%Page20,ex1
(5) 等于[exp(1),exp(2);exp(3),exp(4)]
(7) 3=1*3, 8=2*4
(8) a为各列最小值，b为最小值所在的行号
(10) 1>=4,false, 2>=3,false, 3>=2, ture, 4>=1,ture
(11) 答案表明：编址第2元素满足不等式(30>=20)和编址第4元素满足不等式(40>=10)
(12) 答案表明：编址第2行第1列元素满足不等式(30>=20)和编址第2行第2列元素满足不等式(40>=10)
%Page20, ex2
(1)a, b, c的值尽管都是1， 但数据类型分别为数值，字符， 逻辑， 注意a与c相等，他们不等于b
(2)double(fun)输出的分别是字符a,b,s,(,x,)的ASCII码
%Page20,ex3
>> r=2;p=0.5;n=12;
>> T=log(r)/n/log(1+0.01*p)
T =
11.5813
%Page20,ex4
>> x=-2:0.05:2;f=x.^4-2.^x;
>> [fmin,min_index]=min(f)
fmin =
-1.3907 %最小值
min_index =
54 %最小值点编址
>> x(min_index)
ans =
0.6500 %最小值点
>> [f1,x1_index]=min(abs(f)) %求近似根--绝对值最小的点
f1 =
0.0328
x1_index =
24
>> x(x1_index)
ans =
-0.8500
>> x(x1_index)=[];f=x.^4-2.^x; %删去绝对值最小的点以求函数绝对值次小的点 >> [f2,x2_index]=min(abs(f)) %求另一近似根--函数绝对值次小的点
f2 =
0.0630
x2_index =
65 但
>> x(x2_index)
ans =
1.2500
%Page20,ex5
>> z=magic(10)
z =
92 99 1 8 15 67 74 51 58 40
98 80 7 14 16 73 55 57 64 41
4 81 88 20 22 54 56 63 70 47
85 87 19 21 3 60 62 69 71 28
86 93 25 2 9 61 68 75 52 34
17 24 76 83 90 42 49 26 33 65
23 5 82 89 91 48 30 32 39 66
79 6 13 95 97 29 31 38 45 72
10 12 94 96 78 35 37 44 46 53
11 18 100 77 84 36 43 50 27 59
>> sum(z)
ans =
505 505 505 505 505 505 505 505 505 505
>> sum(diag(z))
ans =
505
>> z(:,2)/sqrt(3)
ans =
57.1577
46.1880
46.7654
50.2295
53.6936
13.8564
2.8868
3.4641
6.9282
10.3923
>> z(8,:)=z(8,:)+z(3,:)
z =
92 99 1 8 15 67 74 51 58 40
98 80 7 14 16 73 55 57 64 41
4 81 88 20 22 54 56 63 70 47
85 87 19 21 3 60 62 69 71 28
86 93 25 2 9 61 68 75 52 34
17 24 76 83 90 42 49 26 33 65
23 5 82 89 91 48 30 32 39 66
83 87 101 115 119 83 87 101 115 119
10 12 94 96 78 35 37 44 46 53
11 18 100 77 84 36 43 50 27 59
%Page 40 ex1
先在编辑器窗口写下列M函数，保存为eg2_1.m
function [xbar,s]=ex2_1(x)
n=length(x);
xbar=sum(x)/n;
s=sqrt((sum(x.^2)-n*xbar^2)/(n-1));
例如
>>x=[81 70 65 51 76 66 90 87 61 77];
>>[xbar,s]=ex2_1(x)
xbar =
72.4000
s =
12.1124
%Page 40 ex2
s=log(1);n=0;
while s
n=n+1;
s=s+log(1+n);
end
m=n
计算结果m=37
%Page 40 ex3
clear;
F(1)=1;F(2)=1;k=2;x=0;
e=1e-8; a=(1+sqrt(5))/2;
while abs(x-a)>e
k=k+1;F(k)=F(k-1)+F(k-2); x=F(k)/F(k-1);
end
a,x,k
计算至k=21可满足精度
%Page 40 ex4
clear;tic;s=0;
for i=1:1000000
s=s+sqrt(3)/2^i;
end
s,toc
tic;s=0;i=1;
while i
s=s+sqrt(3)/2^i;i=i+1;
end
s,toc
tic;s=0;
i=1:1000000;
s=sqrt(3)*sum(1./2.^i);
s,toc
%Page 40 ex5
t=0:24;
c=[15 14 14 14 14 15 16 18 20 22 23 25 28 ...
31 32 31 29 27 25 24 22 20 18 17 16];
plot(t,c)
%Page 40 ex6
%(1)
x=-2:0.1:2;y=x.^2.*sin(x.^2-x-2);plot(x,y)
y=inline('x^2*sin(x^2-x-2)');fplot(y,[-2 2])
%(2)参数方法
t=linspace(0,2*pi,100);
x=2*cos(t);y=3*sin(t); plot(x,y)
%(3)
x=-3:0.1:3;y=x;
[x,y]=meshgrid(x,y);
z=x.^2+y.^2;
surf(x,y,z)
%(4)
x=-3:0.1:3;y=-3:0.1:13;
[x,y]=meshgrid(x,y);
z=x.^4+3*x.^2+y.^2-2*x-2*y-2*x.^2.*y+6;
surf(x,y,z)
%(5)
t=0:0.01:2*pi;
x=sin(t);y=cos(t);z=cos(2*t);
plot3(x,y,z)
%(6)
theta=linspace(0,2*pi,50);fai=linspace(0,pi/2,20);
[theta,fai]=meshgrid(theta,fai);
x=2*sin(fai).*cos(theta);
y=2*sin(fai).*sin(theta);z=2*cos(fai);
surf(x,y,z)
%(7)
x=linspace(0,pi,100);
y1=sin(x);y2=sin(x).*sin(10*x);y3=-sin(x);
plot(x,y1,x,y2,x,y3)
%page41, ex7
x=-1.5:0.05:1.5;
y=1.1*(x>1.1)+x.*(x=-1.1)-1.1*(x
plot(x,y)
%page41,ex8
分别使用which trapz, type trapz, dir C:\MATLAB7\toolbox\matlab\datafun\ %page41,ex9
clear;close;
x=-2:0.1:2;y=x;
[x,y]=meshgrid(x,y);
a=0.5457;b=0.7575;
p=a*exp(-0.75*y.^2-3.75*x.^2-1.5*x).*(x+y>1);
p=p+b*exp(-y.^2-6*x.^2).*(x+y>-1).*(x+y
p=p+a*exp(-0.75*y.^2-3.75*x.^2+1.5*x).*(x+y
mesh(x,y,p)
%page41, ex10
lookfor lyapunov
help lyap
>> A=[1 2 3;4 5 6;7 8 0];C=[2 -5 -22;-5 -24 -56;-22 -56 -16];
>> X=lyap(A,C)
X =
1.0000 -1.0000 -0.0000
-1.0000 2.0000 1.0000
-0.0000 1.0000 7.0000
%Chapter 3
%Exercise 1
>> a=[1,2,3];b=[2,4,3];a./b,a.\b,a/b,a\b
ans =
0.5000 0.5000 1.0000
ans =
2 2 1
ans =
0.6552 %一元方程组x[2,4,3]=[1,2,3]的近似解
ans =
0 0 0
0 0 0
0.6667 1.3333 1.0000
%矩阵方程[1,2,3][x11,x12,x13;x21,x22,x23;x31,x32,x33]=[2,4,3]的特解 %Exercise 2(1)
>> A=[4 1 -1;3 2 -6;1 -5 3];b=[9;-2;1];
>> rank(A), rank([A,b]) %[A,b]为增广矩阵
ans =
3
ans =
3 %可见方程组唯一解
>> x=A\b
x =
2.3830
1.4894
2.0213
%Exercise 2(2)
>> A=[4 -3 3;3 2 -6;1 -5 3];b=[-1;-2;1];
>> rank(A), rank([A,b])
ans =
3
ans =
3 %可见方程组唯一解
>> x=A\b
x =
-0.4706
-0.2941
%Exercise 2(3)
>> A=[4 1;3 2;1 -5];b=[1;1;1];
>> rank(A), rank([A,b])
ans =
2
ans =
3 %可见方程组无解
>> x=A\b
x =
0.3311
-0.1219 %最小二乘近似解
%Exercise 2(4)
>> a=[2,1,-1,1;1,2,1,-1;1,1,2,1];b=[1 2 3]';%注意b的写法 >> rank(a),rank([a,b])
ans =
3
ans =
3 %rank(a)==rank([a,b])
>> a\b
ans =
1
1
0 %一个特解
%Exercise 3
>> a=[2,1,-1,1;1,2,1,-1;1,1,2,1];b=[1,2,3]';
>> x=null(a),x0=a\b
x =
-0.6255
0.6255
-0.2085
0.4170
x0 =
1
1
%通解kx+x0
%Exercise 4
>> x0=[0.2 0.8]';a=[0.99 0.05;0.01 0.95];
>> x1=a*x, x2=a^2*x, x10=a^10*x
>> x=x0;for i=1:1000,x=a*x;end,x
x =
0.8333
0.1667
>> x0=[0.8 0.2]';
>> x=x0;for i=1:1000,x=a*x;end,x
x =
0.8333
0.1667
>> [v,e]=eig(a)
v =
0.9806 -0.7071
0.1961 0.7071
e =
1.0000 0
0 0.9400
>> v(:,1)./x
ans =
1.1767
1.1767 %成比例，说明x是最大特征值对应的特征向量 %Exercise 5
%用到公式(3.11)(3.12)
>> B=[6,2,1;2.25,1,0.2;3,0.2,1.8];x=[25 5 20]';
>> C=B/diag(x)
C =
0.2400 0.4000 0.0500
0.1200 0.0400 0.0900
>> A=eye(3,3)-C
A =
0.7600 -0.4000 -0.0500
-0.0900 0.8000 -0.0100
-0.1200 -0.0400 0.9100
>> D=[17 17 17]';x=A\D
x =
37.5696
25.7862
24.7690
%Exercise 6(1)
>> a=[4 1 -1;3 2 -6;1 -5 3];det(a),inv(a),[v,d]=eig(a) ans =
-94
ans =
0.2553 -0.0213 0.0426
0.1596 -0.1383 -0.2234
0.1809 -0.2234 -0.0532
v =
0.0185 -0.9009 -0.3066
-0.7693 -0.1240 -0.7248
-0.6386 -0.4158 0.6170
d =
-3.0527 0 0
0 3.6760 0
0 0 8.3766
%Exercise 6(2)
>> a=[1 1 -1;0 2 -1;-1 2 0];det(a),inv(a),[v,d]=eig(a) ans =
1
ans =
2.0000 -2.0000 1.0000
1.0000 -1.0000 1.0000
2.0000 -3.0000 2.0000
v =
-0.5773 0.5774 + 0.0000i 0.5774 - 0.0000i
-0.5773 0.5774 0.5774
-0.5774 0.5773 - 0.0000i 0.5773 + 0.0000i
d =
1.0000 0 0
0 1.0000 + 0.0000i 0
%Exercise 6(3)
>> A=[5 7 6 5;7 10 8 7;6 8 10 9;5 7 9 10]
A =
5 7 6 5
7 10 8 7
6 8 10 9
5 7 9 10
>> det(A),inv(A), [v,d]=eig(A)
ans =
1
ans =
68.0000 -41.0000 -17.0000 10.0000
-41.0000 25.0000 10.0000 -6.0000
-17.0000 10.0000 5.0000 -3.0000
10.0000 -6.0000 -3.0000 2.0000
v =
0.8304 0.0933 0.3963 0.3803
-0.5016 -0.3017 0.6149 0.5286
-0.2086 0.7603 -0.2716 0.5520
0.1237 -0.5676 -0.6254 0.5209
d =
0.0102 0 0 0
0 0.8431 0 0
0 0 3.8581 0
0 0 0 30.2887
%Exercise 6(4)(以n=5为例)
%关键是矩阵的定义
%方法一（三个for）
n=5;
for i=1:n, a(i,i)=5;end
for i=1:(n-1),a(i,i+1)=6;end
for i=1:(n-1),a(i+1,i)=1;end
a
%方法二（一个for）
n=5;a=zeros(n,n);
a(1,1:2)=[5 6];
for i=2:(n-1),a(i,[i-1,i,i+1])=[1 5 6];end
a(n,[n-1 n])=[1 5];
a
%方法三（不用for）
n=5;a=diag(5*ones(n,1));
b=diag(6*ones(n-1,1));
c=diag(ones(n-1,1));
a=a+[zeros(n-1,1),b;zeros(1,n)]+[zeros(1,n);c,zeros(n-1,1)] %下列计算
>> det(a)
ans =
665
>> inv(a)
ans =
0.3173 -0.5865 1.0286 -1.6241 1.9489
-0.0977 0.4887 -0.8571 1.3534 -1.6241
0.0286 -0.1429 0.5429 -0.8571 1.0286
-0.0075 0.0376 -0.1429 0.4887 -0.5865
0.0015 -0.0075 0.0286 -0.0977 0.3173
>> [v,d]=eig(a)
v =
-0.7843 -0.7843 -0.9237 0.9860 -0.9237
0.5546 -0.5546 -0.3771 -0.0000 0.3771
-0.2614 -0.2614 0.0000 -0.1643 0.0000
0.0924 -0.0924 0.0628 -0.0000 -0.0628
-0.0218 -0.0218 0.0257 0.0274 0.0257
d =
0.7574 0 0 0 0
0 9.2426 0 0 0
0 0 7.4495 0 0
0 0 0 5.0000 0
0 0 0 0 2.5505
%Exercise 7(1)
>> a=[4 1 -1;3 2 -6;1 -5 3];[v,d]=eig(a)
v =
0.0185 -0.9009 -0.3066
-0.7693 -0.1240 -0.7248
-0.6386 -0.4158 0.6170
d =
-3.0527 0 0
0 3.6760 0
0 0 8.3766
>> det(v)
ans =
-0.9255 %v行列式正常, 特征向量线性相关，可对角化 >> inv(v)*a*v %验算
ans =
-3.0527 0.0000 -0.0000
0.0000 3.6760 -0.0000
-0.0000 -0.0000 8.3766
>> [v2,d2]=jordan(a) %也可用jordan
v2 =
0.0798 0.0076 0.9127
0.1886 -0.3141 0.1256
-0.1605 -0.2607 0.4213 %特征向量不同
d2 =
8.3766 0 0
0 -3.0527 - 0.0000i 0
0 0 3.6760 + 0.0000i
>> v2\a*v2
ans =
8.3766 0 0.0000
0.0000 -3.0527 0.0000
0.0000 0.0000 3.6760
>> v(:,1)./v2(:,2) %对应相同特征值的特征向量成比例
ans =
2.4491
2.4491
2.4491
%Exercise 7(2)
>> a=[1 1 -1;0 2 -1;-1 2 0];[v,d]=eig(a)
v =
-0.5773 0.5774 + 0.0000i 0.5774 - 0.0000i
-0.5773 0.5774 0.5774
-0.5774 0.5773 - 0.0000i 0.5773 + 0.0000i
d =
1.0000 0 0
0 1.0000 + 0.0000i 0
0 0 1.0000 - 0.0000i
>> det(v)
ans =
-5.0566e-028 -5.1918e-017i %v的行列式接近0, 特征向量线性相关，不可对角化 >> [v,d]=jordan(a)
v =
1 0 1
1 0 0
1 -1 0
d =
1 1 0
0 1 1
0 0 1 %jordan标准形不是对角的，所以不可对角化
%Exercise 7(3)
>> A=[5 7 6 5;7 10 8 7;6 8 10 9;5 7 9 10]
A =
5 7 6 5
7 10 8 7
6 8 10 9
5 7 9 10
>> [v,d]=eig(A)
v =
0.8304 0.0933 0.3963 0.3803
-0.5016 -0.3017 0.6149 0.5286
-0.2086 0.7603 -0.2716 0.5520
0.1237 -0.5676 -0.6254 0.5209
d =
0.0102 0 0 0
0 0.8431 0 0
0 0 3.8581 0
0 0 0 30.2887
>> inv(v)*A*v
ans =
0.0102 0.0000 -0.0000 0.0000
0.0000 0.8431 -0.0000 -0.0000
-0.0000 0.0000 3.8581 -0.0000
-0.0000 -0.0000 0 30.2887
%本题用jordan不行, 原因未知
%Exercise 7(4)参考6(4)和7(1), 略
%Exercise 8 只有(3)对称, 且特征值全部大于零, 所以是正定矩阵.
%Exercise 9(1)
>> a=[4 -3 1 3;2 -1 3 5;1 -1 -1 -1;3 -2 3 4;7 -6 -7 0]
>> rank(a)
ans =
3
>> rank(a(1:3,:))
ans =
2
>> rank(a([1 2 4],:)) %1,2,4行为最大无关组
ans =
3
>> b=a([1 2 4],:)';c=a([3 5],:)';
>> b\c %线性表示的系数
ans =
0.5000 5.0000
-0.5000 1.0000
0 -5.0000
%Exercise 10
>> a=[1 -2 2;-2 -2 4;2 4 -2]
>> [v,d]=eig(a)
v =
0.3333 0.9339 -0.1293
0.6667 -0.3304 -0.6681
-0.6667 0.1365 -0.7327
d =
-7.0000 0 0
0 2.0000 0
0 0 2.0000
>> v'*v
ans =
1.0000 0.0000 0.0000
0.0000 1.0000 0
0.0000 0 1.0000 %v确实是正交矩阵
%Exercise 11
%设经过6个电阻的电流分别为i1, ..., i6. 列方程组如下
%20-2i1=a; 5-3i2=c; a-3i3=c; a-4i4=b; c-5i5=b; b-3i6=0;
%i1=i3+i4;i5=i2+i3;i6=i4+i5;
%计算如下
>> A=[1 0 0 2 0 0 0 0 0;0 0 1 0 3 0 0 0 0;1 0 -1 0 0 -3 0 0 0; 1 -1 0 0 0 0 -4 0 0;
0 -1 1 0 0 0 0 -5 0;0 1 0 0 0 0 0 0 -3; 0 0 0 1 0 -1 -1 0 0;0 0 0 0 -1 -1 0 1 0;
0 0 0 0 0 0 -1 -1 1];
>>b=[20 5 0 0 0 0 0 0 0]'; A\b
ans =
13.3453
6.4401
8.5420
3.3274
-1.1807
1.6011
1.7263
0.4204
2.1467
%Exercise 12
>> A=[1 2 3;4 5 6;7 8 0];
>> left=sum(eig(A)), right=sum(trace(A))
left =
6.0000
right =
6
>> left=prod(eig(A)), right=det(A) %原题有错, (-1)^n应删去
left =
27.0000
right =
27
>> fA=(A-p(1)*eye(3,3))*(A-p(2)*eye(3,3))*(A-p(3)*eye(3,3))
fA =
1.0e-012 *
0.0853 0.1421 0.0284
0.1421 0.1421 0
-0.0568 -0.1137 0.1705
>> norm(fA) %f(A)范数接近0
ans =
2.9536e-013
%Exercise 1(1)
roots([1 1 1])
%Exercise 1(2)
roots([3 0 -4 0 2 -1])
%Exercise 1(3)
p=zeros(1,24);
p([1 17 18 22])=[5 -6 8 -5];
roots(p)
%Exercise 1(4)
p1=[2 3];
p2=conv(p1, p1);
p3=conv(p1, p2);
p3(end)=p3(end)-4; %原p3最后一个分量-4
roots(p3)
%Exercise 2
fun=inline('x*log(sqrt(x^2-1)+x)-sqrt(x^2-1)-0.5*x');
fzero(fun,2)
%Exercise 3
fun=inline('x^4-2^x');
fplot(fun,[-2 2]);grid on;
fzero(fun,-1),fzero(fun,1),fminbnd(fun,0.5,1.5)
%Exercise 4
fun=inline('x*sin(1/x)','x');
fplot(fun, [-0.1 0.1]);
x=zeros(1,10);for i=1:10, x(i)=fzero(fun,(i-0.5)*0.01);end;
x=[x,-x]
%Exercise 5
fun=inline('[9*x(1)^2+36*x(2)^2+4*x(3)^2-36;x(1)^2-2*x(2)^2-20*x(3);16*x(1)-x(1)^3-2*x(2)^2-16*x(3)^2]','x');
[a,b,c]=fsolve(fun,[0 0 0])
%Exercise 6
fun=@(x)[x(1)-0.7*sin(x(1))-0.2*cos(x(2)),x(2)-0.7*cos(x(1))+0.2*sin(x(2))];
[a,b,c]=fsolve(fun,[0.5 0.5])
%Exercise 7
clear; close; t=0:pi/100:2*pi;
x1=2+sqrt(5)*cos(t); y1=3-2*x1+sqrt(5)*sin(t);
x2=3+sqrt(2)*cos(t); y2=6*sin(t);
plot(x1,y1,x2,y2); grid on; %作图发现4个解的大致位置，然后分别求解
y1=fsolve('[(x(1)-2)^2+(x(2)-3+2*x(1))^2-5,2*(x(1)-3)^2+(x(2)/3)^2-4]',[1.5,2])
y2=fsolve('[(x(1)-2)^2+(x(2)-3+2*x(1))^2-5,2*(x(1)-3)^2+(x(2)/3)^2-4]',[1.8,-2])
y3=fsolve('[(x(1)-2)^2+(x(2)-3+2*x(1))^2-5,2*(x(1)-3)^2+(x(2)/3)^2-4]',[3.5,-5])
y4=fsolve('[(x(1)-2)^2+(x(2)-3+2*x(1))^2-5,2*(x(1)-3)^2+(x(2)/3)^2-4]',[4,-4])
%Exercise 8(1)
clear;
fun=inline('x.^2.*sin(x.^2-x-2)');
fplot(fun,[-2 2]);grid on; %作图观察
x(1)=-2;
x(3)=fminbnd(fun,-1,-0.5);
x(5)=fminbnd(fun,1,2);
fun2=inline('-x.^2.*sin(x.^2-x-2)');
x(2)=fminbnd(fun2,-2,-1);
x(4)=fminbnd(fun2,-0.5,0.5);
x(6)=2
feval(fun,x)
%答案: 以上x(1)(3)(5)是局部极小，x(2)(4)(6)是局部极大，从最后一句知道x(1)全局最小， x(2)最大。
%Exercise 8(2)
clear;
fun=inline('3*x.^5-20*x.^3+10');
fplot(fun,[-3 3]);grid on;%作图观察
x(1)=-3;
x(3)=fminsearch(fun,2.5);
fun2=inline('-(3*x.^5-20*x.^3+10)');
x(2)=fminsearch(fun2,-2.5);
x(4)=3;
feval(fun,x)
%Exercise 8(3)
fun=inline('abs(x^3-x^2-x-2)');
fplot(fun,[0 3]);grid on;%作图观察
fminbnd(fun,1.5,2.5)
fun2=inline('-abs(x^3-x^2-x-2)');
fminbnd(fun2,0.5,1.5)
%Exercise 9
close;
x=-2:0.1:1;y=-7:0.1:1;
[x,y]=meshgrid(x,y);
z=y.^3/9+3*x.^2.*y+9*x.^2+y.^2+x.*y+9;
mesh(x,y,z);grid on;%作图观察
fun=inline('x(2)^3/9+3*x(1)^2*x(2)+9*x(1)^2+x(2)^2+x(1)*x(2)+9');
x=fminsearch(fun,[0 0])%求极小值
fun2=inline('-(x(2)^3/9+3*x(1)^2*x(2)+9*x(1)^2+x(2)^2+x(1)*x(2)+9)');
x=fminsearch(fun2,[0 -5])%求极大值
%Exercise 10
clear;t=0:24;
c=[15 14 14 14 14 15 16 18 20 22 23 25 28 ...
31 32 31 29 27 25 24 22 20 18 17 16];
p2=polyfit(t,c,2)
p3=polyfit(t,c,3)
fun=inline('a(1)*exp(a(2)*(t-14).^2)','a','t');
a=lsqcurvefit(fun,[0 0],t,c)%初值可以试探
f=feval(fun, a,t)
norm(f-c)%拟合效果
plot(t,c,t,f) %作图检验
fun2=inline('b(1)*sin(pi/12*t+b(2))+20','b','t');%原题修改f(x)+20
b=lsqcurvefit(fun2,[0 0],t,c)
figure
f2=feval(fun2, b,t)
norm(f2-c)%拟合效果
plot(t,c,t,f2) %作图检验
%Exercise 11
fun=inline('(1-x)*sqrt(10.52+x)-3.06*x*sqrt(1+x)*sqrt(5)');
x=fzero(fun, 0, 1)
%Exercise 12
r=5.04/12/100;N=20*12;
x=7500*180 %房屋总价格
y=x*0.3 %首付款额
x0=x-y%贷款总额
a=(1+r)^N*r*x0/((1+r)^N-1)%月付还款额
r1=4.05/12/100;x1=10*10000;%公积金贷款
a1=(1+r1)^N*r1*x1/((1+r1)^N-1)
x2=x0-x1%商业贷款
a2=(1+r)^N*r*x2/((1+r)^N-1)
a=a1+a2
%Exercise 13
%列方程th*R^2+(pi-2*th)*r^2-R*r*sin(th)=pi*r^2/2
%化简得sin(2*th)-2*th*cos(2*th)=pi/2
%以下Matlab计算
clear;fun= inline('sin(2*th)-2*th*cos(2*th)-pi/2','th')
th=fsolve(fun,pi/4)
R=20*cos(th)
%Exercise 14
%先在Editor窗口写M函数保存
function x=secant(fname,x0,x1,e)
while abs(x0-x1)>e,
x=x1-(x1-x0)*feval(fname,x1)/(feval(fname,x1)-feval(fname,x0));
x0=x1;x1=x;
end
%再在指令窗口
fun=inline('x*log(sqrt(x^2-1)+x)-sqrt(x^2-1)-0.5*x');
secant(fun,1,2,1e-8)
%Exercise 15
%作系数为a,初值为xo,从第m步到第n步迭代过程的M函数：
function f=ex4_15fun(a,x0,m,n)
x(1)=x0; y(1)=a*x(1)+1;x(2)=y(1);
if m
for i=2:n
y(i)=a*x(i)+1; x(i+1)=y(i);
if i>m, plot([x(i),x(i),x(i+1)],[y(i-1),y(i),y(i)]); end
end
hold off;
%M脚本文件
subplot(2,2,1);ex4_15fun(0.9,1,1,20);
subplot(2,2,2);ex4_15fun(-0.9,1,1,20);
subplot(2,2,3);ex4_15fun(1.1,1,1,20);
subplot(2,2,4);ex4_15fun(-1.1,1,1,20);
%Exercise 16
%设夹角t, 问题转化为 min f=5/sin(t)+10/cos(t)
%取初始值pi/4, 计算如下
fun=@(t)5/sin(t)+10/cos(t);
[t,f]=fminsearch(fun, pi/4)
t =
0.6709
f =
20.8097
%Exercise 17
%提示：x(k+2)=f(x(k))=a^2*x(k)*(1-x(k))*(1-a*x(k)*(1-x(k)))
%计算平衡点x
%|f'(x)|
%Exercise 18
%先写M文件
function f=ex4_18(a,x0,n)
x=zeros(1,n);y=x;
x(1)=x0;
y(1)=a*x(1)+1;
x(2)=y(1);
plot([x(1),x(1),x(2)],[0,y(1),y(1)],'r');
hold on;
for i=2:n
y(i)=a*x(i)+1;
x(i+1)=y(i);
plot([x(i),x(i),x(i+1)],[y(i-1),y(i),y(i)])
end
hold off;
%再执行指令
>> ex4_18(0.9,1,20)
>> ex4_18(-0.9,1,20)
>> ex4_18(1.1,1,20)
>> ex4_18(-1.1,1,20)
%Exercise 19
clear; close; x(1)=0; y(1)=0;
for k=1:3000
x(k+1)=1+y(k)-1.4*x(k)^2; y(k+1)=0.3*x(k);
end
plot(x(1000:1500),y(1000:1500),'+g');hold on
plot(x(1501:2000),y(1501:2000),'.b');
plot(x(2001:2500),y(2001:2500),'*y');
plot(x(2501:3001),y(2501:3001),'.r');
%Exercise 1
x=[0 4 10 12 15 22 28 34 40];
y=[0 1 3 6 8 9 5 3 0];
trapz(x,y)
%Exercise 2
x=[0 4 10 12 15 22 28 34 40];
y=[0 1 3 6 8 9 5 3 0];
diff(y)./diff(x)
%Exercise 3
xa=-1:0.1:1;ya=0:0.1:2;
[x,y]=meshgrid(xa,ya);
z=x.*exp(-x.^2 -y.^3);
[px,py] = gradient(z,xa,ya);
px
%Exercise 4
t=0:0.01:1.5;
x=log(cos(t));
y=cos(t)-t.*sin(t);
dydx=gradient(y,x)
[x_1,id]=min(abs(x-(-1)));%找最接近x=-1的点
dydx(id)
%Exercise 5(2)
fun=inline('exp(2*x).*cos(x).^3');
quadl(fun,0,2*pi)
或用trapz
x=linspace(0,2*pi,100);
y=exp(2*x).*cos(x).^3;
trapz(x,y)
%Exercise 5(3)
fun=@(x)x.*log(x.^4).*asin(1./x.^2);
quadl(fun,1,3)
或用trapz
x=1:0.01:3;
y=feval(fun,x);
trapz(x,y)
%Exercise 5(4)
fun=@(x)sin(x)./x;
quadl(fun,1e-10,1) %注意由于下限为0，被积函数没有意义，用很小的1e-10代替 %Exercise 5(5)
%参考Exercise 5(4)
%Exercise 5(6)
fun=inline('sqrt(1+r.^2.*sin(th))','r','th');
dblquad(fun,0,1,0,2*pi)
%Exercise 5(7)
首先建立84页函数dblquad2
clear;
fun=@(x,y)1+x+y.^2;
clo=@(x)-sqrt(2*x-x.^2);
dup=@(x)sqrt(2*x-x.^2);
dblquad2(fun,0,2,clo,dhi,100)
%Exercise 6
t=linspace(0,2*pi,100);
x=2*cos(t);y=3*sin(t);
dx=gradient(x,t);dy=gradient(y,t);
f=sqrt(dx.^2+dy.^2);
trapz(t,f)
%Exercise 7
xa=-1:0.1:1;ya=0:0.1:2;
[x,y]=meshgrid(xa,ya);
z=x.*exp(x.^2+y.^2);
[zx,zy]=gradient(z,xa,ya);
f=sqrt(1+zx.^2+zy.^2);
s=0;
for i=2:length(xa)
for j=2:length(ya)
s=s+(xa(i)-xa(i-1))*(ya(j)-ya(j-1))*(f(i,j)+f(i-1,j)+f(i,j-1)+f(i-1,j-1))/4; end
end
s
%Exercise 8
funl=inline('-(-x).^0.2.*cos(x)');
funr=inline('x.^0.2.*cos(x)');
quadl(funl,-1,0)+quadl(funr,0,1)
%Exercise 9 (以I32为例)
fun=@(x)abs(sin(x));
h=0.1;x=0:h:32*pi;y=feval(fun,x);t1=trapz(x,y)
h=pi;x=0:h:32*pi;y=feval(fun,x);t2=trapz(x,y)%步长与周期一致，结果失真 q1=quad(fun,0,32*pi)
q2=quadl(fun,0,32*pi)
%Exercise 10(2)
%先在程序编辑器，写下列函数，保存为ex5_10_2f
function d=ex5_10_2f(fname,a,h0,e)
h=h0;d=(feval(fname,a+h)-2*feval(fname,a)+feval(fname,a-h))/(h*h); d0=d+2*e;
while abs(d-d0)>e
d0=d;h0=h;h=h0/2;
d=(feval(fname,a+h)-2*feval(fname,a)+feval(fname,a-h))/(h*h);
end
%再在指令窗口执行
fun=inline('x.^2*sin(x.^2-x-2)','x');
d=ex5_10_2f(fun,1.4,0.1,1e-3)
%Exercise 11
%提示：f上升时，f'>0;f下降时，f'
%Exercise 12
在程序编辑器，写下列函数，保存为ex5_12f
function I=ex5_12(fname,a,b,n)
x=linspace(a,b,n+1);
y=feval(fname,x);
I=(b-a)/n/3*(y(1)+y(n+1)+2*sum(y(3:2:n))+4*sum(y(2:2:n)));
%再在指令窗口执行
ex5_12(inline('1/sqrt(2*pi)*exp(-x.^2/2)'),0,1,50)
%Exercise 13
fun=inline('5400*v./(8.276*v.^2+2000)','v');
quadl(fun,15,30)
%Exercise 14
重心不超过凳边沿。1/2, 2/3, 3/4, ...,n/(n+1)
%Exercise15
利润函数fun=inline('(p-c0+k*log(M*exp(-a*p)))*M*exp(-a*p)','p');
求p使fun最大
%Exercise 16
clear; x=-3/4:0.01:3/4;
y=(3/4+x)*2.*sqrt(1-16/9.*x.^2)*9.8;
P=trapz(x,y) %单位：千牛
%Exercise 17
clear; close;
fplot('17-t^(2/3)-5-2*t^(2/3)',[0,20]); grid;
t=fzero('17-x^(2/3)-5-2*x^(2/3)',7)
t=0:0.1:8; y=17-t.^(2/3)-5-2*t.^(2/3);
trapz(t,y)-20 %单位：百万元
%Exercise 18
%曲面面积计算
%Excercise 1(1)
fun=inline('x+y','x','y');
[t,y]=ode45(fun,[0 1 2 3],1) %注意由于初值为y(0)=1,[0 1 2 3]中0不可缺
%Excercise 1(3)
%令y(1)=y,y(2)=y',化为方程组
%y(1)'=y(2),y(2)'=0.01*y(2)^2-2*y(1)+sin(t)
%运行下列指令
clear;close;
fun=@(t,y)[y(2);0.01*y(2)^2-2*y(1)+sin(t)];
[t,y]=ode45(fun,[0 5],[0;1]);
plot(t,y(:,1))
%Excercise 1(5)
%令y(1)=y,y(2)=y',化为方程组
%y(1)'=y(2),y(2)'=-mu*(y(1)^2-1)*y(2)-y(1)
%运行下列指令,注意参数mu的处理
clear;close;
fun=@(t,y,mu)[y(2);-mu*(y(1)^2-1)*y(2)-y(1)];
[t,y]=ode45(fun,[0 20],[2;0],[],1);
plot(y(:,1),y(:,2));hold on;
[t,y]=ode45(fun,[0 20],[2;0],[],2);
plot(y(:,1),y(:,2),'r');hold off;
%Excercise 2
roots([1 10 54 132 137 50])
%通解A1*exp(-3*t)*cos(4*t)+A2*exp(-3*t)*sin(4*t)+A3*exp(-2*t)+A4*exp(-t)+A5*t*exp(-t) %Excercise 3
dfun=inline('[-1000.25*y(1)+999.75*y(2)+0.5;999.75*y(1)-1000.25*y(2)+0.5]','x','y');
[x,y]=ode45(dfun,[0,50],[1;-1]);length(x)
%所用节点很多
[x,y]=ode15s(dfun,[0,50],[1;-1]);length(x)
%所用节点很少
%Excercise 4
clear;
dfun=inline('[x(2);2*x(3)+x(1)-((1-1/82.45)*(x(1)+1/82.45))/(sqrt((x(1)+1/82.45)^2+x(3)^2))^3-(1/82.45*(x(1)-1+1/82.45))/(sqrt((x(1)+1-1/82.45)^2+x(3)^2))^3;
x(4);-2*x(2)+x(3)-((1-1/82.45)*x(3))/(sqrt((x(1)+1/82.45)^2+x(3)^2))^3-(1/82.45*x(3))/(sqrt((x(1)+1-1/82.45)^2+x(3)^2))^3]','t','x');
[t,x]=ode45(dfun,[0 24],[1.2; 0; 0; -1.04935371]);
plot(x(:,1),x(:,3));
%Excercise 5
%方程y'=2x+y^2,y(0)=0
clear;close;
fun=inline('2*x+y^2','x','y');
[x,y]=ode45(fun,[0 1.57],0); %x的上界再增加,解会
plot(x,y)
%Excercise 6
clear;close;
fun=@(t,x,a,b)a*x+b;
[t,x]=ode45(fun,[0 10],0.1,[],1,1);
subplot(2,4,1);plot(t,x)
[t,x]=ode45(fun,[0 10],-0.1,[],1,1);
subplot(2,4,2);plot(t,x)
[t,x]=ode45(fun,[0 10],0.1,[],1,-1);
subplot(2,4,3);plot(t,x)
[t,x]=ode45(fun,[0 10],-0.1,[],1,-1);
subplot(2,4,4);plot(t,x)
[t,x]=ode45(fun,[0 10],0.1,[],-1,1);
subplot(2,4,5);plot(t,x)
[t,x]=ode45(fun,[0 10],-0.1,[],-1,1);
subplot(2,4,6);plot(t,x)
[t,x]=ode45(fun,[0 10],0.1,[],-1,-1);
subplot(2,4,7);plot(t,x)
[t,x]=ode45(fun,[0 10],-0.1,[],-1,-1);
subplot(2,4,8);plot(t,x)
%Excercise 7
%微分方程 T'=k(c-T),T(0)=20
dsolve('DT=k*(c-T)','T(0)=20','t')
%得c+exp(-k*t)*(-c+20)
%利用T(10)=25.2, T(20)=28.32拟合(或者解非线性方程)
fun=inline('c(1)+exp(-c(2)*t)*(-c(1)+20)','c','t')
lsqcurvefit(fun,[30 1],[10 20],[25.2 28.32])
%解得户外温度c=33,比例系数k=0.05.
%Excercise 8
%微分方程 x'/x=0.5*(1-x),x(0)=0.05
fun=inline('0.5*(1-x)*x','t','x');
[t,x]=ode45(fun,[0 10],0.05);
plot(t,x)
id=min(find(x>0.8));
t(id)
%Excercise 9
%微分方程组 V'(t)=K(t)*V(t)^a,K'(t)=-b*K(t)
%答案(1)exp(20);(2)0.353;(3)30;(4)451,0.4,9.6
%Chapter 7
%Exercise 1
syms ph th;
a=sin(ph)*cos(th)-cos(ph)*sin(th)-sin(ph-th);
simple(a)
%化简后差的结果为0
%Exercise 2
syms x;s=x^4-5*x^3+5*x^2+5*x-6;
factor(s)
%Exercise 3
syms a;A=[1 2;2 a];
iA=inv(A),[v,d]=eig(A)
%Exercise 4
syms x y;
limit((3^x+9^x)^(1/x),x,inf)
s1=limit(x*y/(sqrt(x*y+1)-1),x,0);s2=limit(s1,y,0)
%Exercise 5
syms k n x;s1=symsum(k^2,k,1,n);s1=simple(s1)
s2=symsum(k^(-2),k,1,inf);s2=simple(s2)
s3=symsum(1/(2*n+1)/(2*x+1)^(2*n+1),n,0,inf);s3=simple(s3) %Exercise 6
syms x y z;s=sin(x^2*y*z);
s=diff(s,x,2);
s=diff(s,y,1);
s=subs(s,{x,y,z},{1,1,3})
%Exercise 7
syms x;s=log(x+sqrt(1+x^2));taylor(s,8,0,x)
%Exercise 8 (以第四章习题9为例)
先用符号运算求偏导数
syms x y;f=y^3/9+3*x^2*y+9*x^2+y^2+x*y+9;
fx=diff(f,x),fy=diff(f,y)
根据计算结果得方程组. 求解方程组
[sx,sy]=solve(6*x*y+18*x+y,1/3*y^2+3*x^2+2*y+x,x,y)
得四个解(0,0),(-1/3,-6),(-7/6,-7/2),(5/6,-5/2).计算Hesse矩阵 fh2=[diff(fx,x),diff(fx,y);diff(fy,x),diff(fy,y)]
计算
eig(subs(fh2,[x,y],[0,0]))
得知正定，所以是极小值点.极小值用
subs(f,[x,y],[0,0])
求得。同理可得(-1/3,-6)为极大值点，其它两个为鞍点。 %Exercise 9(以第一小题为例)
syms y;f=exp(2*y)/(exp(y)+2);
fi=int(f,y)
s=simple(diff(fi)-f)
%Exercise 10
syms x y;f=(x-y)^3*sin(x+2*y);Ix=simple(int(f,y,-x,x))
%Exercise 11(3)
syms x;f=x*log(x^4)*asin(1/x);Ix=int(f,x,1,3);vpa(Ix)
%Exercise 12
%1(3)
syms x;solve(5*x^23-6*x^7+8*x^6-5*x^2)
%6
syms a b;s=solve(a-0.7*sin(a)-0.2*cos(b),b-0.7*cos(a)+0.2*sin(b)); s.a,s.b
%Exercise 13
%1(3)
dsolve('D2y-0.01*Dy^2+2*y=sin(t)','y(0)=0','Dy(0)=1','t')(解不出) %1(4)
dsolve('2*D2x-5*Dx+3*x=45*exp(2*t)','x(0)=2','Dx(0)=1','t') %Exercise 14
%6(ii)
ezplot('x^2/4+y^2/9=1')
%6(vi)
ezmesh('2*sin(ph)*cos(th)','2*sin(ph)*sin(th)','2*cos(ph)',[0 pi/2 0 2*pi])
%Page20,ex1
(5) 等于[exp(1),exp(2);exp(3),exp(4)]
(7) 3=1*3, 8=2*4
(8) a为各列最小值，b为最小值所在的行号
(10) 1>=4,false, 2>=3,false, 3>=2, ture, 4>=1,ture
(11) 答案表明：编址第2元素满足不等式(30>=20)和编址第4元素满足不等式(40>=10)
(12) 答案表明：编址第2行第1列元素满足不等式(30>=20)和编址第2行第2列元素满足不等式(40>=10)
%Page20, ex2
(1)a, b, c的值尽管都是1， 但数据类型分别为数值，字符， 逻辑， 注意a与c相等，他们不等于b
(2)double(fun)输出的分别是字符a,b,s,(,x,)的ASCII码
%Page20,ex3
>> r=2;p=0.5;n=12;
>> T=log(r)/n/log(1+0.01*p)
T =
11.5813
%Page20,ex4
>> x=-2:0.05:2;f=x.^4-2.^x;
>> [fmin,min_index]=min(f)
fmin =
-1.3907 %最小值
min_index =
54 %最小值点编址
>> x(min_index)
ans =
0.6500 %最小值点
>> [f1,x1_index]=min(abs(f)) %求近似根--绝对值最小的点
f1 =
0.0328
x1_index =
24
>> x(x1_index)
ans =
-0.8500
>> x(x1_index)=[];f=x.^4-2.^x; %删去绝对值最小的点以求函数绝对值次小的点 >> [f2,x2_index]=min(abs(f)) %求另一近似根--函数绝对值次小的点
f2 =
0.0630
x2_index =
65 但
>> x(x2_index)
ans =
1.2500
%Page20,ex5
>> z=magic(10)
z =
92 99 1 8 15 67 74 51 58 40
98 80 7 14 16 73 55 57 64 41
4 81 88 20 22 54 56 63 70 47
85 87 19 21 3 60 62 69 71 28
86 93 25 2 9 61 68 75 52 34
17 24 76 83 90 42 49 26 33 65
23 5 82 89 91 48 30 32 39 66
79 6 13 95 97 29 31 38 45 72
10 12 94 96 78 35 37 44 46 53
11 18 100 77 84 36 43 50 27 59
>> sum(z)
ans =
505 505 505 505 505 505 505 505 505 505
>> sum(diag(z))
ans =
505
>> z(:,2)/sqrt(3)
ans =
57.1577
46.1880
46.7654
50.2295
53.6936
13.8564
2.8868
3.4641
6.9282
10.3923
>> z(8,:)=z(8,:)+z(3,:)
z =
92 99 1 8 15 67 74 51 58 40
98 80 7 14 16 73 55 57 64 41
4 81 88 20 22 54 56 63 70 47
85 87 19 21 3 60 62 69 71 28
86 93 25 2 9 61 68 75 52 34
17 24 76 83 90 42 49 26 33 65
23 5 82 89 91 48 30 32 39 66
83 87 101 115 119 83 87 101 115 119
10 12 94 96 78 35 37 44 46 53
11 18 100 77 84 36 43 50 27 59
%Page 40 ex1
先在编辑器窗口写下列M函数，保存为eg2_1.m
function [xbar,s]=ex2_1(x)
n=length(x);
xbar=sum(x)/n;
s=sqrt((sum(x.^2)-n*xbar^2)/(n-1));
例如
>>x=[81 70 65 51 76 66 90 87 61 77];
>>[xbar,s]=ex2_1(x)
xbar =
72.4000
s =
12.1124
%Page 40 ex2
s=log(1);n=0;
while s
n=n+1;
s=s+log(1+n);
end
m=n
计算结果m=37
%Page 40 ex3
clear;
F(1)=1;F(2)=1;k=2;x=0;
e=1e-8; a=(1+sqrt(5))/2;
while abs(x-a)>e
k=k+1;F(k)=F(k-1)+F(k-2); x=F(k)/F(k-1);
end
a,x,k
计算至k=21可满足精度
%Page 40 ex4
clear;tic;s=0;
for i=1:1000000
s=s+sqrt(3)/2^i;
end
s,toc
tic;s=0;i=1;
while i
s=s+sqrt(3)/2^i;i=i+1;
end
s,toc
tic;s=0;
i=1:1000000;
s=sqrt(3)*sum(1./2.^i);
s,toc
%Page 40 ex5
t=0:24;
c=[15 14 14 14 14 15 16 18 20 22 23 25 28 ...
31 32 31 29 27 25 24 22 20 18 17 16];
plot(t,c)
%Page 40 ex6
%(1)
x=-2:0.1:2;y=x.^2.*sin(x.^2-x-2);plot(x,y)
y=inline('x^2*sin(x^2-x-2)');fplot(y,[-2 2])
%(2)参数方法
t=linspace(0,2*pi,100);
x=2*cos(t);y=3*sin(t); plot(x,y)
%(3)
x=-3:0.1:3;y=x;
[x,y]=meshgrid(x,y);
z=x.^2+y.^2;
surf(x,y,z)
%(4)
x=-3:0.1:3;y=-3:0.1:13;
[x,y]=meshgrid(x,y);
z=x.^4+3*x.^2+y.^2-2*x-2*y-2*x.^2.*y+6;
surf(x,y,z)
%(5)
t=0:0.01:2*pi;
x=sin(t);y=cos(t);z=cos(2*t);
plot3(x,y,z)
%(6)
theta=linspace(0,2*pi,50);fai=linspace(0,pi/2,20);
[theta,fai]=meshgrid(theta,fai);
x=2*sin(fai).*cos(theta);
y=2*sin(fai).*sin(theta);z=2*cos(fai);
surf(x,y,z)
%(7)
x=linspace(0,pi,100);
y1=sin(x);y2=sin(x).*sin(10*x);y3=-sin(x);
plot(x,y1,x,y2,x,y3)
%page41, ex7
x=-1.5:0.05:1.5;
y=1.1*(x>1.1)+x.*(x=-1.1)-1.1*(x
plot(x,y)
%page41,ex8
分别使用which trapz, type trapz, dir C:\MATLAB7\toolbox\matlab\datafun\ %page41,ex9
clear;close;
x=-2:0.1:2;y=x;
[x,y]=meshgrid(x,y);
a=0.5457;b=0.7575;
p=a*exp(-0.75*y.^2-3.75*x.^2-1.5*x).*(x+y>1);
p=p+b*exp(-y.^2-6*x.^2).*(x+y>-1).*(x+y
p=p+a*exp(-0.75*y.^2-3.75*x.^2+1.5*x).*(x+y
mesh(x,y,p)
%page41, ex10
lookfor lyapunov
help lyap
>> A=[1 2 3;4 5 6;7 8 0];C=[2 -5 -22;-5 -24 -56;-22 -56 -16];
>> X=lyap(A,C)
X =
1.0000 -1.0000 -0.0000
-1.0000 2.0000 1.0000
-0.0000 1.0000 7.0000
%Chapter 3
%Exercise 1
>> a=[1,2,3];b=[2,4,3];a./b,a.\b,a/b,a\b
ans =
0.5000 0.5000 1.0000
ans =
2 2 1
ans =
0.6552 %一元方程组x[2,4,3]=[1,2,3]的近似解
ans =
0 0 0
0 0 0
0.6667 1.3333 1.0000
%矩阵方程[1,2,3][x11,x12,x13;x21,x22,x23;x31,x32,x33]=[2,4,3]的特解 %Exercise 2(1)
>> A=[4 1 -1;3 2 -6;1 -5 3];b=[9;-2;1];
>> rank(A), rank([A,b]) %[A,b]为增广矩阵
ans =
3
ans =
3 %可见方程组唯一解
>> x=A\b
x =
2.3830
1.4894
2.0213
%Exercise 2(2)
>> A=[4 -3 3;3 2 -6;1 -5 3];b=[-1;-2;1];
>> rank(A), rank([A,b])
ans =
3
ans =
3 %可见方程组唯一解
>> x=A\b
x =
-0.4706
-0.2941
%Exercise 2(3)
>> A=[4 1;3 2;1 -5];b=[1;1;1];
>> rank(A), rank([A,b])
ans =
2
ans =
3 %可见方程组无解
>> x=A\b
x =
0.3311
-0.1219 %最小二乘近似解
%Exercise 2(4)
>> a=[2,1,-1,1;1,2,1,-1;1,1,2,1];b=[1 2 3]';%注意b的写法 >> rank(a),rank([a,b])
ans =
3
ans =
3 %rank(a)==rank([a,b])
>> a\b
ans =
1
1
0 %一个特解
%Exercise 3
>> a=[2,1,-1,1;1,2,1,-1;1,1,2,1];b=[1,2,3]';
>> x=null(a),x0=a\b
x =
-0.6255
0.6255
-0.2085
0.4170
x0 =
1
1
%通解kx+x0
%Exercise 4
>> x0=[0.2 0.8]';a=[0.99 0.05;0.01 0.95];
>> x1=a*x, x2=a^2*x, x10=a^10*x
>> x=x0;for i=1:1000,x=a*x;end,x
x =
0.8333
0.1667
>> x0=[0.8 0.2]';
>> x=x0;for i=1:1000,x=a*x;end,x
x =
0.8333
0.1667
>> [v,e]=eig(a)
v =
0.9806 -0.7071
0.1961 0.7071
e =
1.0000 0
0 0.9400
>> v(:,1)./x
ans =
1.1767
1.1767 %成比例，说明x是最大特征值对应的特征向量 %Exercise 5
%用到公式(3.11)(3.12)
>> B=[6,2,1;2.25,1,0.2;3,0.2,1.8];x=[25 5 20]';
>> C=B/diag(x)
C =
0.2400 0.4000 0.0500
0.1200 0.0400 0.0900
>> A=eye(3,3)-C
A =
0.7600 -0.4000 -0.0500
-0.0900 0.8000 -0.0100
-0.1200 -0.0400 0.9100
>> D=[17 17 17]';x=A\D
x =
37.5696
25.7862
24.7690
%Exercise 6(1)
>> a=[4 1 -1;3 2 -6;1 -5 3];det(a),inv(a),[v,d]=eig(a) ans =
-94
ans =
0.2553 -0.0213 0.0426
0.1596 -0.1383 -0.2234
0.1809 -0.2234 -0.0532
v =
0.0185 -0.9009 -0.3066
-0.7693 -0.1240 -0.7248
-0.6386 -0.4158 0.6170
d =
-3.0527 0 0
0 3.6760 0
0 0 8.3766
%Exercise 6(2)
>> a=[1 1 -1;0 2 -1;-1 2 0];det(a),inv(a),[v,d]=eig(a) ans =
1
ans =
2.0000 -2.0000 1.0000
1.0000 -1.0000 1.0000
2.0000 -3.0000 2.0000
v =
-0.5773 0.5774 + 0.0000i 0.5774 - 0.0000i
-0.5773 0.5774 0.5774
-0.5774 0.5773 - 0.0000i 0.5773 + 0.0000i
d =
1.0000 0 0
0 1.0000 + 0.0000i 0
%Exercise 6(3)
>> A=[5 7 6 5;7 10 8 7;6 8 10 9;5 7 9 10]
A =
5 7 6 5
7 10 8 7
6 8 10 9
5 7 9 10
>> det(A),inv(A), [v,d]=eig(A)
ans =
1
ans =
68.0000 -41.0000 -17.0000 10.0000
-41.0000 25.0000 10.0000 -6.0000
-17.0000 10.0000 5.0000 -3.0000
10.0000 -6.0000 -3.0000 2.0000
v =
0.8304 0.0933 0.3963 0.3803
-0.5016 -0.3017 0.6149 0.5286
-0.2086 0.7603 -0.2716 0.5520
0.1237 -0.5676 -0.6254 0.5209
d =
0.0102 0 0 0
0 0.8431 0 0
0 0 3.8581 0
0 0 0 30.2887
%Exercise 6(4)(以n=5为例)
%关键是矩阵的定义
%方法一（三个for）
n=5;
for i=1:n, a(i,i)=5;end
for i=1:(n-1),a(i,i+1)=6;end
for i=1:(n-1),a(i+1,i)=1;end
a
%方法二（一个for）
n=5;a=zeros(n,n);
a(1,1:2)=[5 6];
for i=2:(n-1),a(i,[i-1,i,i+1])=[1 5 6];end
a(n,[n-1 n])=[1 5];
a
%方法三（不用for）
n=5;a=diag(5*ones(n,1));
b=diag(6*ones(n-1,1));
c=diag(ones(n-1,1));
a=a+[zeros(n-1,1),b;zeros(1,n)]+[zeros(1,n);c,zeros(n-1,1)] %下列计算
>> det(a)
ans =
665
>> inv(a)
ans =
0.3173 -0.5865 1.0286 -1.6241 1.9489
-0.0977 0.4887 -0.8571 1.3534 -1.6241
0.0286 -0.1429 0.5429 -0.8571 1.0286
-0.0075 0.0376 -0.1429 0.4887 -0.5865
0.0015 -0.0075 0.0286 -0.0977 0.3173
>> [v,d]=eig(a)
v =
-0.7843 -0.7843 -0.9237 0.9860 -0.9237
0.5546 -0.5546 -0.3771 -0.0000 0.3771
-0.2614 -0.2614 0.0000 -0.1643 0.0000
0.0924 -0.0924 0.0628 -0.0000 -0.0628
-0.0218 -0.0218 0.0257 0.0274 0.0257
d =
0.7574 0 0 0 0
0 9.2426 0 0 0
0 0 7.4495 0 0
0 0 0 5.0000 0
0 0 0 0 2.5505
%Exercise 7(1)
>> a=[4 1 -1;3 2 -6;1 -5 3];[v,d]=eig(a)
v =
0.0185 -0.9009 -0.3066
-0.7693 -0.1240 -0.7248
-0.6386 -0.4158 0.6170
d =
-3.0527 0 0
0 3.6760 0
0 0 8.3766
>> det(v)
ans =
-0.9255 %v行列式正常, 特征向量线性相关，可对角化 >> inv(v)*a*v %验算
ans =
-3.0527 0.0000 -0.0000
0.0000 3.6760 -0.0000
-0.0000 -0.0000 8.3766
>> [v2,d2]=jordan(a) %也可用jordan
v2 =
0.0798 0.0076 0.9127
0.1886 -0.3141 0.1256
-0.1605 -0.2607 0.4213 %特征向量不同
d2 =
8.3766 0 0
0 -3.0527 - 0.0000i 0
0 0 3.6760 + 0.0000i
>> v2\a*v2
ans =
8.3766 0 0.0000
0.0000 -3.0527 0.0000
0.0000 0.0000 3.6760
>> v(:,1)./v2(:,2) %对应相同特征值的特征向量成比例
ans =
2.4491
2.4491
2.4491
%Exercise 7(2)
>> a=[1 1 -1;0 2 -1;-1 2 0];[v,d]=eig(a)
v =
-0.5773 0.5774 + 0.0000i 0.5774 - 0.0000i
-0.5773 0.5774 0.5774
-0.5774 0.5773 - 0.0000i 0.5773 + 0.0000i
d =
1.0000 0 0
0 1.0000 + 0.0000i 0
0 0 1.0000 - 0.0000i
>> det(v)
ans =
-5.0566e-028 -5.1918e-017i %v的行列式接近0, 特征向量线性相关，不可对角化 >> [v,d]=jordan(a)
v =
1 0 1
1 0 0
1 -1 0
d =
1 1 0
0 1 1
0 0 1 %jordan标准形不是对角的，所以不可对角化
%Exercise 7(3)
>> A=[5 7 6 5;7 10 8 7;6 8 10 9;5 7 9 10]
A =
5 7 6 5
7 10 8 7
6 8 10 9
5 7 9 10
>> [v,d]=eig(A)
v =
0.8304 0.0933 0.3963 0.3803
-0.5016 -0.3017 0.6149 0.5286
-0.2086 0.7603 -0.2716 0.5520
0.1237 -0.5676 -0.6254 0.5209
d =
0.0102 0 0 0
0 0.8431 0 0
0 0 3.8581 0
0 0 0 30.2887
>> inv(v)*A*v
ans =
0.0102 0.0000 -0.0000 0.0000
0.0000 0.8431 -0.0000 -0.0000
-0.0000 0.0000 3.8581 -0.0000
-0.0000 -0.0000 0 30.2887
%本题用jordan不行, 原因未知
%Exercise 7(4)参考6(4)和7(1), 略
%Exercise 8 只有(3)对称, 且特征值全部大于零, 所以是正定矩阵.
%Exercise 9(1)
>> a=[4 -3 1 3;2 -1 3 5;1 -1 -1 -1;3 -2 3 4;7 -6 -7 0]
>> rank(a)
ans =
3
>> rank(a(1:3,:))
ans =
2
>> rank(a([1 2 4],:)) %1,2,4行为最大无关组
ans =
3
>> b=a([1 2 4],:)';c=a([3 5],:)';
>> b\c %线性表示的系数
ans =
0.5000 5.0000
-0.5000 1.0000
0 -5.0000
%Exercise 10
>> a=[1 -2 2;-2 -2 4;2 4 -2]
>> [v,d]=eig(a)
v =
0.3333 0.9339 -0.1293
0.6667 -0.3304 -0.6681
-0.6667 0.1365 -0.7327
d =
-7.0000 0 0
0 2.0000 0
0 0 2.0000
>> v'*v
ans =
1.0000 0.0000 0.0000
0.0000 1.0000 0
0.0000 0 1.0000 %v确实是正交矩阵
%Exercise 11
%设经过6个电阻的电流分别为i1, ..., i6. 列方程组如下
%20-2i1=a; 5-3i2=c; a-3i3=c; a-4i4=b; c-5i5=b; b-3i6=0;
%i1=i3+i4;i5=i2+i3;i6=i4+i5;
%计算如下
>> A=[1 0 0 2 0 0 0 0 0;0 0 1 0 3 0 0 0 0;1 0 -1 0 0 -3 0 0 0; 1 -1 0 0 0 0 -4 0 0;
0 -1 1 0 0 0 0 -5 0;0 1 0 0 0 0 0 0 -3; 0 0 0 1 0 -1 -1 0 0;0 0 0 0 -1 -1 0 1 0;
0 0 0 0 0 0 -1 -1 1];
>>b=[20 5 0 0 0 0 0 0 0]'; A\b
ans =
13.3453
6.4401
8.5420
3.3274
-1.1807
1.6011
1.7263
0.4204
2.1467
%Exercise 12
>> A=[1 2 3;4 5 6;7 8 0];
>> left=sum(eig(A)), right=sum(trace(A))
left =
6.0000
right =
6
>> left=prod(eig(A)), right=det(A) %原题有错, (-1)^n应删去
left =
27.0000
right =
27
>> fA=(A-p(1)*eye(3,3))*(A-p(2)*eye(3,3))*(A-p(3)*eye(3,3))
fA =
1.0e-012 *
0.0853 0.1421 0.0284
0.1421 0.1421 0
-0.0568 -0.1137 0.1705
>> norm(fA) %f(A)范数接近0
ans =
2.9536e-013
%Exercise 1(1)
roots([1 1 1])
%Exercise 1(2)
roots([3 0 -4 0 2 -1])
%Exercise 1(3)
p=zeros(1,24);
p([1 17 18 22])=[5 -6 8 -5];
roots(p)
%Exercise 1(4)
p1=[2 3];
p2=conv(p1, p1);
p3=conv(p1, p2);
p3(end)=p3(end)-4; %原p3最后一个分量-4
roots(p3)
%Exercise 2
fun=inline('x*log(sqrt(x^2-1)+x)-sqrt(x^2-1)-0.5*x');
fzero(fun,2)
%Exercise 3
fun=inline('x^4-2^x');
fplot(fun,[-2 2]);grid on;
fzero(fun,-1),fzero(fun,1),fminbnd(fun,0.5,1.5)
%Exercise 4
fun=inline('x*sin(1/x)','x');
fplot(fun, [-0.1 0.1]);
x=zeros(1,10);for i=1:10, x(i)=fzero(fun,(i-0.5)*0.01);end;
x=[x,-x]
%Exercise 5
fun=inline('[9*x(1)^2+36*x(2)^2+4*x(3)^2-36;x(1)^2-2*x(2)^2-20*x(3);16*x(1)-x(1)^3-2*x(2)^2-16*x(3)^2]','x');
[a,b,c]=fsolve(fun,[0 0 0])
%Exercise 6
fun=@(x)[x(1)-0.7*sin(x(1))-0.2*cos(x(2)),x(2)-0.7*cos(x(1))+0.2*sin(x(2))];
[a,b,c]=fsolve(fun,[0.5 0.5])
%Exercise 7
clear; close; t=0:pi/100:2*pi;
x1=2+sqrt(5)*cos(t); y1=3-2*x1+sqrt(5)*sin(t);
x2=3+sqrt(2)*cos(t); y2=6*sin(t);
plot(x1,y1,x2,y2); grid on; %作图发现4个解的大致位置，然后分别求解
y1=fsolve('[(x(1)-2)^2+(x(2)-3+2*x(1))^2-5,2*(x(1)-3)^2+(x(2)/3)^2-4]',[1.5,2])
y2=fsolve('[(x(1)-2)^2+(x(2)-3+2*x(1))^2-5,2*(x(1)-3)^2+(x(2)/3)^2-4]',[1.8,-2])
y3=fsolve('[(x(1)-2)^2+(x(2)-3+2*x(1))^2-5,2*(x(1)-3)^2+(x(2)/3)^2-4]',[3.5,-5])
y4=fsolve('[(x(1)-2)^2+(x(2)-3+2*x(1))^2-5,2*(x(1)-3)^2+(x(2)/3)^2-4]',[4,-4])
%Exercise 8(1)
clear;
fun=inline('x.^2.*sin(x.^2-x-2)');
fplot(fun,[-2 2]);grid on; %作图观察
x(1)=-2;
x(3)=fminbnd(fun,-1,-0.5);
x(5)=fminbnd(fun,1,2);
fun2=inline('-x.^2.*sin(x.^2-x-2)');
x(2)=fminbnd(fun2,-2,-1);
x(4)=fminbnd(fun2,-0.5,0.5);
x(6)=2
feval(fun,x)
%答案: 以上x(1)(3)(5)是局部极小，x(2)(4)(6)是局部极大，从最后一句知道x(1)全局最小， x(2)最大。
%Exercise 8(2)
clear;
fun=inline('3*x.^5-20*x.^3+10');
fplot(fun,[-3 3]);grid on;%作图观察
x(1)=-3;
x(3)=fminsearch(fun,2.5);
fun2=inline('-(3*x.^5-20*x.^3+10)');
x(2)=fminsearch(fun2,-2.5);
x(4)=3;
feval(fun,x)
%Exercise 8(3)
fun=inline('abs(x^3-x^2-x-2)');
fplot(fun,[0 3]);grid on;%作图观察
fminbnd(fun,1.5,2.5)
fun2=inline('-abs(x^3-x^2-x-2)');
fminbnd(fun2,0.5,1.5)
%Exercise 9
close;
x=-2:0.1:1;y=-7:0.1:1;
[x,y]=meshgrid(x,y);
z=y.^3/9+3*x.^2.*y+9*x.^2+y.^2+x.*y+9;
mesh(x,y,z);grid on;%作图观察
fun=inline('x(2)^3/9+3*x(1)^2*x(2)+9*x(1)^2+x(2)^2+x(1)*x(2)+9');
x=fminsearch(fun,[0 0])%求极小值
fun2=inline('-(x(2)^3/9+3*x(1)^2*x(2)+9*x(1)^2+x(2)^2+x(1)*x(2)+9)');
x=fminsearch(fun2,[0 -5])%求极大值
%Exercise 10
clear;t=0:24;
c=[15 14 14 14 14 15 16 18 20 22 23 25 28 ...
31 32 31 29 27 25 24 22 20 18 17 16];
p2=polyfit(t,c,2)
p3=polyfit(t,c,3)
fun=inline('a(1)*exp(a(2)*(t-14).^2)','a','t');
a=lsqcurvefit(fun,[0 0],t,c)%初值可以试探
f=feval(fun, a,t)
norm(f-c)%拟合效果
plot(t,c,t,f) %作图检验
fun2=inline('b(1)*sin(pi/12*t+b(2))+20','b','t');%原题修改f(x)+20
b=lsqcurvefit(fun2,[0 0],t,c)
figure
f2=feval(fun2, b,t)
norm(f2-c)%拟合效果
plot(t,c,t,f2) %作图检验
%Exercise 11
fun=inline('(1-x)*sqrt(10.52+x)-3.06*x*sqrt(1+x)*sqrt(5)');
x=fzero(fun, 0, 1)
%Exercise 12
r=5.04/12/100;N=20*12;
x=7500*180 %房屋总价格
y=x*0.3 %首付款额
x0=x-y%贷款总额
a=(1+r)^N*r*x0/((1+r)^N-1)%月付还款额
r1=4.05/12/100;x1=10*10000;%公积金贷款
a1=(1+r1)^N*r1*x1/((1+r1)^N-1)
x2=x0-x1%商业贷款
a2=(1+r)^N*r*x2/((1+r)^N-1)
a=a1+a2
%Exercise 13
%列方程th*R^2+(pi-2*th)*r^2-R*r*sin(th)=pi*r^2/2
%化简得sin(2*th)-2*th*cos(2*th)=pi/2
%以下Matlab计算
clear;fun= inline('sin(2*th)-2*th*cos(2*th)-pi/2','th')
th=fsolve(fun,pi/4)
R=20*cos(th)
%Exercise 14
%先在Editor窗口写M函数保存
function x=secant(fname,x0,x1,e)
while abs(x0-x1)>e,
x=x1-(x1-x0)*feval(fname,x1)/(feval(fname,x1)-feval(fname,x0));
x0=x1;x1=x;
end
%再在指令窗口
fun=inline('x*log(sqrt(x^2-1)+x)-sqrt(x^2-1)-0.5*x');
secant(fun,1,2,1e-8)
%Exercise 15
%作系数为a,初值为xo,从第m步到第n步迭代过程的M函数：
function f=ex4_15fun(a,x0,m,n)
x(1)=x0; y(1)=a*x(1)+1;x(2)=y(1);
if m
for i=2:n
y(i)=a*x(i)+1; x(i+1)=y(i);
if i>m, plot([x(i),x(i),x(i+1)],[y(i-1),y(i),y(i)]); end
end
hold off;
%M脚本文件
subplot(2,2,1);ex4_15fun(0.9,1,1,20);
subplot(2,2,2);ex4_15fun(-0.9,1,1,20);
subplot(2,2,3);ex4_15fun(1.1,1,1,20);
subplot(2,2,4);ex4_15fun(-1.1,1,1,20);
%Exercise 16
%设夹角t, 问题转化为 min f=5/sin(t)+10/cos(t)
%取初始值pi/4, 计算如下
fun=@(t)5/sin(t)+10/cos(t);
[t,f]=fminsearch(fun, pi/4)
t =
0.6709
f =
20.8097
%Exercise 17
%提示：x(k+2)=f(x(k))=a^2*x(k)*(1-x(k))*(1-a*x(k)*(1-x(k)))
%计算平衡点x
%|f'(x)|
%Exercise 18
%先写M文件
function f=ex4_18(a,x0,n)
x=zeros(1,n);y=x;
x(1)=x0;
y(1)=a*x(1)+1;
x(2)=y(1);
plot([x(1),x(1),x(2)],[0,y(1),y(1)],'r');
hold on;
for i=2:n
y(i)=a*x(i)+1;
x(i+1)=y(i);
plot([x(i),x(i),x(i+1)],[y(i-1),y(i),y(i)])
end
hold off;
%再执行指令
>> ex4_18(0.9,1,20)
>> ex4_18(-0.9,1,20)
>> ex4_18(1.1,1,20)
>> ex4_18(-1.1,1,20)
%Exercise 19
clear; close; x(1)=0; y(1)=0;
for k=1:3000
x(k+1)=1+y(k)-1.4*x(k)^2; y(k+1)=0.3*x(k);
end
plot(x(1000:1500),y(1000:1500),'+g');hold on
plot(x(1501:2000),y(1501:2000),'.b');
plot(x(2001:2500),y(2001:2500),'*y');
plot(x(2501:3001),y(2501:3001),'.r');
%Exercise 1
x=[0 4 10 12 15 22 28 34 40];
y=[0 1 3 6 8 9 5 3 0];
trapz(x,y)
%Exercise 2
x=[0 4 10 12 15 22 28 34 40];
y=[0 1 3 6 8 9 5 3 0];
diff(y)./diff(x)
%Exercise 3
xa=-1:0.1:1;ya=0:0.1:2;
[x,y]=meshgrid(xa,ya);
z=x.*exp(-x.^2 -y.^3);
[px,py] = gradient(z,xa,ya);
px
%Exercise 4
t=0:0.01:1.5;
x=log(cos(t));
y=cos(t)-t.*sin(t);
dydx=gradient(y,x)
[x_1,id]=min(abs(x-(-1)));%找最接近x=-1的点
dydx(id)
%Exercise 5(2)
fun=inline('exp(2*x).*cos(x).^3');
quadl(fun,0,2*pi)
或用trapz
x=linspace(0,2*pi,100);
y=exp(2*x).*cos(x).^3;
trapz(x,y)
%Exercise 5(3)
fun=@(x)x.*log(x.^4).*asin(1./x.^2);
quadl(fun,1,3)
或用trapz
x=1:0.01:3;
y=feval(fun,x);
trapz(x,y)
%Exercise 5(4)
fun=@(x)sin(x)./x;
quadl(fun,1e-10,1) %注意由于下限为0，被积函数没有意义，用很小的1e-10代替 %Exercise 5(5)
%参考Exercise 5(4)
%Exercise 5(6)
fun=inline('sqrt(1+r.^2.*sin(th))','r','th');
dblquad(fun,0,1,0,2*pi)
%Exercise 5(7)
首先建立84页函数dblquad2
clear;
fun=@(x,y)1+x+y.^2;
clo=@(x)-sqrt(2*x-x.^2);
dup=@(x)sqrt(2*x-x.^2);
dblquad2(fun,0,2,clo,dhi,100)
%Exercise 6
t=linspace(0,2*pi,100);
x=2*cos(t);y=3*sin(t);
dx=gradient(x,t);dy=gradient(y,t);
f=sqrt(dx.^2+dy.^2);
trapz(t,f)
%Exercise 7
xa=-1:0.1:1;ya=0:0.1:2;
[x,y]=meshgrid(xa,ya);
z=x.*exp(x.^2+y.^2);
[zx,zy]=gradient(z,xa,ya);
f=sqrt(1+zx.^2+zy.^2);
s=0;
for i=2:length(xa)
for j=2:length(ya)
s=s+(xa(i)-xa(i-1))*(ya(j)-ya(j-1))*(f(i,j)+f(i-1,j)+f(i,j-1)+f(i-1,j-1))/4; end
end
s
%Exercise 8
funl=inline('-(-x).^0.2.*cos(x)');
funr=inline('x.^0.2.*cos(x)');
quadl(funl,-1,0)+quadl(funr,0,1)
%Exercise 9 (以I32为例)
fun=@(x)abs(sin(x));
h=0.1;x=0:h:32*pi;y=feval(fun,x);t1=trapz(x,y)
h=pi;x=0:h:32*pi;y=feval(fun,x);t2=trapz(x,y)%步长与周期一致，结果失真 q1=quad(fun,0,32*pi)
q2=quadl(fun,0,32*pi)
%Exercise 10(2)
%先在程序编辑器，写下列函数，保存为ex5_10_2f
function d=ex5_10_2f(fname,a,h0,e)
h=h0;d=(feval(fname,a+h)-2*feval(fname,a)+feval(fname,a-h))/(h*h); d0=d+2*e;
while abs(d-d0)>e
d0=d;h0=h;h=h0/2;
d=(feval(fname,a+h)-2*feval(fname,a)+feval(fname,a-h))/(h*h);
end
%再在指令窗口执行
fun=inline('x.^2*sin(x.^2-x-2)','x');
d=ex5_10_2f(fun,1.4,0.1,1e-3)
%Exercise 11
%提示：f上升时，f'>0;f下降时，f'
%Exercise 12
在程序编辑器，写下列函数，保存为ex5_12f
function I=ex5_12(fname,a,b,n)
x=linspace(a,b,n+1);
y=feval(fname,x);
I=(b-a)/n/3*(y(1)+y(n+1)+2*sum(y(3:2:n))+4*sum(y(2:2:n)));
%再在指令窗口执行
ex5_12(inline('1/sqrt(2*pi)*exp(-x.^2/2)'),0,1,50)
%Exercise 13
fun=inline('5400*v./(8.276*v.^2+2000)','v');
quadl(fun,15,30)
%Exercise 14
重心不超过凳边沿。1/2, 2/3, 3/4, ...,n/(n+1)
%Exercise15
利润函数fun=inline('(p-c0+k*log(M*exp(-a*p)))*M*exp(-a*p)','p');
求p使fun最大
%Exercise 16
clear; x=-3/4:0.01:3/4;
y=(3/4+x)*2.*sqrt(1-16/9.*x.^2)*9.8;
P=trapz(x,y) %单位：千牛
%Exercise 17
clear; close;
fplot('17-t^(2/3)-5-2*t^(2/3)',[0,20]); grid;
t=fzero('17-x^(2/3)-5-2*x^(2/3)',7)
t=0:0.1:8; y=17-t.^(2/3)-5-2*t.^(2/3);
trapz(t,y)-20 %单位：百万元
%Exercise 18
%曲面面积计算
%Excercise 1(1)
fun=inline('x+y','x','y');
[t,y]=ode45(fun,[0 1 2 3],1) %注意由于初值为y(0)=1,[0 1 2 3]中0不可缺
%Excercise 1(3)
%令y(1)=y,y(2)=y',化为方程组
%y(1)'=y(2),y(2)'=0.01*y(2)^2-2*y(1)+sin(t)
%运行下列指令
clear;close;
fun=@(t,y)[y(2);0.01*y(2)^2-2*y(1)+sin(t)];
[t,y]=ode45(fun,[0 5],[0;1]);
plot(t,y(:,1))
%Excercise 1(5)
%令y(1)=y,y(2)=y',化为方程组
%y(1)'=y(2),y(2)'=-mu*(y(1)^2-1)*y(2)-y(1)
%运行下列指令,注意参数mu的处理
clear;close;
fun=@(t,y,mu)[y(2);-mu*(y(1)^2-1)*y(2)-y(1)];
[t,y]=ode45(fun,[0 20],[2;0],[],1);
plot(y(:,1),y(:,2));hold on;
[t,y]=ode45(fun,[0 20],[2;0],[],2);
plot(y(:,1),y(:,2),'r');hold off;
%Excercise 2
roots([1 10 54 132 137 50])
%通解A1*exp(-3*t)*cos(4*t)+A2*exp(-3*t)*sin(4*t)+A3*exp(-2*t)+A4*exp(-t)+A5*t*exp(-t) %Excercise 3
dfun=inline('[-1000.25*y(1)+999.75*y(2)+0.5;999.75*y(1)-1000.25*y(2)+0.5]','x','y');
[x,y]=ode45(dfun,[0,50],[1;-1]);length(x)
%所用节点很多
[x,y]=ode15s(dfun,[0,50],[1;-1]);length(x)
%所用节点很少
%Excercise 4
clear;
dfun=inline('[x(2);2*x(3)+x(1)-((1-1/82.45)*(x(1)+1/82.45))/(sqrt((x(1)+1/82.45)^2+x(3)^2))^3-(1/82.45*(x(1)-1+1/82.45))/(sqrt((x(1)+1-1/82.45)^2+x(3)^2))^3;
x(4);-2*x(2)+x(3)-((1-1/82.45)*x(3))/(sqrt((x(1)+1/82.45)^2+x(3)^2))^3-(1/82.45*x(3))/(sqrt((x(1)+1-1/82.45)^2+x(3)^2))^3]','t','x');
[t,x]=ode45(dfun,[0 24],[1.2; 0; 0; -1.04935371]);
plot(x(:,1),x(:,3));
%Excercise 5
%方程y'=2x+y^2,y(0)=0
clear;close;
fun=inline('2*x+y^2','x','y');
[x,y]=ode45(fun,[0 1.57],0); %x的上界再增加,解会
plot(x,y)
%Excercise 6
clear;close;
fun=@(t,x,a,b)a*x+b;
[t,x]=ode45(fun,[0 10],0.1,[],1,1);
subplot(2,4,1);plot(t,x)
[t,x]=ode45(fun,[0 10],-0.1,[],1,1);
subplot(2,4,2);plot(t,x)
[t,x]=ode45(fun,[0 10],0.1,[],1,-1);
subplot(2,4,3);plot(t,x)
[t,x]=ode45(fun,[0 10],-0.1,[],1,-1);
subplot(2,4,4);plot(t,x)
[t,x]=ode45(fun,[0 10],0.1,[],-1,1);
subplot(2,4,5);plot(t,x)
[t,x]=ode45(fun,[0 10],-0.1,[],-1,1);
subplot(2,4,6);plot(t,x)
[t,x]=ode45(fun,[0 10],0.1,[],-1,-1);
subplot(2,4,7);plot(t,x)
[t,x]=ode45(fun,[0 10],-0.1,[],-1,-1);
subplot(2,4,8);plot(t,x)
%Excercise 7
%微分方程 T'=k(c-T),T(0)=20
dsolve('DT=k*(c-T)','T(0)=20','t')
%得c+exp(-k*t)*(-c+20)
%利用T(10)=25.2, T(20)=28.32拟合(或者解非线性方程)
fun=inline('c(1)+exp(-c(2)*t)*(-c(1)+20)','c','t')
lsqcurvefit(fun,[30 1],[10 20],[25.2 28.32])
%解得户外温度c=33,比例系数k=0.05.
%Excercise 8
%微分方程 x'/x=0.5*(1-x),x(0)=0.05
fun=inline('0.5*(1-x)*x','t','x');
[t,x]=ode45(fun,[0 10],0.05);
plot(t,x)
id=min(find(x>0.8));
t(id)
%Excercise 9
%微分方程组 V'(t)=K(t)*V(t)^a,K'(t)=-b*K(t)
%答案(1)exp(20);(2)0.353;(3)30;(4)451,0.4,9.6
%Chapter 7
%Exercise 1
syms ph th;
a=sin(ph)*cos(th)-cos(ph)*sin(th)-sin(ph-th);
simple(a)
%化简后差的结果为0
%Exercise 2
syms x;s=x^4-5*x^3+5*x^2+5*x-6;
factor(s)
%Exercise 3
syms a;A=[1 2;2 a];
iA=inv(A),[v,d]=eig(A)
%Exercise 4
syms x y;
limit((3^x+9^x)^(1/x),x,inf)
s1=limit(x*y/(sqrt(x*y+1)-1),x,0);s2=limit(s1,y,0)
%Exercise 5
syms k n x;s1=symsum(k^2,k,1,n);s1=simple(s1)
s2=symsum(k^(-2),k,1,inf);s2=simple(s2)
s3=symsum(1/(2*n+1)/(2*x+1)^(2*n+1),n,0,inf);s3=simple(s3) %Exercise 6
syms x y z;s=sin(x^2*y*z);
s=diff(s,x,2);
s=diff(s,y,1);
s=subs(s,{x,y,z},{1,1,3})
%Exercise 7
syms x;s=log(x+sqrt(1+x^2));taylor(s,8,0,x)
%Exercise 8 (以第四章习题9为例)
先用符号运算求偏导数
syms x y;f=y^3/9+3*x^2*y+9*x^2+y^2+x*y+9;
fx=diff(f,x),fy=diff(f,y)
根据计算结果得方程组. 求解方程组
[sx,sy]=solve(6*x*y+18*x+y,1/3*y^2+3*x^2+2*y+x,x,y)
得四个解(0,0),(-1/3,-6),(-7/6,-7/2),(5/6,-5/2).计算Hesse矩阵 fh2=[diff(fx,x),diff(fx,y);diff(fy,x),diff(fy,y)]
计算
eig(subs(fh2,[x,y],[0,0]))
得知正定，所以是极小值点.极小值用
subs(f,[x,y],[0,0])
求得。同理可得(-1/3,-6)为极大值点，其它两个为鞍点。 %Exercise 9(以第一小题为例)
syms y;f=exp(2*y)/(exp(y)+2);
fi=int(f,y)
s=simple(diff(fi)-f)
%Exercise 10
syms x y;f=(x-y)^3*sin(x+2*y);Ix=simple(int(f,y,-x,x))
%Exercise 11(3)
syms x;f=x*log(x^4)*asin(1/x);Ix=int(f,x,1,3);vpa(Ix)
%Exercise 12
%1(3)
syms x;solve(5*x^23-6*x^7+8*x^6-5*x^2)
%6
syms a b;s=solve(a-0.7*sin(a)-0.2*cos(b),b-0.7*cos(a)+0.2*sin(b)); s.a,s.b
%Exercise 13
%1(3)
dsolve('D2y-0.01*Dy^2+2*y=sin(t)','y(0)=0','Dy(0)=1','t')(解不出) %1(4)
dsolve('2*D2x-5*Dx+3*x=45*exp(2*t)','x(0)=2','Dx(0)=1','t') %Exercise 14
%6(ii)
ezplot('x^2/4+y^2/9=1')
%6(vi)
ezmesh('2*sin(ph)*cos(th)','2*sin(ph)*sin(th)','2*cos(ph)',[0 pi/2 0 2*pi])