.488787, 22.050556 3.054e+001 3, 4.500000, 4.625000 4.750000, -8.488787, 3.173272 1.166e+001 4, 4.625000, 4.687500 4.750000, -3.451716, 3.173272 6.625e+000 5, 4.687500, 4.718750 4.750000, -0.351215, 3.173272 3.524e+000 6, 4.687500, 4.703125 4.718750, -0.351215, 1.356333 1.708e+000 7, 4.687500, 4.695313 4.703125, -0.351215, 0.489097 8.403e-001 8, 4.687500, 4.691406 4.695313, -0.351215, 0.065602 4.168e-001 9, 4.691406, 4.693359 4.695313, -0.143638, 0.065602 2.092e-001 10, 4.693359, 4.694336 4.695313, -0.039226, 0.065602 1.048e-001 11, 4.693359, 4.693848 4.694336, -0.039226, 0.013136 5.236e-002 12, 4.693848, 4.694092 4.694336, -0.013058, 0.013136 2.619e-002 13, 4.693848, 4.693970 4.694092, -0.013058, 0.000036 1.309e-002 14, 4.693970, 4.694031 4.694092, -0.006512, 0.000036 6.548e-003 15, 4.694031, 4.694061 4.694092, -0.003238, 0.000036 3.274e-003 16, 4.694061, 4.694077 4.694092, -0.001601, 0.000036 1.637e-003 17, 4.694077, 4.694084 4.694092, -0.000783, 0.000036 8.186e-004 18, 4.694084, 4.694088 4.694092, -0.000374, 0.000036 4.093e-004 19, 4.694088, 4.694090 4.694092, -0.000169, 0.000036 2.046e-004 20, 4.694090, 4.694091 4.694092, -0.000067, 0.000036 1.023e-004 21, 4.694091, 4.694091 4.694092, -0.000016, 0.000036 5.116e-005Tolerance is satisfied.Final result:Root= 4.694091 1, 7.000000, 7.500000 8.000000,414.377449,-215.864768 6.302e+002 2, 7.500000, 7.750000 8.000000,314.365774,-215.864768 5.302e+002 3, 7.750000, 7.875000 8.000000,121.483081,-215.864768 3.373e+002 4, 7.750000, 7.812500 7.875000,121.483081,-26.644331 1.481e+002 5, 7.812500, 7.843750 7.875000, 52.242111,-26.644331 7.889e+001 6, 7.843750, 7.859375 7.875000, 14.043821,-26.644331 4.069e+001 7, 7.843750, 7.851563 7.859375, 14.043821, -5.984110 2.003e+001 8, 7.851563, 7.855469 7.859375, 4.108278, -5.984110 1.009e+001 9, 7.851563, 7.853516 7.855469, 4.108278, -0.918234 5.027e+000 10, 7.853516, 7.854492 7.855469, 1.599933, -0.918234 2.518e+000 11, 7.854492, 7.854980 7.855469, 0.342079, -0.918234 1.260e+000 12, 7.854492, 7.854736 7.854980, 0.342079, -0.287770 6.298e-001 13, 7.854736, 7.854858 7.854980, 0.027231, -0.287770 3.150e-001 14, 7.854736, 7.854797 7.854858, 0.027231, -0.130250 1.575e-001 15, 7.854736, 7.854767 7.854797, 0.027231, -0.051505 7.874e-002 16, 7.854736, 7.854752 7.854767, 0.027231, -0.012136 3.937e-002 17, 7.854752, 7.854759 7.854767, 0.007548, -0.012136 1.968e-002 18, 7.854752, 7.854755 7.854759, 0.007548, -0.002294 9.842e-003 19, 7.854755, 7.854757 7.854759, 0.002627, -0.002294 4.921e-003 20, 7.854757, 7.854758 7.854759, 0.000167, -0.002294 2.460e-003 21, 7.854757, 7.854758 7.854758, 0.000167, -0.001063 1.230e-003Tolerance is satisfied.Final result:Root= 7.8547587.5（a） 1, 0.000000, 0.200000 0.400000, Inf, -0.062080 Inf 2, 0.200000, 0.300000 0.400000, 0.096172, -0.062080 1.583e-001 3, 0.200000, 0.250000 0.300000, 0.096172, -0.000130 9.630e-002 4, 0.250000, 0.275000 0.300000, 0.042062, -0.000130 4.219e-002 5, 0.275000, 0.287500 0.300000, 0.019760, -0.000130 1.989e-002 6, 0.287500, 0.293750 0.300000, 0.009540, -0.000130 9.670e-003 7, 0.293750, 0.296875 0.300000, 0.004639, -0.000130 4.769e-003 8, 0.296875, 0.298438 0.300000, 0.002239, -0.000130 2.368e-003 9, 0.298438, 0.299219 0.300000, 0.001051, -0.000130 1.180e-003 10, 0.299219, 0.299609 0.300000, 0.000459, -0.000130 5.891e-004 11, 0.299609, 0.299805 0.300000, 0.000165, -0.000130 2.943e-004 12, 0.299805, 0.299902 0.300000, 0.000017, -0.000130 1.471e-004 13, 0.299805, 0.299854 0.299902, 0.000017, -0.000056 7.356e-005 14, 0.299805, 0.299829 0.299854, 0.000017, -0.000019 3.678e-005 15, 0.299805, 0.299817 0.299829, 0.000017, -0.000001 1.839e-005 16, 0.299817, 0.299823 0.299829, 0.000008, -0.000001 9.196e-006 17, 0.299823, 0.299826 0.299829, 0.000004, -0.000001 4.598e-006 18, 0.299826, 0.299828 0.299829, 0.000001, -0.000001 2.299e-006 19, 0.299828, 0.299828 0.299829, 0.000000, -0.000001 1.150e-006 20, 0.299828, 0.299828 0.299828, 0.000000, -0.000000 5.748e-007Tolerance is satisfied.Final result:Root= 0.299828（b）x = 0.1499y = 0.0891实验八 数据的曲线拟合【实验目的】1、掌握线性曲线拟合的几种算法； 2、掌握Matlab的实现方法； 3、掌握用最小二乘法做直线拟合、指数拟合，并能给出散点图和拟合。【实验内容】【实验所使用的仪器设备与软件平台】1PIV2.8/256M计算机；2Matlab6.0。【实验方法与步骤】8.2x=0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0; y=9.9,9.2,8.4,6.6,5.9,5.0,4.1,3.1,1.9,1.1; c=polyfit(x,y,1)8.5 i=1 2 3 4 5 6; xi=0 1 2 3 4 5; yi=0 2.3 4.2 5.7 6.5 6.9; p=polyfit(xi,yi,2) plot(xi,yi,*r) plot(xi,yi,-b) xlabel(x轴) ylabel(y轴) title(拟合二次图像)x=0:0.1:20; y=0.5*exp(x/3)-sin(x); plot(x,y,x,zeros(size(x);x=0:0.1:20; y=log(1+x)-x.*x; plot(x,y,x,zeros(size(x)8.6 i=1 2 3 4 5 6;xi=0 1 2 3 4 5;yi=0 2.3 4.2 5.7 6.5 6.9;p=polyfit(xi,yi,1) plot(xi,yi,-r)xlabel(x轴)ylabel(y轴) title(拟合一次图像) i=1 2 3 4 5 6;xi=0 1 2 3 4 5;yi=0 2.3 4.2 5.7 6.5 6.9; p=polyfit(xi,yi,3) plot(xi,yi,-r)xlabel(x轴)ylabel(y轴) title(拟合的三次图像)【实验结果】8.2c = -10.0121 11.02678.5p = -0.2482 2.6296 -0.03218.6p = 1.3886 0.7952P = -0.0083 -0.1857 2.5155 -0.0071实验九 样条函数与非线性插值【实验目的】1、熟悉三次c样条插值方法的基本思想；2、编写Matlab的实现程序。【实验内容】【实验所使用的仪器设备与软件平台】1PIV2.8/256M计算机；2Matlab6.0。【实验方法与步骤】9.1 clear,clf,hold offxx=0,-0.5,0,1,2,3,3.5,3;yy=0,1,2,2,2,2,1,0;s=1:length(xx);sp=1:(length(xx)/100):length(xx);xp=spline(s,xx,sp);yp=spline(s,yy,sp);clfplot(xp,yp);hold onplot(xx,yy,o);xlabel(X);ylabel(Y);hold off;axis(-0.5,3.5,0,2)9.2 clear,clf,hold offxx=0,1,2,3,3.5,3.7,3.5,3,2,1,0;yy=0,1,1.5,1.5,1,0,-1,-1.5,-1.5,-1,0;s=1:length(xx);sp=1:(length(xx)/100):length(xx);xp=spline(s,xx,sp);yp=spline(s,yy,sp);clfplot(xp,yp);hold onplot(xx,yy,o);xlabel(X);ylabel(Y);hold off;axis(0,3.7,-1.5,1.5)9.3 clear;clf;hold offy=0,0,0,1,1,0.5,0.5,2,2,0,0,0;x=0,0,0,0,1,1,2,2,4,4,4,4;m=length(y);plot(-1,5,-1,3,.);hold onxlabel(x);ylabel(y);plot(x,y,o);for k=1:mz=int2str(k);xk=x(k);yk=y(k);text(xk+0.1,yk,z)endt=0:0.2:1;t2=t.2;t3=t.3;for i=2:m-2yb=1/6*(1-t).3*y(i-1)+(3*t3-6*t2+4)*y(i)+(-3*t3+3*t2+3*t+1)*y(i+1)+t3*y(i+2);xb=1/6*(1-t).3*x(i-1)+(3*t3-6*t2+4)*x(i)+(-3*t3+3*t2+3*t+1)*x(i+1)+t3*x(i+2);plot(xb,yb)endtitle(Cubic b-spline curve on x-y plane)【实验结果】9.19.29.3 实验十 常微分方程的初值问题【实验目的】1掌握几种Euler方法和龙格-库塔方法的基本原理； 2编写Matlab对于Euler方法和龙格-库塔方法的实现程序。【实验内容】【实验所使用的仪器设备与软件平台】1PIV2.8/256M计算机；2Matlab6.0。【实验方法与步骤】10.1(1)clear,clf,hold offt=0;n=0;y=1;h=0.1;t_rec(1)=t;y_rec(1)=y;while t=0.5 n=n+1; y=y+h*(1-t*y); t=t+h; y_rec(n+1)=y; t_rec(n+1)=t;endplot(t_rec,y_rec)xlabel(x)ylabel(y)(2)clear,clf,hold offt=0;n=0;y=1;h=0.1;t_rec(1)=t;y_rec(1)=y;while t=0.5 n=n+1; y=y+h*(exp(-t)-3*y); t=t+h; y_rec(n+1)=y; t_rec(n+1)=t;endplot(t_rec,y_rec)xlabel(x)ylabel(y)(3)clear,clf,hold offt=0;n=0;y=0.5;h=0.1;t_rec(1)=t;y_rec(1)=y;while t=0.5 n=n+1; y=y+h*(t2-y); t=t+h; y_rec(n+1)=y; t_rec(n+1)=t;endplot(t_rec,y_rec)xlabel(x)ylabel(y)(4)clear,clf,hold offt=0;n=0;y=1;h=0.1;t_rec(1)=t;y_rec(1)=y;while t=0.5 n=n+1; y=y+h*(-y*abs(y); t=t+h; y_rec(n+1)=y; t_rec(n+1)=t;endplot(t_rec,y_rec)xlabel(x)ylabel(y)(5)clear,clf,hold offt=0;n=0;y=1;h=0.1;