例1-1 >> r=2;V=4/3*pi*r^3 V =
33.5103 例2-1 计算s=...>> s=0; >> for n=1:100 s=s+1/n/n; end >> s s =
1.6350 例2-5 两个一元函数y=x3-x-1,y=\\x\\0.2sin(5x)在区间-1
y=abs(x).^0.2.*sin(5*x); plot(x,y,\':ro\'); hold off;
曲面图 >> xa=6:8;ya=1:4; >> [x,y]=meshgrid(xa,ya); >> z=x.^2+y.^2; >> mesh(x,y,z) >> [x,y,z] ans =
37
50
65
40
53
68
45
58
73
52
65
80
例2-6 二元函数图z=xexp(-x2-y2).xa=-2:0.2:2;ya=xa;
[x,y]=meshgrid(xa,ya); z=x.*exp(-x.^2-y.^2); mesh(x,y,z);pause; surf(x,y,z);pause;
contour(x,y,z,[0.1,0.1]);pause mesh(x,y,z);
Page40 1.先在编辑器窗口写下列M函数，保存为ex2_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));
>> x=[81 70 65 51 76 66 90 87 61 77]; >> [xbar,s]=ex2_1(x) xbar =
72.4000 s =
12.1124 2.s=log(1);n=0; while s
s=s+log(1+n); end
m=n 3.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 m =
37 a =
1.6180 x =
1.6180 k =
21 4.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
s =
1.7321 Elapsed time is 2.038973 seconds.s =
1.7321 Elapsed time is 2.948968 seconds.s =
1.7321 Elapsed time is 0.453414 seconds 5.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)
6.(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) 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)
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
Page59 1.>> a=[1 2 3];b=[2 4 3]; >> a./b ans =
0.5000
0.5000
1.0000 >> a.\\b ans =
1 >> a/b ans =
0.6552 >> a\\b ans =
0
0
0
0
0
0
0.6667
1.3333
1.0000 2.(1) >> a=[4 1 -1;3 2 -6;1 -5 3];b=[9;-2;1]; >> a\\b ans =
2.3830
1.4894 2.0213 (2) >> a=[4 -3 3;3 2 -6;1 -5 3],b=[-1;-2;1] a =
-3
-6
-5
3 b =
-1
-2
1 >> a\\b ans =
-0.4706
-0.2941
0 (3) >> a=[4 1;3 2;1 -5],b=[1;1;1] a =
-5 b =
1 >> a\\b ans =
0.3311
-0.1219 (4) >> a=[2 1 -1 1;1 2 1 -1;1 1 2 1],b=[1;2;3] a =
-1
-1
1 b =
3 >> a\\b ans =
0
0 6.(1) >> a=[4 1 -1;3 2 -6;1 -5 3]; >> b=det(a),inv(a),[V,D]=eig(a) b =
-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 (2) >> a=[1 1 -1;0 2 -1;-1 2 0];b=det(a),inv(a),[V,D]=eig(a) b =
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
0.57740.0000i
0.5773 + 0.0000i D =
1.0000
0
0
0
1.0000 + 0.0000i
0
0
0
1.00000.0000i
-0.5773
0.5774
0.5774
-0.5774
0.57730.0000i >> 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 (3) >> 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 8
对称阵A为正定的充分必要条件是：A的特征值全为正。 只有(3)对称, 且特征值全部大于零, 所以(3)是正定矩阵. 例4.2用2次多项式拟合下列数据。 >> clear;x=[0.1,0.2,0.15,0,-0.2,0.3]; >> y=[0.95,0.84,0.86,1.06,1.50,0.72]; >> p=polyfit(x,y,2) p =
1.7432
-1.6959
1.0850 得到二次拟合式：1.7432x^2-1.6959x+1.0850 >> xi=-0.2:0.01:0.3; >> yi=polyval(p,xi);plot(x,y,\'o\',xi,yi);
例4.3 求函数y=x*sin(x^2-x-1)在（-2，-0.1）内的零点。 >> fun=inline(\'x*sin(x^2-x-1)\',\'x\') fun =
Inline function:
fun(x) = x*sin(x^2-x-1) >> fzero(fun,[-2,-0.1]) ??? Error using ==> fzero at 292 The function values at the interval endpoints must differ in sign.>> fplot(fun,[-2,-0.1]);grid on;
>> [x,f,h]=fsolve(fun,-1.6),[x,f,h]=fsolve(fun,-0.6) Optimization terminated: first-order optimality is le than options.TolFun.x =
-1.5956 f =
1.4909e-009 h =
1
Optimization terminated: first-order optimality is le than options.TolFun.x =
-0.6180 f =
-3.3152e-012 h =
1 例4.4求下列方程组在原点附近的解
>> fun=inline(\'[4*x(1)-x(2)+exp(x(1))/10-1,-x(1)+4*x(2)+x(1)^2/8]\',\'x\'); [x,f,h]=fsolve(fun,[0,0]) Optimization terminated: first-order optimality is le than options.TolFun.x =
0.2326
0.0565 f =
1.0e-006 *
0.0908
0.1798 h =
1 例4.5 求二元函数f(x,y)=5-x^4-y^4+4*x*y在原点附近的极大值。 （等价于求-f(x,y)的极小值）
>> fun=inline(\'x(1)^4+x(2)^4-4*x(1)*x(2)-5\'); >> [x,g]=fminsearch(fun,[0,0]) x =
1.0000
1.0000 g =
-7.0000 例4.6 用Newton迭代法求下列方程的正根，要求精度为10的-6次 X^2-3x+e^x=2 >> fun=inline(\'x^2-3*x+exp(x)-2\'); >> fplot(fun,[0,2]); >> grid on;
%M函数 newton.m function x =newton(fname,dfname,x0,e) if nargine
x0=x;x=x0-feval(fname,x0)/feval(dfname,x0); end
>> dfun=inline(\'2*x-3+exp(x)\');format long; newton(fun,dfun,1.5,1e-6),format short ans =
1.446238685966427 例4.7 用函数y=a*e^（b*x）拟合例4.2的数据。 >> fun=inline(\'c(1)*exp(c(2)*x)\',\'c\',\'x\'); >> x=[0.1,0.2,0.15,0,-0.2,0.3];y=[0.95,0.84,0.86,1.06,1.50,0.72]; >> c=lsqcurvefit(fun,[0,0],x,y) Optimization terminated: relative function value changing by le than OPTIONS.TolFun.c =
1.0997
-1.4923
PAGE 77 1.%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) 2.%Exercise 2
fun=inline(\'x*log(sqrt(x^2-1)+x)-sqrt(x^2-1)-0.5*x\'); fzero(fun,2) 3.%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)
4.%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] x =
Columns 1 through 11
0.0050
0.0152
0.0245
0.0354
0.0455
0.0531
0.0637
0.0796
0.0796
0.1061
-0.0050
Columns 12 through 20
-0.0152
-0.0245
-0.0354
-0.0455
-0.0531
-0.0637
-0.0796
-0.0796
-0.1061
5.%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]) 6.%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]) 7.%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])
8.%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) x =
-2.0000
-1.5326
-0.7315
-0.0000
1.5951
2.0000 ans =
-3.0272
2.2364
-0.3582
-0.0000
-2.2080
0
%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) ans =
-179
74
-54
199
%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) ans =
2.0000 ans =
1.0000
9.%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
x =
0
0 x =
-0.3333
-6.0000
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
Page 94 chapter5 1.x=[0 4 10 12 15 22 28 34 40]; y=[0 1 3 6 8 9 5 3 0]; trapz(x,y) ans =
178.5000 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)
ans =
0.2500
0.3333
1.5000
0.6667
0.1429
-0.6667
-0.3333
-0.5000 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 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) 5.(1) (2) fun=inline(\'exp(2*x).*cos(x).^3\'); quadl(fun,0,2*pi) (3) fun=@(x)x.*log(x.^4).*asin(1./x.^2); quadl(fun,1,3) (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 (5) (6) fun=inline(\'sqrt(1+r.^2.*sin(th))\',\'r\',\'th\'); dblquad(fun,0,1,0,2*pi) (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) 10 (1) (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) 13.fun=inline(\'5400*v./(8.276*v.^2+2000)\',\'v\'); quadl(fun,15,30)
