本内容及供参考
实验一
第一部分
5、>> a1=[5 12 47;13 41 2;9 6 72];
>> a2=[12 9;6 5;7 21];
>> a1*a2
ans =
461 1092
416 364
648 1623
>> a1(:,1:2).*a2
ans =
60 108
78 205
63 126
>> a1^2
ans =
604 834 3643
616 1849 837
771 786 5619 >> a1(:).^2
ans =
25
169
81
144
1681
36
2209
4
5184
7、>> b1=[a1;rot90(a2)]
b1 =
5 12 47
13 41 2
9 6 72
9 5 21
12 6 7
>> c1=b1(4,1)
c1 =
9
>> c2=b1(5,3)
c2 =
7
>> b1(3:4,:)*a2
ans =
648 1623
285 547
13、>> d1=1:16;
>> d2=reshape(d1,4,4);
>> d3=fliplr(d2);
>> c=rot90(d3)
c =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
>> c^(-4)
ans =
1.0e+010 *
0.5286 -1.7180 1.8502 -0.6608
-1.5858 3.4359 -2.1145 0.2644
1.5858 -1.7180 -1.3215 1.4537
-0.5286 0.0000 1.5858 -1.0572 >> (c^3)^(-1)
ans =
1.0e+012 *
-0.3665 0.5500 -0.0005 -0.1830
0.5498 -0.5502 -0.5490 0.5494
0 -0.5498 1.0995 -0.5498
-0.1833 0.5499 -0.5500 0.1834 >> (3*c+5*c^(-1))/5
ans =
1.0e+015 *
3.9406 -
4.5036 -2.8147 3.3777
-4.1283 4.5036 3.3777 -3.7530
-3.5653 4.5036 1.6888 -2.6271
3.7530 -
4.5036 -2.2518 3.0024 18、>> a=[1 2 3];
>> b=[2 3 4];
>> dot(a,b)
ans =
20
>> cross(a,b)
ans =
-1 2 -1
第二部分
6、>> syms x
>> expand((x+1)^3+(x-1)^3-2*x^3)
ans =
6*x
7、>> syms x
>> diff(log(cos(1/x))+(x^2-2*x^2+3)*exp(-x)+sin(x^3),1)
ans =
sin(1/x)/x^2/cos(1/x)-2*x*exp(-x)-(-x^2+3)*exp(-x)+3*cos(x^3)*x^2 >> diff(log(cos(1/x))+(x^2-2*x^2+3)*exp(-x)+sin(x^3),2)
ans =
-1/x^4-2*sin(1/x)/x^3/cos(1/x)-sin(1/x)^2/x^4/cos(1/x)^2-2*exp(-x)+4*x*exp(-x)+(-x^2+3) *exp(-x)-9*sin(x^3)*x^4+6*cos(x^3)*x
8、(1)>> syms x y dz dx dy
>> A=diff(sin(x*cos(y)),'x',1)
A =
cos(x*cos(y))*cos(y)
>> B=diff(sin(x*cos(y)),'y',1)
B =
-cos(x*cos(y))*x*sin(y)
>> dz=A*dx+B*dy
dz =
cos(x*cos(y))*cos(y)*dx-cos(x*cos(y))*x*sin(y)*dy
(2)>> syms x y z du dx dy dz
>> a=diff(asin(z/sqrt(x^2+y^2)),'x',1)
a =
-z/(x^2+y^2)^(3/2)*x/(1-z^2/(x^2+y^2))^(1/2)
>> b=diff(asin(z/sqrt(x^2+y^2)),'y',1)
b =
-z/(x^2+y^2)^(3/2)*y/(1-z^2/(x^2+y^2))^(1/2)
>> c=diff(asin(z/sqrt(x^2+y^2)),'z',1)
c =
1/(x^2+y^2)^(1/2)/(1-z^2/(x^2+y^2))^(1/2)
>> du=a*dx+b*dy+c*dz
du =
-z/(x^2+y^2)^(3/2)*x/(1-z^2/(x^2+y^2))^(1/2)*dx-z/(x^2+y^2)^(3/2)*y/(1-z^2/(x^2+y^
2))^(1/2)*dy+1/(x^2+y^2)^(1/2)/(1-z^2/(x^2+y^2))^(1/2)*dz
9、>> syms x a
>> symsum(a/(x^2),x,1,10)
ans =
1968329/1270080*a
>> symsum(a/(x^2),x,1,inf)
ans =
1/6*a*pi^2
10、>> syms n
>> symsum(3^(n+1)/(n+1),'n',1,5)
ans =
4077/20
>> symsum(5^(n+1)/(n+1),'n',1,5)
ans =
41275/12
11、>> syms x a
>> taylor(sin(x^2)+cos(x),5,'x',0)
ans =
1+1/2*x^2+1/24*x^4
>> taylor(sin(x^2)+cos(x),5,'x',3)
ans =
sin(9)+cos(3)+(6*cos(9)-sin(3))*(x-3)+(-1/2*cos(3)+cos(9)-18*sin(9))*(x-3)^2+(1/6*sin(
3)-36*cos(9)-6*sin(9))*(x-3)^3+(-18*cos(9)+107/2*sin(9)+1/24*cos(3))*(x-3)^4
>> taylor(log(x+a),5,'x',0)
ans =
log(a)+1/a*x-1/2/a^2*x^2+1/3/a^3*x^3-1/4/a^4*x^4
>> taylor(log(x+a),5,'x',3)
ans =
log(3+a)+1/(3+a)*(x-3)-1/2/(3+a)^2*(x-3)^2+1/3/(3+a)^3*(x-3)^3-1/4/(3+a)^4*(x-3)^4 >> taylor(sinh(x),5,'x',0)
ans =
x+1/6*x^3
>> taylor(sinh(x),5,'x',3)
ans =
1/2*exp(3)-1/2/exp(3)+(1/2*exp(3)+1/2/exp(3))*(x-3)+(-1/4/exp(3)+1/4*exp(3))*(x-3)^2 +(1/12/exp(3)+1/12*exp(3))*(x-3)^3+(1/48*exp(3)-1/48/exp(3))*(x-3)^4
实验二
第一部分
1、>>x1=-5:0.01:6;y1=0;
y2=-6:0.01:3;x2=0;
x3=-3:0.01:5;y3=1.5-0.5*x3;
plot(x1,y1,x2,y2,x3,y3)
或者
>> x1=-5:0.01:6;y1=0;
plot(x1,y1)
hold on
y2=-6:0.01:3;x2=0;
plot(x2,y2)
hold on
x3=-3:0.01:5;y3=1.5-0.5*x3;
plot(x3,y3)
或者
>> x=-5:0.01:6;y=0;
plot(x,y)
hold on
y=-6:0.01:3;x=0;
plot(x,y)
hold on
x=-3:0.01:5;y=1.5-0.5*x3;
plot(x,y)
图像
2、>> X=[-1 0 1 -1];
Y=[0 1 0 0];
plot(X,Y,'r--+','linewidth',3) 图像
3(1)
>> ezplot('y^2-2*x-1')
hold on
ezplot('y-x+4')
hold on
ezplot('y-x^2')
hold on
ezplot('x^2+y^2-1')
图像
3、(2)
>> clf
ezplot('6*x^2+5*y^2-30')
hold on
ezplot('(x^2+y^2)^2-x^2+y^2')
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3、(3)
>> ezplot('x^2+y^2+x-sqrt(x^2+y^2)') hold on
t=0:0.01:pi;
x=2*(cos(t)).^3;
y=2*(sin(t)).^3;
plot(x,y)
图像
4、a=-3;b=13;n=1000;
h=(b-a)/n;x=a;
for i=1:n
if x<0
y=4*(cos(x-pi/4))^2;
else if x<5
y=x/5+4;
else
y=x*exp(5-x);
end
X(i)=x;Y(i)=y;
x=x+h;
end
plot(X,Y)
图像
或者
a=-3;b=13;n=1000;
h=(b-a)/n;
for i=1:n
x(i)=a+h*(i-1);
if x(i)<0
y(i)=4*(cos(x(i)-pi/4))^2;
else if x(i)<5
y(i)=x(i)/5+4;
else
y(i)=x(i)*exp(5-x(i));
end
end
end
plot(x,y)
图像
5、
fplot('cos(tan(pi*x))',[0 2],1e-3) 图像
6、ezplot('exp(y)+cos(x)/x+y')
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第二部分
3、
>> [x y z]=ellipsoid(0,0,1,2,3,4,40);
subplot(1,2,1);
surf(x,y,z);
t=-1:0.01:1;
[x y z]=cylinder(1+t.^2,40);
subplot(1,2,2);
surf(x,y,z);
图像
综合题
1、A=1;fi=0;
w1=1;w2=2;
t=0:pi/100:2*pi;
x1=A*sin(w1*t+fi);
x2=A*sin(w2*t+fi);
plot(t,x1,t,x2)
text(2.5,0.7,'x1=A*sin(w1*t+fi)');
text(5,0.6,'x2=A*sin(w2*t+fi)');
legend('x1','x2')
图像
2、
clear
clf
A=1;w=1;
fi3=pi/6;fi4=pi/3;
t=0:pi/100:2*pi;
x3=A*sin(w*t+fi3);
x4=A*sin(w*t+fi4);
plot(t,x3,t,x4)
text(2.5,0.7,'x3=sin(t+pi/6)');
text(5,0.6,'x4=sin(t+pi/3)');
legend('x3','x4')
图像
3、
clear
clf
A5=1;A6=2;
w=1;fi=0;
t=0:pi/100:2*pi;
x5=A5*sin(w*t+fi); x6=A6*sin(w*t+fi); plot(t,x5,t,x6)
text(1,0.5,'x5=sin(t)'); text(3,1,'x6=2*sin(t)'); legend('x5','x6')
图像
实验五
龙格库塔法
二阶
a=2;b=4;
n=1000;
h=(b-a)/n;
y2(1)=1;x2(1)=a;
for i=1:n
k=h*f(x2(i),y2(i));
y2(i+1)=y2(i)+h*f(x2(i)+h/2,y2(i)+h/2);
x2(i+1)=x2(i)+h;
end
t=2:0.1:4;
T=exp(2-t.^2/2);
plot(t,T,'*',x2,y2)
图像
三阶
clear
clf
a=2;b=4;
n=1000;
h=(b-a)/n;
y3(1)=1;x3(1)=a;
for i=1:n
k1=h*f(x3(i),y3(i));
k2=h*f(x3(i)+h/2,y3(i)+k1/2);
k3=h*f(x3(i)+h,y3(i)-k1+2*k2);
y3(i+1)=y3(i)+(k1+4*k2+k3)/6;
x3(i+1)=x3(i)+h;
end
t=2:0.1:4;
T=exp(2-t.^2/2);
plot(t,T,'*',x3,y3)
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四阶
clear
clf
a=2;b=4;
n=1000;
h=(b-a)/n;
y4(1)=1;x4(1)=a;
for i=1:n
k1=h*f(x4(i),y4(i));
k2=h*f(x4(i)+h/2,y4(i)+k1/2);
k3=h*f(x4(i)+h/2,y4(i)+k2/2);
k4=h*f(x4(i)+h,y4(i)+k3);
y4(i+1)=y4(i)+(k1+2*k2+2*k3+k4)/6;
x4(i+1)=x4(i)+h;
end
t=2:0.1:4;
T=exp(2-t.^2/2);
plot(t,T,'*',x4,y4)
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2、Numerov算法
clear
clf
a=0;b=1;N=1000;h=(b-a)/N;
y(1)=2;y(2)=2;x(1)=a;x(2)=a+h;
for n=2:N
y(n+1)=2*(1-5*h^2/12*4*pi*pi)/(1+h^2/12*4*pi*pi)*y(n)-y(n-1); x(n+1)=x(n)+h;
end
plot(x,y)
图像