目录
第一章................................................................................................................................. 1 第二章................................................................................................................................. 5 第三章............................................................................................................................... 12 第四章............................................................................................................................... 32 第五章............................................................................................................................... 47 第六章............................................................................................................................... 54 补充题 欧拉法，龙格库塔法解方程，黑板上的题..................................................... 57
第一章
1.创建表达式
%可以用syms先符号运算再带入值 x=1; y=2;
z=(sqrt(4*x^2+1)+0.5457*exp(-0.75*x^2-3.75*y^2-1.5*x))/(2*sin(3*y)-1)
z =
-1.4345
2.计算复数
x=(-1+sqrt(-5))/4; y=x+8+10j
y =
7.7500 +10.5590i
3.help命令学三维曲线
x=-5:0.1:5; y=x;
[X,Y]=meshgrid(x,y);
Z=(sin(sqrt(X.^2+Y.^2)))./(sqrt(X.^2+Y.^2)); subplot(221); surf(X,Y,Z); colormap(cool); subplot(222);
plot3(X,Y,Z,'linewidth',4); %绘制三维曲线，也可以随意给定一个三维曲线的函数。如果画这个曲面，那么将绘出一族三维曲线 grid on;
subplot(223);
meshz(X,Y,Z); %地毯绘图 subplot(224);
meshc(X,Y,Z); %等高线绘图
4.peaks等高线（更改原函数）
subplot(221);
contour(peaks1,20); subplot(222);
contour3(peaks1,10); %可以定义等高线条数 subplot(223);
contourf(peaks1,10); subplot(224); peaks1;
z = 3*(1-x).^2.*exp(-(x.^2) - (y+1).^2) ...
- 10*(x/5 - x.^3 - y.^5).*exp(-x.^2-y.^2) ... - 1/3*exp(-(x+1).^2 - y.^2)
5. LOGO绘制
membrane
logo
第一章书后习题 1.合法性
不合法 合法 不合法 不合法 合法
2.运行命令及探讨
a=sqrt(2)
a =
1.4142
答：不是精确的2。是一个近似。可通过改变format进行位数显示调整。 例如：
format long; a=sqrt(2)
format short;
a =
1.414213562373095
或可使用digits任意指定输出位数。 例如：
digits(50); a=sqrt(2); vpa(a)
ans =
1.4142135623730950488016887242096980785696718753769 常见情况下毋需太高精度。
3.运行结果讨论
format long; w1=a^(2/3)
w2=a^2^(1/3) w3=(a^(1/3))^2
w1 =
1.259921049894873 w2 =
1.259921049894873 w3 =
1.259921049894873
测试结果为相同，说明MATLAB程序执行时经过的过程相同。
4.clear clf clc
clear 为从内存中清除变量和函数
clf 为清除figure中的已绘图形以及子图形 clc 为清除命令行窗口
5.产生二维数组
显然第一第二个方法可以实现。 例如：
s=[1 2 3;4 5 6;7 8 9]
s =
1 2 3 4 5 6 7 8 9
即是一个简便的键入矩阵的方法。
第二章 1 数据类型
class(3/7+0.1)
class(sym(3/7+0.1))
class(vpa(sym(3/7+0.1),4)) class(vpa(sym(3/7+0.1)))
ans = double ans = sym ans = sym ans = sym
2 哪些精准？
a1=sin(sym(pi/4)+exp(sym(0.7)+sym(pi/3)));
a2=sin(sym(pi/4)+exp(sym(0.7))*exp(sym(pi/3)));
a3=sin(sym('pi/4')+exp(sym('0.7'))*exp(sym('pi/3'))); a4=sin(sym('pi/4')+exp(sym('0.7+pi/3'))); a5=sin(sym(pi/4)+exp(sym(0.7+pi/3))); a6=sin(sym(pi/4)+sym(exp(0.7+pi/3))); a7=sin(sym(pi/4+exp(0.7+pi/3))); a8=sym(sin(pi/4+exp(0.7+pi/3)));
digits(64);
vpa(a2-a1) vpa(a3-a1)
vpa(a4-a1) %为精确值 vpa(a5-a1) vpa(a6-a1) vpa(a7-a1) vpa(a8-a1)
ans =
8.772689107613377606024459313047548287536202098197290121158158175e-72 ans =
8.772689107613377606024459313047548287536202098197290121158158175e-72 ans = 0.0 ans =
-0.0000000000000008874822716959584619522637254014249128254875650208152937300697045 ans =
-0.000000000000001489122128176563341755713716272780778030227615022223735634526288 ans =
-0.000000000000001518855593927822635897082947744411794950714383466168364259064934 ans =
-0.00000000000000151859755909122793880734918235619076228065004813152159311456667
可以看到，除了a4为精确，其余均存在很小的误差。其中a2与a3的误差较小，小于eps精度，故可认为为精确的。
3 独立自由变量
a1=sym('sin(w*t)') ; a2=sym('a*exp(-X)' ); a3=sym('z*exp(j*th)'); symvar(a1,1) symvar(a2,1) symvar(a3,1)
ans = w
ans = a
ans = z
6 符号解
syms x k; f1=x.^k;
s1=symsum(f1,k,0,inf); s2=subs(f1,x,(-1/3)); s3=subs(f1,x,(1/pi)); s4=subs(f1,x,3); symsum(s2,k,0,inf)
double(symsum(s3,k,0,inf)) symsum(s4,k,0,inf)
ans = 3/4 ans =
1.4669 ans = Inf
7 限定性假设
reset(symengine); syms k;
syms x positive;
f1=(2/(2*k+1))*((x-1)/(x+1))^(2*k+1); f1_s=symsum(f1,k,0,inf);
simplify(f1_s,'steps',27,'IgnoreAnalyticConstraints',true)
ans = log(x)
8 符号计算
syms t;
yt=abs(sin(t)); dydt=diff(yt,t)
dydt0=limit(dydt,t,0,'left') dydtpi=subs(dydt,t,(pi/2))
dydt =
sign(sin(t))*cos(t) dydt0 = -1
dydtpi = 0
9 积分值
syms x;
fx=exp(-abs(x))*abs(sin(x)) fxint=int(fx,-5*pi,1.7*pi); vpa(fxint,64)
fx =
abs(sin(x))*exp(-x) ans =
3617514.635647088707100018393465500554242735057835123431773680704
10二重积分
syms x y; fxy=x^2+y^2;
int(int(fxy,y,1,x^2),x,1,2)
ans =
1006/105
11 绘出曲线
syms t x;
fx=int((sin(t)./t),t,0,x); ezplot(fx)
fx4=subs(fx,x,4.5)
fx4 =
sinint(9/2) sinint(x)21.510.50-0.5-1-1.5-2-6-4-20x246
12 积分表达式
syms x;
syms n positive;
yn=int((sin(x)).^n,x,0,pi/2) yn3=subs(yn,n,1/3); vpa(yn3,32)
yn =
beta(1/2, n/2 + 1/2)/2 ans =
1.2935547796148952674767575125656
13 序列卷积
syms a b n;
syms k positive; xk=a.^k; hk=b.^k;
kn=subs(xk,k,k-n)*subs(hk,k,n); yk=symsum(kn,n,0,k)
yk =
piecewise([a == b and b ~= 0, b^k*(k + 1)], [a ~= b or b == 0, (a*a^k - b*b^k)/(a - b)])
所以答案为a*a^k - b*b^k)/(a - b)
20求解solve
reset(symengine) syms x y;
s=solve('x^2+y^2-1','x*y-2','x','y') s.x s.y
s =
x: [4x1 sym] y: [4x1 sym] ans =
((15^(1/2)*i)/2 + 1/2)^(1/2)/2 - ((15^(1/2)*i)/2 + 1/2)^(3/2)/2 - ((15^(1/2)*i)/2 + 1/2)^(1/2)/2 + ((15^(1/2)*i)/2 + 1/2)^(3/2)/2 (1/2 - (15^(1/2)*i)/2)^(1/2)/2 - (1/2 - (15^(1/2)*i)/2)^(3/2)/2 - (1/2 - (15^(1/2)*i)/2)^(1/2)/2 + (1/2 - (15^(1/2)*i)/2)^(3/2)/2 ans =
((15^(1/2)*i)/2 + 1/2)^(1/2) -((15^(1/2)*i)/2 + 1/2)^(1/2) (1/2 - (15^(1/2)*i)/2)^(1/2) -(1/2 - (15^(1/2)*i)/2)^(1/2)
23 求通解
clear all;
yso=simplify(dsolve('Dy*y*0.1+0.3*x=0','x'))
yso =
(- 3*x^2 + 2*C3)^(1/2) -(- 3*x^2 + 2*C3)^(1/2) %此题存疑
hold on;clear all; reset(symengine); syms x;
y1=(- 3*x^2 + 2*1)^(1/2); y2=-(- 3*x^2 + 2*1)^(1/2);
h1=ezplot(y1,x,[-2 2 -2 2],1); h2=ezplot(y2,x,[-2 2 -2 2],1);
grid on;title('');warning off;axis([-2 2 -2 2]); set(h1,'color','r','linewidth',2); set(h2,'color','r','linewidth',2); xlabel('Y');ylabel('x');
21.510.50-0.5-1-1.5-2-2x-1.5-1-0.50Y0.511.52
%对于第二章存在问题的习题的探讨 2.23
clear all; syms x;
yso=simplify(dsolve('Dy*y*0.1+0.3*x=0','x'))
%此题存疑
hold on;clear all; reset(symengine); syms x;
y1=(- 3*x^2 + 2*1)^(1/2); y2=-(- 3*x^2 + 2*1)^(1/2); h1=ezplot(y1,x,[-2 2 -2 2],1); h2=ezplot(y2,x,[-2 2 -2 2],1);
grid on;title('');warning off;axis([-2 2 -2 2]); set(h1,'color','r','linewidth',2); set(h2,'color','r','linewidth',2); xlabel('Y');ylabel('x');
yso =
(- 3*x^2 + 2*C3)^(1/2) -(- 3*x^2 + 2*C3)^(1/2)
21.510.50-0.5-1-1.5-2-2x-1.5-1-0.50Y0.511.52
%以上方法可以绘出正常的横坐标为y纵坐标为x的图像，但发现在y=0处x延伸至正负无穷。
h1=ezplot(y1,[-2 2 -2 2],1); h2=ezplot(y2,[-2 2 -2 2],1); -(2 - 3 x2)1/221.510.50-0.5-1-1.5-2-2x-1.5-1-0.50x0.511.52 reset(symengine); syms x y S;
%以上方法绘出的图像存在一个空隙，且默认为y-x图像。
S = dsolve('Dy*y/5+x/4=0','x');
ezplot(subs(y^2-(S(1))^2, 'C3', 1),[-2,2 -2,2],2); grid on;
%用椭圆方程绘图不产生间隙
(5 x2)/4 + y2 - 221.510.50-0.5-1-1.5-2-2y-1.5-1-0.50x0.511.52
24 一阶微分方程
syms a b;
ys=dsolve('Dy-a*x^2-b*x=0','y(0)=2',x)
ys =
(x^2*(3*b + 2*a*x))/6 + 2
25 边值问题
fs=dsolve('Df-3*f=4*g,Dg+4*f=3*g','f(0)=0,g(0)=1')
fs =
g: [1x1 sym] f: [1x1 sym]
fs.g fs.f
ans =
cos(4*t)*exp(3*t) ans =
sin(4*t)*exp(3*t)
第三章
3.行下标列下标
rng('default'); A=rand(3,5); L=A>0.5
L =
1 1 0 1 1 1 1 1 0 0 0 0 1 1 1 [a,b]=find(L==1)
IND=sub2ind(size(A),a,b)
IND = 1 2 4 5 8 9 10 12 13 15
4.循环运算、数组运算
t=0:0.1:10; N=length(t);
y1=zeros(size(t)); for k=1:N
y1(k)=1-exp(-0.5*t(k))*cos(2*t(k)); end
plot(t,y1); xlabel('t'); ylabel('y1');
1.51y10.50012345t678910 y2=1-exp(-0.5*t).*cos(2*t); plot(t,y2); xlabel('t'); ylabel('y2');
1.51y20.50012345t678910 5.回答问题
clear all;
A=magic(3);B=rand(3); A*B B*A
ans =
5.4072 11.5771 3.0037 6.3884 10.3215 4.9680 2.7058 7.5337 4.8496 ans =
2.5916 3.8303 5.2097 3.4833 5.6313 3.6800 10.9646 9.0086 12.3554
相同，对于矩阵而言对位相乘无差异
不相同，点乘与矩阵乘法进行的不是同一种运算。 不相同，左乘右乘运算不同。
相同，A左点除B等同于B右点除A，均是对位计算。 不相同，左除右除运算亦不相同。
A*A\\B-B A*(A\\B)-B
A*(A*inv(B))-B
ans =
-0.0562 -0.6902 -0.0436 -0.1051 -0.3282 -0.4311 -0.8011 -0.9350 -0.3763 ans =
1.0e-15 *
0 -0.1110 -0.0278 0 0 0 0 0.1110 0 ans =
-80.2971 65.0383 107.2212 -8.0299 91.2626 70.5679 -66.2535 153.4898 66.4342
不相同。第二个更接近0。具体原理需要参考线性代数书……有点忘了。
A\\eye(3) eye(3)/A
ans =
0.1472 -0.1444 0.0639 -0.0611 0.0222 0.1056 -0.0194 0.1889 -0.1028 ans =
0.1472 -0.1444 0.0639 -0.0611 0.0222 0.1056 -0.0194 0.1889 -0.1028
相同。因为对于对角阵，，二者均可化为同一形式。 6.结果不同
A=[1 2; 3 4]; B1=A.^(0.5) B2=0.5.^A B3=A^(0.5) B4=0.5^A
B1 =
1.0000 1.4142 1.7321 2.0000 B2 =
0.5000 0.2500 0.1250 0.0625 B3 =
0.5537 + 0.4644i 0.8070 - 0.2124i 1.2104 - 0.3186i 1.7641 + 0.1458i B4 =
0.9910 -0.4422 -0.6634 0.3276 A1=B1.*B1 A3=B3*B3
norm(A1-A3,'fro')
A1 =
1.0000 2.0000
3.0000 4.0000 A3 =
1.0000 + 0.0000i 2.0000 + 0.0000i 3.0000 - 0.0000i 4.0000 + 0.0000i ans =
1.2831e-15
可见误差在eps量级，可以认为相等。
7.绘出图形
x=-3*pi:pi/15:3*pi; y=x;
[X,Y]=meshgrid(x,y); warning off;
Z=sin(X).*sin(Y)./X./Y;
共有10个非数数据。
surf(X,Y,Z)
shading interp
x=-3*pi:pi/15:3*pi; Lx=(x==0);
xx=x+Lx*realmin; y=xx;
[X,Y]=meshgrid(xx,y); warning off;
Z=sin(X).*sin(Y)./X./Y; surf(X,Y,Z)
shading interp
即消除零点处的断点即可
8.两种思路 %第二种思路
function z=zpoly_z(x,y) if x+y<=-1
z=0.546*exp(-0.75*y.^2-3.75*x.^2+1.5*x); elseif x+y>-1 & x+y<=1
z=0.758*exp(-y.^2-6*x.^2); else
z=0.546*exp(-0.75*y.^2-3.75*x.^2-1.5*x); end
x=-1.5:0.1:1.5; y=-3:0.1:3;
[X,Y]=meshgrid(x,y); Z=zpoly_z(X,Y); surf(X,Y,Z);
%第一种思路
x=-1.5:0.1:1.5; y=-3:0.2:3; LX=length(x); LY=length(y); for ii=1:LX for jj=1:LY
if x(ii)+y(jj)<=-1
z=0.546*exp(-0.75*y.^2-3.75*x.^2+1.5*x); elseif x(ii)+y(jj)>-1 & x(ii)+y(jj)<=1 z=0.758*exp(-y.^2-6*x.^2); else
z=0.546*exp(-0.75*y.^2-3.75*x.^2-1.5*x); end end end
[X,Y]=meshgrid(x,y); Z=zpoly_z(X,Y); surf(X,Y,Z);
%其实for循环完全无意义……
9.矩阵计算
%第一问老师取消
rng default
A=randn(50,70)+1i*randn(50,70); B=randn(70,60)+1i*randn(70,60); C=randn(50,60)+1i*randn(50,60); D=randn(60,1)+1i*randn(60,1); G=(A*B-C)*D
Gr=real(G),70,70 Gi=imag(G) Gn=norm(G,2)
G =
1.0e+02 *
-0.1776 + 1.9914i 0.6088 + 0.3316i -0.1340 - 0.8615i 0.0752 - 0.0759i -0.1171 - 1.8169i 0.2005 - 1.4540i -1.4501 + 0.1897i 0.6445 + 0.1657i -1.0651 + 0.1191i 0.3301 - 0.0450i -1.4338 + 0.8707i -0.9491 + 1.4840i 1.1314 + 1.2751i -0.5158 - 0.0725i -0.2746 + 0.2518i -1.0279 - 0.8409i
-1.1161 - 2.3362i 0.1346 + 1.3500i 0.4220 - 1.2839i 0.2650 - 0.2849i -1.0212 + 0.5374i 0.0563 + 0.4151i -1.9074 - 0.2448i 0.1645 + 1.2071i 1.1870 + 0.0085i 1.2304 + 0.6672i 0.3303 - 1.6027i -0.5728 - 0.5519i 0.3738 + 0.2863i -0.6682 - 0.7565i 1.6063 + 1.2886i 0.6994 - 1.3377i 0.6523 + 0.0318i -0.2143 - 2.8209i 1.7026 - 0.1371i 0.9285 + 1.5852i -0.7550 - 0.2427i -1.3879 - 1.8978i -0.5266 - 0.8334i -0.0849 + 0.1680i 1.1590 + 0.2109i -1.8938 + 0.6709i 0.3406 - 1.8211i -1.0916 - 1.8076i 0.2062 - 1.4363i 1.3679 + 0.2061i -0.4541 + 0.8056i 1.3574 + 0.8773i -0.1071 + 0.0948i 0.1042 + 2.2812i Gr =
-17.7553 60.8848 -13.4003 7.5175 -11.7073 20.0458 -145.0055 64.4517 -106.5069 33.0077 -143.3779 -94.9055 113.1368 -51.5804 -27.4560 -102.7914 -111.6150 13.4596 42.2009 26.5006 -102.1225 5.6295
-190.7388 16.4525 118.6963 123.0361 33.0336 -57.2817 37.3849 -66.8175 160.6261 69.9436 65.2278 -21.4319 170.2597 92.8549 -75.5045 -138.7923 -52.6574 -8.4902 115.9030 -189.3844 34.0593 -109.1584 20.6169 136.7896 -45.4089 135.7386 -10.7050 10.4240 Gi =
199.1404 33.1590 -86.1452 -7.5887 -181.6856 -145.4039 18.9686 16.5731 11.9053 -4.5021 87.0651 148.4022 127.5072 -7.2483 25.1791 -84.0887 -233.6194 135.0018 -128.3931 -28.4923 53.7385 41.5139 -24.4788 120.7113 0.8532 66.7238 -160.2738 -55.1871
28.6287 -75.6522 128.8596 -133.7671 3.1772 -282.0866 -13.7111 158.5203 -24.2673 -189.7767 -83.3384 16.7992 21.0869 67.0898 -182.1134 -180.7631 -143.6344 20.6149 80.5622 87.7339 9.4764 228.1237 Gn =
1.0253e+03 1.51y10.50012345t678910 y2=1-exp(-0.5*t).*cos(2*t); plot(t,y2); xlabel('t');
ylabel('y2'); 1.51y20.50012345t678910 5.回答问题
clear all;
A=magic(3);B=rand(3); A*B B*A
ans =
5.4072 11.5771 3.0037 6.3884 10.3215 4.9680 2.7058 7.5337 4.8496 ans =
2.5916 3.8303 5.2097 3.4833 5.6313 3.6800 10.9646 9.0086 12.3554
相同，对于矩阵而言对位相乘无差异
不相同，点乘与矩阵乘法进行的不是同一种运算。 不相同，左乘右乘运算不同。
相同，A左点除B等同于B右点除A，均是对位计算。 不相同，左除右除运算亦不相同。
A*A\\B-B
A*(A\\B)-B
A*(A*inv(B))-B
ans =
-0.0562 -0.6902 -0.0436 -0.1051 -0.3282 -0.4311 -0.8011 -0.9350 -0.3763 ans =
1.0e-15 *
0 -0.1110 -0.0278 0 0 0 0 0.1110 0 ans =
-80.2971 65.0383 107.2212 -8.0299 91.2626 70.5679 -66.2535 153.4898 66.4342
不相同。第二个更接近0。具体原理需要参考线性代数书……有点忘了。
A\\eye(3) eye(3)/A
ans =
0.1472 -0.1444 0.0639 -0.0611 0.0222 0.1056 -0.0194 0.1889 -0.1028 ans =
0.1472 -0.1444 0.0639 -0.0611 0.0222 0.1056 -0.0194 0.1889 -0.1028
相同。因为对于对角阵，，二者均可化为同一形式。 6.结果不同
A=[1 2; 3 4]; B1=A.^(0.5) B2=0.5.^A B3=A^(0.5) B4=0.5^A
B1 =
1.0000 1.4142 1.7321 2.0000 B2 =
0.5000 0.2500 0.1250 0.0625 B3 =
0.5537 + 0.4644i 0.8070 - 0.2124i 1.2104 - 0.3186i 1.7641 + 0.1458i B4 =
0.9910 -0.4422 -0.6634 0.3276 A1=B1.*B1 A3=B3*B3
norm(A1-A3,'fro')
A1 =
1.0000 2.0000 3.0000 4.0000 A3 =
1.0000 + 0.0000i 2.0000 + 0.0000i 3.0000 - 0.0000i 4.0000 + 0.0000i ans =
1.2831e-15
可见误差在eps量级，可以认为相等。
7.绘出图形
x=-3*pi:pi/15:3*pi; y=x;
[X,Y]=meshgrid(x,y); warning off;
Z=sin(X).*sin(Y)./X./Y;
共有10个非数数据。
surf(X,Y,Z)
shading interp
x=-3*pi:pi/15:3*pi; Lx=(x==0);
xx=x+Lx*realmin; y=xx;
[X,Y]=meshgrid(xx,y); warning off;
Z=sin(X).*sin(Y)./X./Y; surf(X,Y,Z)
shading interp
即消除零点处的断点即可
8.两种思路 %第二种思路
function z=zpoly_z(x,y) if x+y<=-1
z=0.546*exp(-0.75*y.^2-3.75*x.^2+1.5*x); elseif x+y>-1 & x+y<=1
z=0.758*exp(-y.^2-6*x.^2); else
z=0.546*exp(-0.75*y.^2-3.75*x.^2-1.5*x); end
x=-1.5:0.1:1.5; y=-3:0.1:3;
[X,Y]=meshgrid(x,y); Z=zpoly_z(X,Y); surf(X,Y,Z);
%第一种思路
x=-1.5:0.1:1.5; y=-3:0.2:3; LX=length(x); LY=length(y); for ii=1:LX for jj=1:LY
if x(ii)+y(jj)<=-1
z=0.546*exp(-0.75*y.^2-3.75*x.^2+1.5*x); elseif x(ii)+y(jj)>-1 & x(ii)+y(jj)<=1 z=0.758*exp(-y.^2-6*x.^2); else
z=0.546*exp(-0.75*y.^2-3.75*x.^2-1.5*x); end end end
[X,Y]=meshgrid(x,y); Z=zpoly_z(X,Y); surf(X,Y,Z);
%其实for循环完全无意义……
9.矩阵计算
%第一问老师取消
rng default
A=randn(50,70)+1i*randn(50,70); B=randn(70,60)+1i*randn(70,60); C=randn(50,60)+1i*randn(50,60); D=randn(60,1)+1i*randn(60,1); G=(A*B-C)*D
Gr=real(G),70,70 Gi=imag(G) Gn=norm(G,2)
G =
1.0e+02 *
-0.1776 + 1.9914i 0.6088 + 0.3316i -0.1340 - 0.8615i 0.0752 - 0.0759i -0.1171 - 1.8169i 0.2005 - 1.4540i -1.4501 + 0.1897i 0.6445 + 0.1657i -1.0651 + 0.1191i 0.3301 - 0.0450i -1.4338 + 0.8707i -0.9491 + 1.4840i 1.1314 + 1.2751i -0.5158 - 0.0725i -0.2746 + 0.2518i -1.0279 - 0.8409i
-1.1161 - 2.3362i 0.1346 + 1.3500i 0.4220 - 1.2839i 0.2650 - 0.2849i -1.0212 + 0.5374i 0.0563 + 0.4151i -1.9074 - 0.2448i 0.1645 + 1.2071i 1.1870 + 0.0085i 1.2304 + 0.6672i 0.3303 - 1.6027i -0.5728 - 0.5519i 0.3738 + 0.2863i -0.6682 - 0.7565i 1.6063 + 1.2886i 0.6994 - 1.3377i 0.6523 + 0.0318i -0.2143 - 2.8209i 1.7026 - 0.1371i 0.9285 + 1.5852i -0.7550 - 0.2427i -1.3879 - 1.8978i -0.5266 - 0.8334i -0.0849 + 0.1680i 1.1590 + 0.2109i -1.8938 + 0.6709i 0.3406 - 1.8211i -1.0916 - 1.8076i 0.2062 - 1.4363i 1.3679 + 0.2061i -0.4541 + 0.8056i 1.3574 + 0.8773i -0.1071 + 0.0948i 0.1042 + 2.2812i Gr =
-17.7553 60.8848 -13.4003 7.5175 -11.7073 20.0458 -145.0055 64.4517 -106.5069 33.0077 -143.3779 -94.9055 113.1368 -51.5804 -27.4560 -102.7914 -111.6150 13.4596 42.2009 26.5006 -102.1225 5.6295
-190.7388 16.4525 118.6963 123.0361 33.0336 -57.2817 37.3849 -66.8175 160.6261 69.9436 65.2278 -21.4319 170.2597 92.8549 -75.5045 -138.7923 -52.6574 -8.4902 115.9030 -189.3844 34.0593 -109.1584 20.6169 136.7896 -45.4089 135.7386 -10.7050 10.4240 Gi =
199.1404 33.1590 -86.1452 -7.5887 -181.6856 -145.4039 18.9686 16.5731 11.9053 -4.5021 87.0651 148.4022 127.5072 -7.2483 25.1791 -84.0887 -233.6194 135.0018 -128.3931 -28.4923 53.7385 41.5139 -24.4788 120.7113 0.8532 66.7238 -160.2738 -55.1871
28.6287 -75.6522 128.8596 -133.7671 3.1772 -282.0866 -13.7111 158.5203 -24.2673 -189.7767 -83.3384 16.7992 21.0869 67.0898 -182.1134 -180.7631 -143.6344 20.6149 80.5622 87.7339 9.4764 228.1237 Gn =
1.0253e+03
第四章 1 数值差分
d=0.001;
dxdt_diff=diff(y)/d;
dxdt_grad=gradient(y)/d; hold on;
plot(t,y,'-r');
plot(t(1:end-1),dxdt_diff,'-g'); plot(t,dxdt_grad,'-b');
210-1-2-3-4-5-601234567数值求导后存在很多的毛刺
2 数值计算
d=1e-5; t=0:d:10;
t=t+(t==0)*realmin;%去除影响 fx=sin(t)./t;
y=cumtrapz(fx)*d; plot(t,y);
yt1=find(t==4.5); y45=y(yt1)
y45 =
1.6541
21.81.61.41.210.80.60.40.20012345678910尝试integral d=1e-5; t=0:d:10;
t=t+(t==0)*realmin;%去除影响 fx=@(t)sin(t)./t; y=integral(fx,0,10) %plot(t,y);
%yt1=find(t==4.5); %y45=y(yt1)
y =
1.6583
3 数值积分
d=1e-5; t=0:d:pi;
t=t+(t==0)*realmin;%去除影响 fx=@(t)exp((sin(t)).^3); s=integral(fx,0,pi)
s =
5.1370
符号验算
syms x;
fxx=exp((sin(x))^3); ss=int(fxx,x,0,pi)
ss =
int(exp(sin(x)^3), x, 0, pi)
此时符号运算不如数值运算可以得出解析解
4 求数值积分
clear;
fx=@(x)exp(-abs(x)).*abs(sin(x)); format long ;
s=integral(fx,-5*pi,inf,'Abstol',1e-9)
s =
1.090331100328886
x=linspace(-5*pi,50,1e7); dx=x(2)-x(1);
st=trapz(exp(-abs(x)).*abs(sin(x)))*dx
st =
1.090331328559288
可知，trapz必须为有限数值；同样，quadl也需为有限数值
clear;
fx=@(x)exp(-abs(x)).*abs(sin(x)); format long ;
s=quadl(fx,-5*pi,50,1e-9)
s =
1.090331336243919
5 最小值点
%首先绘出函数曲线，确定大概找最小值范围 clear;
ft=@(t)sin(5*t).^2.*exp(0.06*t.*t)+1.8*abs(t+0.5)-1.5*t.*cos(2*t); t=-5:0.01:5;
ezplot(ft,[-5,5]); sin(5 t)2 exp(0.06 t t)+...-1.5 t cos(2 t)20181614121086420-5-4-3-2-10t12345
%显然，最小值在[-2 -1]区间内。 tt=linspace(-2,-1,101); for k=1:100
[tmin(k),fobj(k)]=fminbnd(ft,tt(k),tt(k+1)); end
mini=min(fobj)
tmini=tmin(find(fobj==mini))
mini =
-0.1860 tmini =
-1.2850
6 ode45初试
format long ts=[0,1]; y0=[1;0];
dydt=@(t,y)[y(2);-2*y(1)+3*y(2)+1]; %匿名函数写成的ode45所需dydt [tt,yy]=ode45(dydt,ts,y0);
y05=interp1(tt,yy(:,1),0.5,'spline') %用一维插值求y(0.5)
y05 =
0.789580207901271
%符号运算法 syms t;
ys=dsolve('D2y-3*Dy+2*y=1','y(0)=1,Dy(0)=0','t') ; ys_05=subs(ys,t,sym('0.5'))
ys_05 =
0.78958035647060552916850705213783
7 值空间基阵（第一天参考）
A=magic(8); format short; [R,jb]=rref(A); B1=A(:,jb) %基向量
B2=orth(A) %单位正交基
B1 =
64 2 3 9 55 54 17 47 46 40 26 27 32 34 35 41 23 22 49 15 14 8 58 59 B2 =
-0.3536 0.5401 0.3536 -0.3536 -0.3858 -0.3536 -0.3536 -0.2315 -0.3536
-0.3536 0.0772 0.3536 -0.3536 -0.0772 0.3536 -0.3536 0.2315 -0.3536 -0.3536 0.3858 -0.3536 -0.3536 -0.5401 0.3536
B1_A=rref([B1,A]) B2_A=rref([B2,A])
B1_A =
1 0 0 1 0 0 1 1 0 0 1 0 1 0 0 1 0 3 4 -3 -4 7 0 0 1 0 0 1 -3 -4 4 5 -7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B2_A =
Columns 1 through 10
1.0000 0 0 -91.9239 -91.9239 -91.9239 -91.9239 -91.9239 -91.9239 -91.9239
0 1.0000 0 51.8459 -51.8459 -51.8459 51.8459 51.8459 -51.8459 -51.8459
0 0 1.0000 9.8995 -7.0711 -4.2426 1.4142 -1.4142 4.2426 7.0711
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 Column 11 -91.9239 51.8459 -9.8995 0 0 0 0 0
8 gallery矩阵
A=gallery(5) [V,D]=eig(A); diag(D)'
A =
-9 11 -21 63 -252 70 -69 141 -421 1684 -575 575 -1149 3451 -13801 3891 -3891 7782 -23345 93365
1024 -1024 2048 -6144 24572 ans =
-0.0405 + 0.0000i -0.0118 - 0.0383i -0.0118 + 0.0383i 0.0320 - 0.0228i 0.0320 + 0.0228i
%验算看分解精度 Areset=V*D/V
eAreset=norm(A-Areset,'fro')/norm(A,'fro')
Areset =
1.0e+04 *
-0.0009 - 0.0000i 0.0011 + 0.0000i -0.0021 -0.0252 - 0.0000i
0.0070 + 0.0000i -0.0069 - 0.0000i 0.0141 0.1684 + 0.0000i
-0.0575 - 0.0000i 0.0575 + 0.0000i -0.1149 -1.3801 - 0.0000i
0.3891 + 0.0000i -0.3891 - 0.0000i 0.7782 9.3365 + 0.0000i
0.1024 + 0.0000i -0.1024 - 0.0000i 0.2048 2.4572 + 0.0000i eAreset =
3.0594e-06
精度在1e-6左右，可以接受。
[V,D,s]=condeig(A)
V =
-0.0000 + 0.0000i -0.0000 + 0.0000i -0.0000 0.0000 - 0.0000i
0.0206 + 0.0000i 0.0206 + 0.0001i 0.0206 0.0207 - 0.0001i -0.1398 + 0.0000i -0.1397 + 0.0001i -0.1397 -0.1397 - 0.0000i
0.9574 + 0.0000i 0.9574 + 0.0000i 0.9574 0.9574 + 0.0000i
0.2519 + 0.0000i 0.2519 - 0.0000i 0.2519 0.2519 + 0.0000i D =
-0.0405 + 0.0000i 0.0000 + 0.0000i 0.0000 0.0000 + 0.0000i
0.0000 + 0.0000i -0.0118 + 0.0383i 0.0000 0.0000 + 0.0000i
0.0000 + 0.0000i 0.0000 + 0.0000i -0.0118 0.0000 + 0.0000i
0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 0.0000 + 0.0000i
0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 0.0320 - 0.0228i s =
1.0e+10 * 2.1969 2.1468 2.1468 2.0688 2.0688
- 0.0000i 0.0063 + 0.0000i -0.0421 - 0.0000i 0.3451 + 0.0000i -2.3345 + 0.0000i -0.6144 - 0.0000i 0.0000 - 0.0001i 0.0207 - 0.0001i -0.1397 + 0.0000i 0.9574 + 0.0000i 0.2519 + 0.0000i 0.0000 + 0.0000i 0.0000 - 0.0383i 0.0000 + 0.0000i 0.0320 + 0.0000i 0.0000 + 0.0000i - 0.0000i + 0.0000i - 0.0000i - 0.0000i + 0.0000i + 0.0001i + 0.0000i + 0.0000i - 0.0000i + 0.0000i + 0.0000i + 0.0000i + 0.0228i + 0.0000i 其中，当s较大时，对应的特征值可信度较差
9 求解矩阵
A=magic(3); b=ones(3,1) x1=A\\b
b = 1 1 1 x1 =
0.0667 0.0667 0.0667
[R,C]=rref([A,b]) %消去法解方程，同时判断解的唯一性
R =
1.0000 0 0 0.0667 0 1.0000 0 0.0667 0 0 1.0000 0.0667 C =
1 2 3
x3=inv(A)*b %左乘A逆解方程
x3 =
0.0667 0.0667 0.0667
三种方法所得解形式一致
10 四阶魔方阵
A=magic(4); b=ones(4,1); x1=A\\b
x2=inv(A)*b
[x3,C]=rref([A,b])
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 1.306145e-17. x1 =
0.0588 0.1176 -0.0588 0
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 1.306145e-17. x2 =
0.0469 0.1875 -0.0625 -0.0156
x3 =
1.0000 0 0 1.0000 0.0588 0 1.0000 0 3.0000 0.1176 0 0 1.0000 -3.0000 -0.0588 0 0 0 0 0 C =
1 2 3
可以观察到，A不满秩，用逆矩阵的方法无法求得正确的解。 b可以由A的前三列线性表出，解不唯一。 验算：
A1=A*x1 A2=A*x2
A3=A(:,C)*x3(C,5)
A1 = 1 1 1 1 A2 =
0.7344 1.5469 1.1719 1.8594 A3 = 1 1 1 1
验算表明，逆矩阵法结果有误，理论上逆矩阵不存在。
xx=null(A) %齐次解
xx =
0.2236 0.6708 -0.6708 -0.2236
f=randn(1,6); %6个随机系数
xf=repmat(x1,1,6)+xx*f; %产生6个不同的特解 A*xf
ans =
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
11 求解矩阵
A=magic(4); b=(1:4)'; x1=A\\b
x2=inv(A)*b
[x3,C]=rref([A,b])
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 1.306145e-17. x1 =
1.0e+15 * -0.5629 -1.6888 1.6888 0.5629
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 1.306145e-17. x2 =
1.0e+15 * -0.5629 -1.6888 1.6888 0.5629 x3 =
1 0 0 1 0 0 1 0 3 0 0 0 1 -3 0 0 0 0 0 1 C =
1 2 3 5
由于该矩阵不满秩，b也不再A的值空间中，因而无法求得准确解。
12 实数解
t=-5:0.01:5;
y=@(t)(-0.5)+t-10*exp(-0.2*t).*abs(sin(sin(t))); plot(t,y(t)) %利用匿名函数求y函数值 grid on;
[tt1,yy1]=ginput(1) %从图形获得近似解
tt1 =
2.7535 yy1 =
0.0365
50-5-10-15-20-25-30-5-4-3-2-1012345 [t,yt]=fzero(y,tt1)%寻找函数零点，接近的地方
t =
2.7341 yt =
-4.4409e-16
13 求解二元函数方程组
%符号法
S=solve('sin(x-y)','cos(x+y)','x','y'); X=S.x Y=S.y
X =
(3*pi)/4 -(3*pi)/4 pi/4 -pi/4 Y =
-pi/4 pi/4 -(3*pi)/4 (3*pi)/4
%图像交点法 x=-5:0.1:5; y=x;
[X,Y]=meshgrid(x,y); z1=sin(X-Y); z2=cos(X+Y); axis square; grid on;
hold on;
contour(X,Y,z1,3) contour(X,Y,z2,3) 543210-1-2-3-4-5-505
14 概率分布曲线
N=100; p=0.157;
P=binopdf(0:N,N,p); stem(0:N,P); grid on;
0.120.10.080.060.040.0200102030405060708090100
15 正态分布随机数组
rng default mu=4;sigma=2;
a=normrnd(mu,sigma,10000,1); subplot(211); hist(a);
subplot(212); histfit(a); 3000200010000-44003002001000-4-2024681012-2024681012
16 程序正确吗？
load prob_data416;%R是一个二维数组 Mx=max(max(R)) Me=mean(mean(R)) St=std(std(R))
Mx =
0.9997 Me =
0.5052 St =
0.0142
其中标准差是各列数据的标准差，与整个数组标准差有区别。
Mx1=max(R(:)) Me1=mean(R(:)) S1=std(R(:))
Mx1 =
0.9997 Me1 =
0.5052 S1 =
0.2919
其中标准差是整个数组标准差
17 求分式多项式，商多项式
N=conv([3 0 1 0],[1 0 0 0.5]);%用离散卷积表示多项式相乘 D=conv([1 2 -2],[5 2 0 1]); [Q,r]=deconv(N,D)
Q =
0.6000 -1.4400 r =
0 0 21.8800 -5.3400 -5.5200 4.5800 -2.8800
yif=conv(D,Q)+r
yif =
3.0000 0 1.0000 1.5000 0 0.5000 0 err=norm(N-yif)
err =
7.1607e-15
数值计算存在误差，可以近似认为相等。
18 五阶拟合多项式
load prob_data418; poly=polyfit(x,y,5); yy=polyval(poly,x);
plot(x,y,'.b'); hold on;
plot(x,yy,'-r'); 0.60.40.20-0.2-0.4-0.6-0.8-1-1.2-1.4-1-0.500.511.522.533.54
19 系统冲激响应
h=[0.05,0.24,0.40,0.24,0.15,-0.1,0.1]; rng default;
u=2*(randn(1,100)>0.5)-1; con=conv(u,h); lu=length(u); lc=length(con);
u1=[u,nan*ones(1,lc-lu)]; subplot(211);
stem(0:lc-1,u1,'filled'); axis([0 lc -1 1]); title('Input u'); subplot(212);
stem(0:lc-1,con,'filled'); axis([0 lc -1 1]); title('Output y')
Input u10.50-0.5-10102030405060708090100Output y10.50-0.5-10102030405060708090100
第五章 1 椭圆绘制
t=0:pi/100:2*pi; a=4;b=2;
x=a*cos(t);y=b*sin(t);
plot(x,y,'.r','markersize',6) xlabel('x');ylabel('y'); axis equal; 3210y-1-2-3-4-3-2-10x1234
2 心脏线
theta=0:pi/100:2*pi; rou=1-cos(theta); h=polar(theta,rou)
set(h,'linewidth',4,'color','r'); axis equal;
title('\\rho =1-cos \\theta') ? =1-cos ?90 2120 1.5150 1 0.518030600210330240270300
直角坐标绘图
t=0:pi/100:2*pi;
x=2*cos(t)-cos(2*t); y=2*sin(t)-sin(2*t);
plot(x,y,'r','linewidth',4) axis equal;grid on;
title('\\rho =1-cos \\theta')
? =1-cos ?2.521.510.50-0.5-1-1.5-2-2.5-4-3-2-1012
3 直方图
x=1:6;x=x';
y=[170,120,180,200,190,220;120,100,110,180,170,180;70,50,80,100,95,120]';
bar(x,y,'stacked'); axis([0,7,0,600]); colormap(cool);
legend('A','B','C','Location','NorthWest'); 600ABC 5004003002001000 123456
4 二阶线性系统的归一化
% exmp504.m 供第4道习题使用的程序 clc,clf,clear; t=(0:0.05:18)'; N=length(t);
zeta=linspace(0.2,1.4,7);%改变产生向量的方式 L=length(zeta); y=zeros(N,L); hold on for k=1:L
zk=zeta(k);
beta=sqrt(abs(1-zk^2)); if zk<1 %满足此条件，绘蓝色线 y=1/beta*exp(-zk*t).*sin(beta*t); plot(t,y,'b') if zk<0.4
text(2.2,0.63,'\\zeta = 0.2') end
elseif zk==1 %满足此条件，绘黑色线;问题在于可能无法取得 y=t.*exp(-t);
plot(t,y,'k','LineWidth',2) zk else %其余，绘红色线
y=(exp(-(zk-beta)*t)-exp(-(zk+beta)*t))/(2*beta); plot(t,y,'r') if zk>1.2
text(0.3,0.14,'\\zeta = 1.4') end end end
text(10,0.7,'\\Delta\\zeta=0.2') axis([0,18,-0.4,0.8]) hold off box on grid on
zk = 1
0.8??=0.20.6? = 0.20.40.2? = 1.40-0.2-0.4024681012141618
事实上，当我们运行以下语句时，都会发现数值运算的精确度较低。
x=0:0.1:1;y=linspace(0,1,11);x==y
ans =
1 1 1 0 1 1 1 1 1 1 1
5 题目说绿实线但图片是蓝实线
t=0:pi/100:4*pi; x=sin(t); y=cos(t); z=t;
plot3(x,y,z,'-b','linewidth',3); box on
15105010.50-0.5-1-1-0.50.501
6 三维透视图
%数值法
x=-3:0.1:3; %间隔过小网格过密 y=-3:0.1:3;
[X,Y]=meshgrid(x,y); Z=4*X.*exp(-X.^2-Y.^2); mesh(X,Y,Z) hidden off
axis([-3,3,-3,3,-2,2])
%符号函数法 syms X Y;
%[X,Y]=meshgrid(x,y); Z=4*X.*exp(-X.^2-Y.^2); ezmesh(Z) hidden off
axis([-3,3,-3,3,-2,2])
12 动画尝试(主要依靠pause)
clf;clear;
theta=0:pi/500:2*pi; rou=sin(3*theta); x=rou.*cos(theta); y=rou.*sin(theta); axis([-1 1 -1 1]); axis equal off;
line('xdata',x,'ydata',y,'Color',[0.7,0.4,0.7],'LineWidth',3);
h=line('xdata',x(1),'ydata',y(1),'Color',[1,0,0],'Marker','.','MarkerSize',40,'EraseMode','xor'); N=length(x); k=2;
while (1)
set(h,'xdata',x(k),'ydata',y(k)) pause(0.005) k=k+1; if k>N
k=2; %无限循环 end end
第六章
1 循环、数值、符号时间比较
%循环for tic S=0;
for ii=0:1000000 S=S+0.2^ii; end S toc
S =
1.2500
Elapsed time is 2.227171 seconds.
%循环while tic
S=0;ii=0
while(ii<1000001) S=S+0.2^ii; ii=ii+1; end S toc
ii = 0 S =
1.2500
Elapsed time is 2.680681 seconds.
%数值法 tic
ii=0:1000000; S=sum(0.2.^ii) toc
S =
1.2500
Elapsed time is 0.294500 seconds.
%符号法 tic
syms ii; f=0.2.^ii;
S=vpa(symsum(f,ii,0,100000)) %有限时间内无法算得1e6 toc
S =
1.2500
Elapsed time is 5.938795 seconds.
2 阈值tao（并非构成while循环）
function [N,S]=sum1(tao) n=1;Sk=0;S=0;p=1; R=zeros(1,1e5); while p for k=1:n
Sk=Sk+k; end
R(n)=1/Sk; S=S+R(n);
if R(n) tao=1e-5;[N,S]=sum1(tao) N = 82 S = 1.4996 3 多反馈函数文件 function ex6_3(n) % ex6_3(n) 画出正n边型 % ex6_3 画出单位圆 % n 应为大于2的自然数或0 % By CHT,2015-7-9 if nargin==0 t=0:pi/100:2*pi; y=sin(t); x=cos(t); str='单位圆'; else if nargin~=0 && n<=2 error('多边形边数需要大于2') end if n-round(n)~=0 error('请输入自然数') end t=0:2*pi/n:2*pi; y=sin(t); x=cos(t); str=['这个正多边形有',int2str(n),'条边']; end plot(x,y,'r','linewidth',4); title(str); axis equal off image; ex6_3(5) 这个正多边形有5条边 4 寻找极小值 Y=@(x)-exp(-x).*abs(sin(cos(x))); [x1, y1]=fminbnd(Y,-1,1) x=-2:0.01:2; y=Y(x); plot(x,y); grid on;zoom on [x2,y2]=ginput(1) zoom off x1 = -0.8634 y1 = -1.4348 x2 = -0.8618 y2 = -1.4342 0-0.5-1-1.5-2-2.5-3-2-1.5-1-0.500.511.52 补充题 欧拉法，龙格库塔法解方程，黑板上的题 1 蔡氏电路方程 1、ode23自带函数法 function dx=chuainitial(t,x) a=9.0; b=14.0; dx=zeros(3,1); dx(1)=a*(x(2)-x(1)-myf(x(1))); dx(2)=x(1)-x(2)+x(3); dx(3)=-b*x(2); end function y=myf(x) a=-1.143;b=-0.714; y=b*x+0.5*(a-b)*(abs(x+1)-abs(x-1)); options = odeset('RelTol',1e-4,'AbsTol',[1e-4 1e-4 1e-4]); [t,x]=ode23(@chuainitial,[2000 3000],[0 0.01 0.01],options); plot(x(:,1),x(:,2));title('x-y平面相图') x-y平面相图0.50.40.30.20.10-0.1-0.2-0.3-0.4-0.5-2.5-2-1.5-1-0.500.511.522.5 2、自编Runge-Kutta (等效ode45法) function y=RK45(f,x0,xn,y0) h=0.01; x=x0:h:xn; n=length(x); y1=x; y1(1:3,1)=y0; for i=1:n-1 K1=f(x(i),y1(:,i)); K2=f(x(i)+h/2,y1(:,i)+h/2*K1); K3=f(x(i)+h/2,y1(:,i)+h/2*K2); K4=f(x(i)+h,y1(:,i)+h*K3); y1(:,i+1)=y1(:,i)+h/6*(K1+2*K2+2*K3+K4); end y=y1; function X=ini2(t,p) a=9.0; b=14.0; X(1)=a*(p(2)-p(1)-myf(p(1))); X(2)=p(1)-p(2)+p(3); X(3)=-b*p(2); X=X'; end x0=[0 0.01 0.01]; X=RK45(@ini2,2000,3000,x0); figure; plot(X(1,:),X(2,:));title('x-y平面相图'); x-y平面相图0.50.40.30.20.10-0.1-0.2-0.3-0.4-0.5-2.5-2-1.5-1-0.500.511.522.5 3、改进欧拉法求解 function y=Euler(f,x0,xn,y0) h=0.01; x=x0:h:xn; n=length(x); y1=x; y1(1:3,1)=y0; for i=1:n-1 y1(:,i+1)=y1(:,i)+h*f(x(i),y1(:,i)); end y=y1; x0=[0 0.01 0.01]; X=Euler(@ini2,2000,3000,x0); figure; plot(X(1,:),X(2,:));title('x-y平面相图'); x-y平面相图0.50.40.30.20.10-0.1-0.2-0.3-0.4-0.5-2.5-2-1.5-1-0.500.511.522.5 可以看出，欧拉法与龙格-库塔法的解明显存在不同，若考虑其它图像则更加明显。例如： 2、洛伦兹方程 洛伦兹方程的解应该是一个蝴蝶翅膀一样的图像。程序证明了这一点。 function dx=lorenz(t,x) beta=8/3;rou=10;sigma=28; dx=zeros(3,1); dx(1)=-beta*x(1)+x(2)*x(3); dx(2)=-rou*x(2)+rou*x(3); dx(3)=-x(1)*x(2)+sigma*x(2)-x(3); [t,x]=ode45(@lorenz,[0 200],[0 0 1e-10]); figure; plot(x(:,2),x(:,3));title('y-z“蝴蝶”图像'); y-z“蝴蝶”图像3020100-10-20-30-20-15-10-505101520 3、课堂上题的解 function ydot=DyDt(t,y) ydot=[y(2);-y(1)]; tspan=[0,30]; y0=[0.2;0]; [tt,yy]=ode45(@DyDt,tspan,y0); subplot(211); plot(tt,yy(:,1)) xlabel('t'),title('x(t)') subplot(212); plot(yy(:,1),yy(:,2)); x(t)0.20.10-0.1-0.2051015t2025300.20.10-0.1-0.2-0.2-0.15-0.1-0.0500.050.10.150.2 使用欧拉法产生的结果明显有误 tspan=[0,300]; y0=[0.2;0]; [tt,X]=Euler2(@DyDt,0,300,y0); subplot(211); plot(tt,X); xlabel('t'),title('x(t)') subplot(212); plot(X(:,1),X(:,2)); x(t)10.50-0.5-1050100150t2002503000.30.20.10-0.100.020.040.060.080.10.120.140.160.180.2