南邮数学实验问题详解.docx
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南邮数学实验问题详解.docx
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南邮数学实验问题详解
第一次练习题
1、求
的所有根。
>>x=-5:
0.01:
5;y=exp(x)-3*x.^2;plot(x,y);gridon
>>fsolve('exp(x)-3*x.^2',-1)
Equationsolved.
fsolvecompletedbecausethevectoroffunctionvaluesisnearzero
asmeasuredbythedefaultvalueofthefunctiontolerance,and
theproblemappearsregularasmeasuredbythegradient.
ans=
-0.4590
>>fsolve('exp(x)-3*x.^2',1)
Equationsolved.
fsolvecompletedbecausethevectoroffunctionvaluesisnearzero
asmeasuredbythedefaultvalueofthefunctiontolerance,and
theproblemappearsregularasmeasuredbythegradient.
ans=
0.9100
>>fsolve('exp(x)-3*x.^2',4)
Equationsolved.
fsolvecompletedbecausethevectoroffunctionvaluesisnearzero
asmeasuredbythedefaultvalueofthefunctiontolerance,and
theproblemappearsregularasmeasuredbythegradient.
ans=
3.7331
2、求下列方程的根。
1)
2)
3)
1)>>p=[1,0,0,0,5,1];r=roots(p)
r=
1.1045+1.0598i
1.1045-1.0598i
-1.0045+1.0609i
-1.0045-1.0609i
-0.1999
2)>>x=-10:
0.01:
10;y=x.*sin(x)-1/2;plot(x,y);gridon
>>fsolve('x.*sin(x)-1/2',-6)
Equationsolved.
fsolvecompletedbecausethevectoroffunctionvaluesisnearzero
asmeasuredbythedefaultvalueofthefunctiontolerance,and
theproblemappearsregularasmeasuredbythegradient.
ans=
-6.3619
>>fsolve('x.*sin(x)-1/2',-4)
Equationsolved.
fsolvecompletedbecausethevectoroffunctionvaluesisnearzero
asmeasuredbythedefaultvalueofthefunctiontolerance,and
theproblemappearsregularasmeasuredbythegradient.
ans=
-2.9726
>>fsolve('x.*sin(x)-1/2',2)
Equationsolved.
fsolvecompletedbecausethevectoroffunctionvaluesisnearzero
asmeasuredbythedefaultvalueofthefunctiontolerance,and
theproblemappearsregularasmeasuredbythegradient.
ans=
0.7408
3)>>x=-3:
0.01:
3;y=sin(x).*cos(x)-x.^2;plot(x,y);gridon
>>fsolve('sin(x).*cos(x)-x.^2',-1)
Equationsolved.
fsolvecompletedbecausethevectoroffunctionvaluesisnearzero
asmeasuredbythedefaultvalueofthefunctiontolerance,and
theproblemappearsregularasmeasuredbythegradient.
ans=
-6.8434e-010
>>fsolve('sin(x).*cos(x)-x.^2',1)
Equationsolved.
fsolvecompletedbecausethevectoroffunctionvaluesisnearzero
asmeasuredbythedefaultvalueofthefunctiontolerance,and
theproblemappearsregularasmeasuredbythegradient.
ans=
0.7022
3、求解下列各题:
1)
2)
3)
4)
5)
6)
1)>>symsx
>>limit((x-sin(x))/x^3)
ans=
1/6
2)>>diff(exp(x)*cos(x),10)
ans=
-32*exp(x)*sin(x)
3)>>int(exp(x^2),x,0,1/2)
ans=
(pi^(1/2)*erfi(1/2))/2
>>vpa(ans,17)
ans=
0.
4)>>int(x^4/(25+4*x^2),x)
ans=
(125*atan((2*x)/5))/32-(25*x)/16+x^3/12
5)>>taylor(sqrt(1+x),9,x,0)
ans=
-(429*x^8)/32768+(33*x^7)/2048-(21*x^6)/1024+(7*x^5)/256-(5*x^4)/128+x^3/16-x^2/8+x/2+1
6)>>diff(exp(sin(1/x)),3)
ans=
(cos(1/x)*exp(sin(1/x)))/x^6-(6*cos(1/x)*exp(sin(1/x)))/x^4+(6*sin(1/x)*exp(sin(1/x)))/x^5-(6*cos(1/x)^2*exp(sin(1/x)))/x^5-(cos(1/x)^3*exp(sin(1/x)))/x^6+(3*cos(1/x)*sin(1/x)*exp(sin(1/x)))/x^6
>>subs(ans,2)
ans=
-0.5826
4、求矩阵
的逆矩阵
及特征值和特征向量。
>>A=[-2,1,1;0,2,0;-4,1,3];
>>inv(A)
ans=
-1.50000.50000.5000
00.50000
-2.00000.50001.0000
>>eig(A)
ans=
-1
2
2
>>[P,D]=eig(A)
P=
-0.7071-0.24250.3015
000.9045
-0.7071-0.97010.3015
D=
-100
020
002
5、已知
分别在下列条件下画出
的图形:
、
(1)>>x=-10:
0.01:
10;
>>y1=1/sqrt(2*pi).*exp(-x.^2/2);
>>y2=1/sqrt(2*pi).*exp(-(x+1).^2/2);
>>y3=1/sqrt(2*pi).*exp(-(x-1).^2/2);
>>plot(x,y1,x,y2,x,y3)
(2)>>x=-10:
0.01:
10;
>>y1=1/sqrt(2*pi).*exp(-x.^2/2);
>>y2=1/(sqrt(2*pi)*2).*exp(-x.^2/(2*2^2));
>>y3=1/(sqrt(2*pi)*4).*exp(-x.^2/(2*4^2));
>>plot(x,y1,x,y2,x,y3)
6、画下列函数的图形:
(1)
(2)
(3)
(1)>>ezmesh('u*sin(t)','u*cos(t)','t/4',[0,20,0,2]);axisequal;
(2)>>ezmesh('x','y','sin(x*y)',[0,3,0,3]);axisequal;
(3)>>ezmesh('sin(t)*(3+cos(u))','cos(t)*(3+cos(u))','sin(u)',[0,2*pi,0,2*pi]);axisequal;
第二次练习题
1、设
,数列
是否收敛?
若收敛,其值为多少?
精确到6位有效数字。
>>f=inline('(x+7/x)/2');
>>x0=3;
>>fori=1:
20
x0=f(x0);
fprintf('%g,%g\n',i,x0);
end
1,2.66667
2,2.64583
3,2.64575
4,2.64575
5,2.64575
6,2.64575
7,2.64575
8,2.64575
9,2.64575
10,2.64575
11,2.64575
12,2.64575
13,2.64575
14,2.64575
15,2.64575
16,2.64575
17,2.64575
18,2.64575
19,2.64575
20,2.64575
2、设
是否收敛?
若收敛,其值为多少?
精确到17位有效数字,
>>f=inline('1/n^8');
>>x1=0;
>>fori=1:
150
x1=x1+f(i);
fprintf('%g,%1.16f\n',i,x1);
end
1,1.00000
2,1.00000
3,1.02759
4,1.93384
5,1.93384
6,1.35192
7,1.00448
8,1.46896
9,1.52626
10,1.52626
11,1.03365
12,1.60168
13,1.19115
14,1.95150
15,1.96993
16,1.25300
17,1.58835
18,1.66281
19,1.55085
20,1.45711
21,1.10101
22,1.92331
23,1.20027
24,1.10874
25,1.76410
26,1.24297
27,1.59704
28,1.86174
29,1.06164
30,1.21406
31,1.33130
32,1.42224
33,1.49334
34,1.54934
35,1.59375
36,1.62919
37,1.65766
38,1.68066
39,1.69933
40,1.71459
41,1.72711
42,1.73744
43,1.74599
44,1.75311
45,1.75906
46,1.76406
47,1.76826
48,1.77181
49,1.77483
50,1.77738
51,1.77956
52,1.78142
53,1.78302
54,1.78440
55,1.78560
56,1.78664
57,1.78753
58,1.78831
59,1.78900
60,1.78960
61,1.79011
62,1.79057
63,1.79097
64,1.79133
65,1.79164
66,1.79193
67,1.79217
68,1.79239
69,1.79259
70,1.79277
71,1.79293
72,1.79306
73,1.79319
74,1.79330
75,1.79339
76,1.79348
77,1.79357
78,1.79364
79,1.79370
80,1.79377
81,1.79381
82,1.79386
83,1.79390
84,1.79395
85,1.79399
86,1.79404
87,1.79406
88,1.79408
89,1.79410
90,1.79413
91,1.79415
92,1.79417
93,1.79419
94,1.79421
95,1.79424
96,1.79426
97,1.79428
98,1.79430
99,1.79430
100,1.79430
101,1.79430
102,1.79430
103,1.79430
104,1.79430
105,1.79430
106,1.79430
107,1.79430
108,1.79430
109,1.79430
110,1.79430
111,1.79430
112,1.79430
113,1.79430
114,1.79430
115,1.79430
116,1.79430
117,1.79430
118,1.79430
119,1.79430
120,1.79430
121,1.79430
122,1.79430
123,1.79430
124,1.79430
125,1.79430
126,1.79430
127,1.79430
128,1.79430
129,1.79430
130,1.79430
131,1.79430
132,1.79430
133,1.79430
134,1.79430
135,1.79430
136,1.79430
137,1.79430
138,1.79430
139,1.79430
140,1.79430
141,1.79430
142,1.79430
143,1.79430
144,1.79430
145,1.79430
146,1.79430
147,1.79430
148,1.79430
149,1.79430
150,1.79430
3、编程判断函数f(x)=
的迭代序列是否收敛。
>>f=inline('(x-1)/(x+1)');
>>x0=2;
>>fori=1:
20
x0=f(x0);
fprintf('%g,%g\n',i,x0);
end
1,0.333333
2,-0.5
3,-3
4,2
5,0.333333
6,-0.5
7,-3
8,2
9,0.333333
10,-0.5
11,-3
12,2
13,0.333333
14,-0.5
15,-3
16,2
17,0.333333
18,-0.5
19,-3
20,2
4、先分别求出线性函数
=
的不动点,再编程判断它们的迭代序列是否收敛。
>>solve('(x-1)/(x+3)-x')
ans=
-1
>>solve('(-x+15)/(x+1)-x')
ans=
3
-5
待定
5、能否找到一个分式线性函数
使它产生的迭代序列收敛到给定的数,用这种方法近似计算
。
>>f=inline('(2+x)/(x+1)');
>>x0=2;
>>fori=1:
20
x0=f(x0);
fprintf('%g,%f\n',i,x0);
end
1,1.333333
2,1.428571
3,1.411765
4,1.414634
5,1.414141
6,1.414226
7,1.414211
8,1.414214
9,1.414213
10,1.414214
11,1.414214
12,1.414214
13,1.414214
14,1.414214
15,1.414214
16,1.414214
17,1.414214
18,1.414214
19,1.414214
20,1.414214
6、函数f(x)=ax(1-x)(0≤x≤1)称为Logistic映射,试从“蜘蛛网”图观察它的取初值为
0.5产生的迭代序列的收敛性,将观察记录填入表中,若出现循环,请指出它的周期。
α
3.3
3.5
3.56
3.568
3.6
3.84
序列收敛情况
>>f=inline('3.3*x*(1-x)');
x=[];
y=[];
x
(1)=0.5;
y
(1)=0;x
(2)=x
(1);y
(2)=f(x
(1));
fori=1:
100
x(1+2*i)=y(2*i);
x(2+2*i)=x(1+2*i);
y(1+2*i)=x(1+2*i);
y(2+2*i)=f(x(2+2*i));
end
plot(x,y,'r');
holdon;
symsx;
ezplot(x,[0,1]);
ezplot(f(x),[0,1]);
axis([0,1,0,1]);
holdoff
>>f=inline('3.5*x*(1-x)');
x=[];
y=[];
x
(1)=0.5;
y
(1)=0;x
(2)=x
(1);y
(2)=f(x
(1));
fori=1:
100
x(1+2*i)=y(2*i);
x(2+2*i)=x(1+2*i);
y(1+2*i)=x(1+2*i);
y(2+2*i)=f(x(2+2*i));
end
plot(x,y,'r');
holdon;
symsx;
ezplot(x,[0,1]);
ezplot(f(x),[0,1]);
axis([0,1,0,1]);
holdoff
>>f=inline('3.56*x*(1-x)');
x=[];
y=[];
x
(1)=0.5;
y
(1)=0;x
(2)=x
(1);y
(2)=f(x
(1));
fori=1:
100
x(1+2*i)=y(2*i);
x(2+2*i)=x(1+2*i);
y(1+2*i)=x(1+2*i);
y(2+2*i)=f(x(2+2*i));
end
plot(x,y,'r');
holdon;
symsx;
ezplot(x,[0,1]);
ezplot(f(x),[0,1]);
axis([0,1,0,1]);
holdoff
>>f=inline('3.568*x*(1-x)');
x=[];
y=[];
x
(1)=0.5;
y
(1)=0;x
(2)=x
(1);y
(2)=f(x
(1));
fori=1:
100
x(1+2*i)=y(2*i);
x(2+2*i)=x(1+2*i);
y(1+2*i)=x(1+2*i);
y(2+2*i)=f(x(2+2*i));
end
plot(x,y,'r');
holdon;
symsx;
ezplot(x,[0,1]);
ezplot(f(x),[0,1]);
axis([0,1,0,1]);
holdoff
>>f=inline('3.6*x*(1-x)');
x=[];
y=[];
x
(1)=0.5;
y
(1)=0;x
(2)=x
(1);y
(2)=f(x
(1));
fori=1:
100
x(1+2*i)=y(2*i);
x(2+2*i)=x(1+2*i);
y(1+2*i)=x(1+2*i);
y(2+2*i)=f(x(2+2*i));
end
plot(x,y,'r');
holdon;
symsx;
ezplot(x,[0,1]);
ezplot(f(x),[0,1]);
axis([0,1,0,1]);
holdoff
>>f=inline('3.84*x*(1-x)');
x=[];
y=[];
x
(1)=0.5;
y
(1)=0;x
(2)=x
(1);y
(2)=f(x
(1));
fori=1:
100
x(1+2*i)=y(2*i);
x(2+2*i)=x(1+2*i);
y(1+2*i)=x(1+2*i);
y(2+2*i)=f(x(2+2*i));
end
plot(x,y,'r');
holdon;
symsx;
ezplot(x,[0,1]);
ezplot(f(x),[0,1]);
axis([0,1,0,1]);
holdoff
7、由函数f(x,y)=y-sgnx
与g(x,y)=a-x构成的二维迭代称为Martin迭代,取
,b=2,c=-300观察图形有什么变化.
functionMartin(a,b,c,N)
f=(x,y)(y-sign(x)*sqrt(abs(b*x-c)));
g=(x)(a-x);
m=[0;0];
forn=1:
N
m(:
n+1)=[f(m(1,n),m(2,n)),g(m(1,n))];
end
plot(m(1,:
),m(2,:
),'kx');
axisequal
>>Martin(4.5,2,-300,5000)
>>Martin(25,2,-300,5000)
>>Martin(55,2,-300,5000)
>>Martin(120,2,-300,5000)
8、取参数a,b,c为其他的值会得到什么图形?
a
b
c
-1000
0.1
-10
0.4
1
0
90
30
10
10
-10
100
-200
-4
-80
-137
17
4
10
100
-10
>>Martin(-1000,0.1,-10,5000)
>>
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