15-16-2《几何与代数》数学实验1

15-16-2《几何与代数》 数学实验报告1

实验一：利用MATLAB用三种不同的方法求解线性方程组Axb。其中，常数项列向量b的分量是你的学号，系数矩阵为 11000000

12100000013100000014100000015100000016100000017100000018

A





方法一：利用Cramer法则求解； 方法二：作为矩阵方程求解； 方法三：利用Gauss消元法求解。 方法一： >> format short b=[0;4;0;1;5;2;0;3]; a1=[1;1;0;0;0;0;0;0]; a2=[1;2;1;0;0;0;0;0]; a3=[0;1;3;1;0;0;0;0]; a4=[0;0;1;4;1;0;0;0]; a5=[0;0;0;1;5;1;0;0]; a6=[0;0;0;0;1;6;1;0]; a7=[0;0;0;0;0;1;7;1]; a8=[0;0;0;0;0;0;1;8]; A =[a1,a2,a3,a4,a5,a6,a7,a8]; A1=[b ,a2,a3,a4,a5,a6,a7,a8]; A2=[a1,b ,a3,a4,a5,a6,a7,a8]; A3=[a1,a2,b ,a4,a5,a6,a7,a8]; A4=[a1,a2,a3,b ,a5,a6,a7,a8]; A5=[a1,a2,a3,a4,b ,a6,a7,a8]; A6=[a1,a2,a3,a4,a5,b ,a7,a8]; A7=[a1,a2,a3,a4,a5,a6,b ,a8]; A8=[a1,a2,a3,a4,a5,a6,a7,b ]; x1=det(A1)/det(A) x2=det(A2)/det(A) x3=det(A3)/det(A) x4=det(A4)/det(A) x5=det(A5)/det(A) x6=det(A6)/det(A) x7=det(A7)/det(A) x8=det(A8)/det(A) x1 = -6.3093

x2 = 6.3093

x3 = -2.3093

x4 = 0.6187

x5 = 0.8346

x6 = 0.2084

x7 = -0.0849

x8 = 0.3856 方法二： >> format short; A= [ 1,1,0,0,0,0,0,0; 1,2,1,0,0,0,0,0; 0,1,3,1,0,0,0,0; 0,0,1,4,1,0,0,0; 0,0,0,1,5,1,0,0; 0,0,0,0,1,6,1,0; 0,0,0,0,0,1,7,1; 0,0,0,0,0,0,1,8 ]; b= [0;4;0;1;5;2;0;3]; X= inv(A)*b

X = -6.3093 6.3093 -2.3093 0.6187 0.8346 0.2084 -0.0849 0.3856 方法三： >> format short; A= [ 1,1,0,0,0,0,0,0; 1,2,1,0,0,0,0,0; 0,1,3,1,0,0,0,0; 0,0,1,4,1,0,0,0; 0,0,0,1,5,1,0,0; 0,0,0,0,1,6,1,0; 0,0,0,0,0,1,7,1; 0,0,0,0,0,0,1,8 ]; b= [0;4;0;1;5;2;0;3]; rref([A,b])

ans = Columns 1 through 8 1.0000 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 1.0000 Column 9 -6.3094 6.3094 -2.3094 0.6187 0.8346 0.2084 -0.0848 0.3856 二、请绘制下面的图形，按照各图标题中说明的要求。注意这里的图示只是一个表示，不是真实的倍数或者角度。

命令： function SmileFace clear all, clf, hold off

dt = pi/20; t=0:dt:2*pi; xf0=cos(t); yf0=sin(t); % face

xe0(1,:)=0.08*xf0-0.3; ye0(1,:)=0.12*yf0+0.2; % left eye xe0(2,:)=0.08*xf0+0.3; ye0(2,:)=ye0(1,:); % right eye

s1 = 3*pi/2-1.1; s2 = 3*pi/2+1.1; s = s1:dt:s2; xm0 = 0.5*cos(s); ym0 = 0.5*sin(s); % mouth

%figure(1) %绘制初始笑脸图 subplot(2,2,1) T0=eye(3); [xf1,yf1,xe1,ye1,xm1,ym1]=DrawSmileFace(xf0,yf0,xe0,ye0,xm0,ym0,T0); title('图1：笑脸初始图'); %figure(2) %绘制笑脸图 subplot(2,2,2) T1=[2.5 0 0;0 1 0;0 0 1]; %横轴2.5倍，纵轴1倍 %T1=[1 0 0;0 2.5 0;0 0 1]; %横轴1倍，纵轴2.5倍 [xf2,yf2,xe2,ye2,xm2,ym2]=DrawSmileFace(xf0,yf0,xe0,ye0,xm0,ym0,T1); title('图2：笑脸横轴缩放2.5倍，纵轴不变');

%figure(3) %绘制笑脸图 subplot(2,2,3) T2=[1 0 1;0 1 1.5;0 0 1]; %横轴平移1；纵轴平移1.5 [xf3,yf3,xe3,ye3,xm3,ym3]=DrawSmileFace(xf2,yf2,xe2,ye2,xm2,ym2,T2); title('笑脸横轴平移1，纵轴平移1.5');

%figure(4) %绘制笑脸图 subplot(2,2,4) theta=pi/4; %逆时针旋转45度 x0=sum(xf3)/length(xf3); y0=sum(yf3)/length(yf3); xy0=[x0;y0]; T3_1=eye(3); T3_1(1:2,3)=-xy0; T3_2=[cos(theta),-sin(theta),0;sin(theta),cos(theta),0;0 0 1]; T3_3=2*eye(3)-T3_1; T3=T3_3*T3_2*T3_1; %T3=[cos(theta),-sin(theta),0;sin(theta),cos(theta),0;0 0 1]; [xf4,yf4,xe4,ye4,xm4,ym4]=DrawSmileFace(xf3,yf3,xe3,ye3,xm3,ym3,T3); title('图4：绕笑脸中心逆时针旋转45度');

%绘制变换笑脸图子程序 function [xf1,yf1,xe1,ye1,xm1,ym1]=DrawSmileFace(xf,yf,xe,ye,xm,ym,T) n1=length(xf); n2=length(xe); n3=length(xm); Pf0=[xf;yf;ones(1,n1)]; Pf1=T*Pf0; xf1=Pf1(1,:); yf1=Pf1(2,:); fill(xf1,yf1,'r'); % face hold on

Pe01=[xe(1,:);ye(1,:);ones(1,n2)]; Pe02=[xe(2,:);ye(2,:);ones(1,n2)]; Pe1=T*Pe01; Pe2=T*Pe02; xe1(1,:)=Pe1(1,:); ye1(1,:)=Pe1(2,:); xe1(2,:)=Pe2(1,:); ye1(2,:)=Pe2(2,:); plot(xe1(1,:),ye1(1,:),'k','linewidth',5) % left eye plot(xe1(2,:),ye1(2,:),'k','linewidth',5) % right eye

Pm0=[xm;ym;ones(1,n3)]; Pm1=T*Pm0; xm1=Pm1(1,:); ym1=Pm1(2,:); plot(xm1,ym1,'k','linewidth',2) % mouth grid on; axis([-3, 3, -2.5, 2.5])