数学实验第2次练习题zj1.docx
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数学实验第2次练习题zj1
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。
输入:
>>fora=1:
1:
100
xn=3;
forn=2:
1:
100+a
xN=xn;
xn=(xN+7/xN)/2;
end
vpa(xn,6);
fprintf('x=%E\n',xn)
end
输出:
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
x=2.645751E+000
>>2、设
是否收敛?
若收敛,其值为多少?
精确到17位有效数字。
注:
学号为单号的取
,学号为双号的取
输入:
>>x=0;xN=0;
>>f=inline('1/(x^8)');
>>forx=1:
1:
100
xN=xN+f(x);
vpa(xN,17);
fprintf('x=%i,xN=%E\n',x,xN);
end
输出:
x=1,xN=1.000000E+000
x=2,xN=1.003906E+000
x=3,xN=1.004059E+000
x=4,xN=1.004074E+000
x=5,xN=1.004076E+000
x=6,xN=1.004077E+000
x=7,xN=1.004077E+000
x=8,xN=1.004077E+000
x=9,xN=1.004077E+000
x=10,xN=1.004077E+000
x=11,xN=1.004077E+000
x=12,xN=1.004077E+000
x=13,xN=1.004077E+000
x=14,xN=1.004077E+000
x=15,xN=1.004077E+000
x=16,xN=1.004077E+000
x=17,xN=1.004077E+000
x=18,xN=1.004077E+000
x=19,xN=1.004077E+000
x=20,xN=1.004077E+000
x=21,xN=1.004077E+000
x=22,xN=1.004077E+000
x=23,xN=1.004077E+000
x=24,xN=1.004077E+000
x=25,xN=1.004077E+000
x=26,xN=1.004077E+000
x=27,xN=1.004077E+000
x=28,xN=1.004077E+000
x=29,xN=1.004077E+000
x=30,xN=1.004077E+000
x=31,xN=1.004077E+000
x=32,xN=1.004077E+000
x=33,xN=1.004077E+000
x=34,xN=1.004077E+000
x=35,xN=1.004077E+000
x=36,xN=1.004077E+000
x=37,xN=1.004077E+000
x=38,xN=1.004077E+000
x=39,xN=1.004077E+000
x=40,xN=1.004077E+000
x=41,xN=1.004077E+000
x=42,xN=1.004077E+000
x=43,xN=1.004077E+000
x=44,xN=1.004077E+000
x=45,xN=1.004077E+000
x=46,xN=1.004077E+000
x=47,xN=1.004077E+000
x=48,xN=1.004077E+000
x=49,xN=1.004077E+000
x=50,xN=1.004077E+000
x=51,xN=1.004077E+000
x=52,xN=1.004077E+000
x=53,xN=1.004077E+000
x=54,xN=1.004077E+000
x=55,xN=1.004077E+000
x=56,xN=1.004077E+000
x=57,xN=1.004077E+000
x=58,xN=1.004077E+000
x=59,xN=1.004077E+000
x=60,xN=1.004077E+000
x=61,xN=1.004077E+000
x=62,xN=1.004077E+000
x=63,xN=1.004077E+000
x=64,xN=1.004077E+000
x=65,xN=1.004077E+000
x=66,xN=1.004077E+000
x=67,xN=1.004077E+000
x=68,xN=1.004077E+000
x=69,xN=1.004077E+000
x=70,xN=1.004077E+000
x=71,xN=1.004077E+000
x=72,xN=1.004077E+000
x=73,xN=1.004077E+000
x=74,xN=1.004077E+000
x=75,xN=1.004077E+000
x=76,xN=1.004077E+000
x=77,xN=1.004077E+000
x=78,xN=1.004077E+000
x=79,xN=1.004077E+000
x=80,xN=1.004077E+000
x=81,xN=1.004077E+000
x=82,xN=1.004077E+000
x=83,xN=1.004077E+000
x=84,xN=1.004077E+000
x=85,xN=1.004077E+000
x=86,xN=1.004077E+000
x=87,xN=1.004077E+000
x=88,xN=1.004077E+000
x=89,xN=1.004077E+000
x=90,xN=1.004077E+000
x=91,xN=1.004077E+000
x=92,xN=1.004077E+000
x=93,xN=1.004077E+000
x=94,xN=1.004077E+000
x=95,xN=1.004077E+000
x=96,xN=1.004077E+000
x=97,xN=1.004077E+000
x=98,xN=1.004077E+000
x=99,xN=1.004077E+000
x=100,xN=1.004077E+000
>>
书上习题:
(实验四)
1,2,4,7
(1),8,12(改为:
对例2,取
观察图形有什么变化.),13,14。
练习1编程判断函数
的迭代序列是否收敛.
输入:
>>f=inline('(x-1)/(x+1)');x0=5;
>>fori=1:
40
x0=f(x0);
fprintf('number=%i,value=%g\n',i,x0);
end
输出:
number=1,value=0.666667
number=2,value=-0.2
number=3,value=-1.5
number=4,value=5
number=5,value=0.666667
number=6,value=-0.2
number=7,value=-1.5
number=8,value=5
number=9,value=0.666667
number=10,value=-0.2
number=11,value=-1.5
number=12,value=5
number=13,value=0.666667
number=14,value=-0.2
number=15,value=-1.5
number=16,value=5
number=17,value=0.666667
number=18,value=-0.2
number=19,value=-1.5
number=20,value=5
number=21,value=0.666667
number=22,value=-0.2
number=23,value=-1.5
number=24,value=5
number=25,value=0.666667
number=26,value=-0.2
number=27,value=-1.5
number=28,value=5
number=29,value=0.666667
number=30,value=-0.2
number=31,value=-1.5
number=32,value=5
number=33,value=0.666667
number=34,value=-0.2
number=35,value=-1.5
number=36,value=5
number=37,value=0.666667
number=38,value=-0.2
number=39,value=-1.5
number=40,value=5
>>
>>f=inline('(x-1)/(x+1)');x0=5;
>>x0=7;
>>fori=1:
40x0=f(x0);
x0=f(x0);fprintf('number=%i,value=%g\n',i,x0);
end
number=1,value=-0.142857
number=2,value=7
number=3,value=-0.142857
number=4,value=7
number=5,value=-0.142857
number=6,value=7
number=7,value=-0.142857
number=8,value=7
number=9,value=-0.142857
number=10,value=7
number=11,value=-0.142857
number=12,value=7
number=13,value=-0.142857
number=14,value=7
number=15,value=-0.142857
number=16,value=7
number=17,value=-0.142857
number=18,value=7
number=19,value=-0.142857
number=20,value=7
number=21,value=-0.142857
number=22,value=7
number=23,value=-0.142857
number=24,value=7
number=25,value=-0.142857
number=26,value=7
number=27,value=-0.142857
number=28,value=7
number=29,value=-0.142857
number=30,value=7
number=31,value=-0.142857
number=32,value=7
number=33,value=-0.142857
number=34,value=7
number=35,value=-0.142857
number=36,value=7
number=37,value=-0.142857
number=38,value=7
number=39,value=-0.142857
number=40,value=7
>>
所以认为不收敛
练习2先分别求出分式线性函数
、
的不动点,再编程判断它们的迭代序列是否收敛.
运用上节的收敛定理可以证明:
如果迭代函数在某不动点处具有连续导数且导数值介于-1与1之间,那末取该不动点附近的点为初值所得到的迭代序列一定收敛到该不动点.
(1)解方程
,得到x=-1,是函数f1(x)的不动点。
输入:
>>f=inline('(x-1)/(x+3)');x0=2;x0=-1;
>>fori=1:
30
x0=f(x0);
fprintf('%g,%g\n',i,x0);
end
输出:
1,-1
2,-1
3,-1
4,-1
5,-1
6,-1
7,-1
8,-1
9,-1
10,-1
11,-1
12,-1
13,-1
14,-1
15,-1
16,-1
17,-1
18,-1
19,-1
20,-1
21,-1
22,-1
23,-1
24,-1
25,-1
26,-1
27,-1
28,-1
29,-1
30,-1
>>
(2)解方程
,得到x=-5和3,是函数f2(x)的不动点。
输入:
(-5)
>>f=inline('(-x+15)/(x+1)');x0=2;x0=-5;
>>fori=1:
20
x0=f(x0);
fprintf('%g,%g\n',i,x0);
end
输出:
1,-5
2,-5
3,-5
4,-5
5,-5
6,-5
7,-5
8,-5
9,-5
10,-5
11,-5
12,-5
13,-5
14,-5
15,-5
16,-5
17,-5
18,-5
19,-5
20,-5
>>
输入:
(3)
>>f=inline('(-x+15)/(x+1)');x0=2;x0=3;
>>fori=1:
20
x0=f(x0);
fprintf('%g,%g\n',i,x0);
end
输出:
1,3
2,3
3,3
4,3
5,3
6,3
7,3
8,3
9,3
10,3
11,3
12,3
13,3
14,3
15,3
16,3
17,3
18,3
19,3
20,3
>>
练习4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?
用这种办法近似计算
.
输入:
>>f=inline('(x+2)/(x+1)');x0=1;
>>fori=1:
20
x0=f(x0);
fprintf('%g,%g\n',i,x0);
end
输出:
1,1.5
2,1.4
3,1.41667
4,1.41379
5,1.41429
6,1.4142
7,1.41422
8,1.41421
9,1.41421
10,1.41421
11,1.41421
12,1.41421
13,1.41421
14,1.41421
15,1.41421
16,1.41421
17,1.41421
18,1.41421
19,1.41421
20,1.41421
>>
练习7下列函数的迭代是否会产生混沌?
(1)
=
输入:
>>f=inline('(2*x)*((x<0.5)&(x>0))+2*(1-x)*((x<1)&(x>0.5))');
>>x0=0.4;
>>fori=1:
20
x0=f(x0);
fprintf('%g,%g\n',i,x0);
end
输出:
1,0.8
2,0.4
3,0.8
4,0.4
5,0.8
6,0.4
7,0.8
8,0.4
9,0.8
10,0.4
11,0.8
12,0.4
13,0.8
14,0.4
15,0.8
16,0.4
17,0.8
18,0.4
19,0.8
20,0.4
>>
练习8函数
=
(0
1)称为Logistic映射,试从“蜘蛛网”图观察它取初值为
=0.5产生的迭代序列的收敛性,将观察记录填人下表,若出现循环,请指出它的周期.
表4.3Logistic迭代的收敛性
3.3
3.5
3.56
3.568
3.6
3.84
序列收敛情况
不收敛
循环
周期为2
不收敛
循环
周期为4
不收敛
循环
周期为8
混沌
混沌
不收敛
循环
周期为3
>>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:
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,[0,20]);
>>ezplot(f(x),[0,20]);
>>axis([0,2,0,2]);
>>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:
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,[0,20]);
>>ezplot(f(x),[0,20]);
>>axis([0,2,0,2]);
>>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:
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,[0,20]);
>>ezplot(f(x),[0,20]);
>>axis([0,2,0,2]);
>>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:
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,[0,20]);
>>ezplot(f(x),[0,20]);
>>axis([0,2,0,2]);
>>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:
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,[0,20]);
>>