MATLAB数学实验课后答案

MATLAB数学实验课后答案

数学实验MATLAB参考答案,重要部分,

P20,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)

P20, 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～ 但他们不等于b

(2)double(fun)输出的分别是字符a,b,s,(,x,)的ASCII码 P20,ex3

>> r=2;p=0.5;n=12;

>> T=log(r)/n/log(1+0.01*p) T =

11.5813

P20,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

>> x(x2_index)

ans =

1.2500

P20,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

P 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

P 40 ex2

s=log(1);n=0;

while s<=100

n=n+1;

s=s+log(1+n);

end

m=n

计算结果m=37

P 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

P 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<=1000000

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

P 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)

P 40 ex6

(1)

clear;

fplot('x^2*sin(x^2-x-2)',[-2,2])

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).*(x>=-1.1)-1.1*(x<-1.1); plot(x,y)

page41,ex8

分别使用which trapz, type trapz, dir

C:MATLAB7toolboxmatlabdatafun

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<=1); p=p+a*exp(-0.75*y.^2-3.75*x.^2+1.5*x).*(x+y<=-1); 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,ab

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=Ab

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=Ab x =

-0.4706

-0.2941

0

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=Ab

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])<4说明有无穷多解 >> ab

ans =

1

0

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=ab

x =

-0.6255 0.6255 -0.2085 0.4170 x0 =

1

0

1

0

%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

0.0900 0.2000 0.0100 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=AD 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 0 0 1.0000 - 0.0000i %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 >> v2a*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],:)';

>> bc %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]'; Ab 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个解的大致位臵～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');

-3 3]);grid on;%作图观察 fplot(fun,[

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<2, plot([x(1),x(1),x(2)],[0,y(1),y(1)]);hold on; end

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)|<1则稳定

%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'<0; f极值～ f'=0.

%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)+A

5*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)

[x,y]=ode15s(dfun,[0,50],[1;-1]);length(x)

%所用节点很多

%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"爆炸"

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])