实验一 MATLAB软件的使用

§ 初等代数

1.1 表达式的运算

1.一元多项式的运算
表1.1一元多项式常用的命令

>> p=[1,2,1];roots(p)ans =-1-1

>> poly([1,2])ans =1    -3     2

>> [r,p,k]=residue([1,2,3,4],[1,-3,2])r =26-10p =21k =1     5

2.一般符号表达式的运算
表1.2

>> syms x y z a b
>> f=(x+y)*(x+2*y);collect(f)
ans =x^2 + 3*x*y + 2*y^2

>> g=(a+b)*(a+2*b);collect(g)ans =a^2 + 3*a*b + 2*b^2

>>  u=sym('(a-b)^2');a=x+y;b=z;v=subs(u)v =(x + y - z)^2

>> [n,d]=numden(x/y+y/x)n =x^2 + y^2d =x*y

1.2 方程求解

表1.3方程组求解的常用命令

>> syms a b c x
>> s=a*x^2+b*x+c;solve(s)ans =-(b + (b^2 - 4*a*c)^(1/2))/(2*a)-(b - (b^2 - 4*a*c)^(1/2))/(2*a)

>> fzero(@sin,3)ans =3.141592653589793

>> q=solve('sin(x)-cos(x)')
q =pi/4

>> s=solve('a*u^2+v^2','u-v=1','a,u')
s = a: [1x1 sym]u: [1x1 sym]

>> s.aans =-v^2/(v + 1)^2

§2 微积分

表1.4

>> syms x n
>> limit(sin(x)/x)ans =1

>> diff(sin(n*x),x,3)ans =-n^3*cos(n*x)

>> int(log(x))ans =x*(log(x) - 1)

>> quad(@(x)4./(1+x.^2),0,1)ans =3.141592682924567

>> dsolve('Dy-y=1')ans =C4*exp(t) - 1

>> dsolve('Dy-y=1','y(0)=1')ans =2*exp(t) - 1

>> taylor(exp(x),x,0,'Order',5)ans =x^4/24 + x^3/6 + x^2/2 + x + 1

§3 线性代数

>> A=[1,2;8,7];inv(A)ans =-0.777777777777778   0.2222222222222220.888888888888889  -0.111111111111111

>> [P,D]=eig(A)P =-0.707106781186547  -0.2425356250363330.707106781186547  -0.970142500145332D =-1     00     9

>> log(A)ans =0   0.6931471805599452.079441541679836   1.945910149055313

>> B=sym([1,2;8,7]);inv(B) %采用符号运算得到精确的结果ans =[ -7/9,  2/9]
[  8/9, -1/9]

>> [P,D]=eig(B)P =[ -1, 1/4]
[  1,   1]D =[ -1, 0]
[  0, 9]

表1.5关于向量与矩阵的一些常用命令

§4 计算方法

4.1 插值

>> x=0:0.1:2;y=sin(x);
xi=0:0.1:2;yi=interp1(x,y,'spline')yi =1 至 11 列-0.1000   -0.0900   -0.0801   -0.0704   -0.0611   -0.0521   -0.0435   -0.0356   -0.0283   -0.0217   -0.015912 至 21 列-0.0109   -0.0068   -0.0036   -0.0015   -0.0003   -0.0000   -0.0008   -0.0026   -0.0054   -0.0091

>> zi=sin(xi);plot(xi,zi-yi)

4.2 拟合
下面的命令用来对数据进行多项式最小二乘拟合

>> x=1:10;y=log(x);f=polyfit(x,y,2)f =-0.03          0.53         -0.36

>> z=polyval(f,x);plot(x,y,'*',x,z,'-')

4.3 最优化

>> [x,y]=fminsearch(@(x)sin(x),5) %求sinx在5附近的最小值x =4.71y =-1.00

>> [x,y]=fminbnd(@(x)cos(x),0,5) %求cosx在【0,5】的最小值x =3.14y =-1.00

§5 MATLAB软件中的作图

5.1 二维作图
1.plot(y)

>> plot([2,3,4,7,11,13])

2.plot(x,y)

>> x=0:pi/50:2*pi;y=sin(x);plot(x,y)

表1.6 plot语句的各种常用选项

3.plot(x1,y1,‘s1’,x2,y2,‘s2’,…)

>> t=0:pi/50:2*pi;
>> x1=cos(t);y1=sin(t);
>> x2=2*cos(t);y2=2*sin(t);
>> x3=3*cos(t);y3=3*sin(t);
>> plot(x1,y1,x2,y2,x3,y3)

4.plot(X,Y,‘s’)

>> t=[0:pi/50:2*pi]';u=1:3;
>> X=cos(t)*u;Y=sin(t)*u;plot(X,Y)

5.图形控制

6.ezplot(‘fun’,[xmin,xmax,ymin,ymax])

ezplot('x^2-2*y^2-1',[-6,6,-4,4])

5.2 三维曲线作图
1.plot3(x,y,z)
若x,y,z为相同长度的向量.该命令在三维空间中生成一条曲线，坐标对应于x,y,z的值.若x,y,z为相同维数的矩阵，则画出若干条空间曲线，数目等于矩阵的列数.

>> t=-8:0.1:8;x=6*cos(t);y=6*sin(t);z=3*t;
>> plot3(x,y,z);grid on

2.ezplot3(‘f’,‘f’,‘h’,[t1,t2])

 ezplot3('6*cos(t)','6*sin(t)','3*t',[-8,8])

5.3 三维曲面作图
1.mesh(X,Y,Z) 与 surf(X,Y,Z)

>> x=-5:0.2:5;y=x;
[X,Y]=meshgrid(x,y);
>> Z=sin(sqrt(X.^2+Y.^2));
>> mesh(X,Y,Z);

2.ezmesh(‘f’,‘g’,‘h’,[u1,u2,v1,v2]) 与ezsurf(‘f’,‘g’,‘h’,[u1,u2,v1,v2])

>> ezmesh('cos(u)*cos(v)','sin(u)*cos(v)','sin(v)',[0,2*pi,-pi,pi]);axis equal;
>> ezsurf('cos(u)*cos(v)','sin(u)*cos(v)','sin(v)',[0,2*pi,-pi,pi]);axis equal;