数值分析习题 (1)

数值分析习题

计算实习2

（1）利用二分法求方程f(x)=(x^2+1)(x-1)^5=0的根x=1，function [x,n]=erfenfa(fun,a,b,eps)

N=log((b-a)/eps)-1;

n=0;

x=1/2*(a+b);

while (feval(fun,x)~=0)&(n<=N)

if feval(fun,x)*feval(fun,a)<0

b=x;

else

a=x;

end

x=1/2*(a+b);

n=n+1;

end

fun=inline('(x^2+1)*(x-1)^5','x');

a=0;

b=1;

eps=1e-5;

[x,n]=erfenfa(fun,a,b,eps);

x

n

x =

0.9998

n =

11

（2）（2）利用牛顿迭代法求f(x)=(x^2+1)(x-1)^5=0的根x=1。function [x,n]=newton(fun,dfun,x0,eps)

x=x0-feval(fun,x0)/feval(dfun,x0);

n=1;

while abs(x-x0)>=eps

x0=x;

x=x0-feval(fun,x0)/feval(dfun,x0);

n=n+1;

end

clear

clc

fun=inline('(x^2+1)*(x-1)^5','x');

dfun=inline('2*x*(x-1)^5+5*(x^2+1)*(x-1)^4','x');

x0=0;

eps=1e-5;

[x,n]=newton(fun,dfun,x0,eps);

x

n

x =

1.0000

n =

45

（3）(3)利用割线法求方程f(x)=(x^2+1)(x-1)^5=0的根x=1。function [x,n]=gexianfa(fun,x0,x1,eps)

x=x1-feval(fun,x1)/(feval(fun,x1)-feval(fun,x0))*(x1-x0);

n=1;

while abs(x-x1)>=eps

x0=x1;

x1=x;

x=x1-feval(fun,x1)/(feval(fun,x1)-feval(fun,x0))*(x1-x0);

n=n+1;

end

fun=inline('(x^2+1)*(x-1)^5','x');

x0=0;

x1=1;

eps=1e-5;

[x,n]=gexianfa(fun,x0,x1,eps);

x

n

x =

1

n =

1

（4）（4）利用史蒂芬森法求方程f(x)=(x^2+1)(x-1)^5=0的根x=1。

function [x,n]=shidifensen(fun,dfun,x0,eps)

x=x0-(x0-feval(fun,x0)/feval(dfun,x0)-x0)^2/(x0-feval(fun,x0)/feval(dfun,x0)-feval(f un...

,x0-feval(fun,x0)/feval(dfun,x0))/feval(dfun,x0-feval(fun,x0)/feval(dfun,x0))-2*(x0-feval(fun,x0)/feval(dfun,x0))+x0);

n=1;

while abs(x-x0)>=eps

x0=x;

x=x0-(x0-feval(fun,x0)/feval(dfun,x0)-x0)^2/((x0-feval(fun,x0)/feval(dfun,x0))-feval( fun...

,(x0-feval(fun,x0)/feval(dfun,x0))/feval(dfun,x0-feval(fun,x0)/feval(dfun,x0))-2*(x0 -feval(fun,x0)/feval(dfun,x0))+x0));

n=n+1;

end

fun=inline('(x^2+1)*(x-1)^5','x');

dfun=inline('2*x*(x-1)^5+5*(x^2+1)*(x-1)^4','x');

x0=0.5;

eps=1e-5;

[x,n]=shidifensen(fun,dfun,x0,eps);

x

n

x =

0.9805

n =

2

计算实习4

（1）用复合辛普森公式求积分dx nx x )cos(1

12?-的结果，并n=1;n=10;n=100,n=500

fun=inline('x.^2.*cos(10.*x)','x');

a=-1;

b=1;

n=10; %相应的将n 改为1，100，500

m=2.*n;

h=(b-a)/m;

x=a:h:b;

sum1=0.0;

sum2=0.0;

for k=1:10

x(2.*k-1)=a+(2.*k-1).*h;

sum1=sum1+feval(fun,x(2.*k-1));

end

for i=1:9

x(2.*i)=a+(2.*i).*h;

sum2=sum2+feval(fun,x(2.*i));

end

S=h/3.*(fun(a)+fun(b))+4.*h/3.*sum1+2.*h/3.*sum2

S =

-0.1401

(2)利用龙贝格积分算法求n=1,10,100,500时，积分dx nx x )cos(112?-的值

fun=inline('x.^2.*cos(x)','x');

a=-1;

b=1;

n=1;%相应的将n 改为10，100，500

esp=1e-6;

h=(b-a)/n;

T(1)=h/2.*(fun(a)+fun(b));

m=1;

sum=0.0;

j=1:2^(m-1);

h=h/2;

sum=sum+h.*feval(fun,a+(2.*j-1).*h); S(1)=1/2.*T(1)+sum;

for i=1:m

S(i+1)=S(i)+(S(i)-T(i))/4.^i-1; end

if abs(S(m+1)-T(m))<=esp,break;

T=S;

m=m+1;

end

S(m+1)

ans =

-0.5948

ans =

-1.1552

ans =

-0.9930

ans =

-1.0035

2,用高斯—勒让德求积公式计算积分

1

s i n x

dx

x

function f=y y=sin(x)/x format long;

fun=inline('sin(x)/x','x');

ga1=2;

x1=0;

ga2=[1,1];

x2=[-0.5773503,0.5773503];

ga3=[0.5555556,0.8888889,0.5555556];

x3=[-0.77445967,0,0.7745967];

ga4=[0.3478548,0.6521452,0.6521452,0.3478548]; x4=[-0.8611363,-0.3399810,0.3399810,0.8611363]; a=0;b=1;

ga=ga3;x=x3;

n=length(ga);

s=0;

for i=1:n

t(i)=(b-a)/2*x(i)+(b+a)/2;

s=s+(b-a)/2*feval(fun,t(i))*ga(i);

end

s

s =

0.94608246570707

计算实验5