高斯求积公式 matlab

1. 分别用三点和四点Gauss-Chebyshev公式计算积分

并与准确积分值2arctan4比较误差。若用同样的三点和四点Gauss-Legendre公式计算，也给出误差比较结果。

2*atan(4)

ans =

2.6516

Gauss-Chebyshev：

function I = gausscheby(f,a,b,n)

syms t;

t= findsym(sym(f));

ta = (b-a)/2;

tb = (a+b)/2;

switch n

case 3

I=pi/n*ta*(subs(sym(f),t,ta*cos(pi/(2*n))+tb)*sqrt(1-cos(pi/(2*n))^2)+...

subs(sym(f),t,ta*cos(3*pi/(2*n))+tb)*sqrt(1-cos(3*pi/(2*n))^2)+...

subs(sym(f),t,ta*cos(5*pi/(2*n))+tb)*sqrt(1-cos(5*pi/(2*n))^2));

case 4

I=pi/n*ta*(subs(sym(f),t,ta*cos(pi/(2*n))+tb)*sqrt(1-cos(pi/(2*n))^2)+...

subs(sym(f),t,ta*cos(3*pi/(2*n))+tb)*sqrt(1-cos(3*pi/(2*n))^2)+...

subs(sym(f),t,ta*cos(5*pi/(2*n))+tb)*sqrt(1-cos(5*pi/(2*n))^2)+...

subs(sym(f),t,ta*cos(7*pi/(2*n))+tb)*sqrt(1-cos(7*pi/(2*n))^2));

end

I=simplify(I);

I=vpa(I,6);

syms x

f=1/(1+x^2);

a=-4;b=4;

n=3;

y=gausscheby(f,a,b,n)

y =

4.511

N=4:

y =

1.90041

Gauss-Legendre：

function I = IntGaussLegen(f,a,b,n)

syms t;

t= findsym(sym(f));

ta = (b-a)/2;

tb = (a+b)/2;

switch n

case 0,

I=2*ta*subs(sym(f),t,tb);

case 1,

I=ta*(subs(sym(f),t,ta*0.5773503+tb)+...

subs(sym(f),t,-ta*0.5773503+tb));

case 2,

I=ta*(0.55555556*subs(sym(f),t,ta*0.7745967+tb)+...

0.55555556*subs(sym(f),t,-ta*0.7745967+tb)+...

0.88888889*subs(sym(f),t,tb));

case 3,

I=ta*(0.3478548*subs(sym(f),t,ta*0.8611363+tb)+...

0.3478548*subs(sym(f),t,-ta*0.8611363+tb)+...

0.6521452*subs(sym(f),t,ta*0.3398810+tb) +...

0.6521452*subs(sym(f),t,-ta*0.3398810+tb));

case 4,

I=ta*(0.2369269*subs(sym(f),t,ta*0.9061793+tb)+...

0.2369269*subs(sym(f),t,-ta*0.9061793+tb)+...

0.4786287*subs(sym(f),t,ta*0.5384693+tb) +...

0.4786287*subs(sym(f),t,-ta*0.5384693+tb)+...

0.5688889*subs(sym(f),t,tb));

case 5,

I=ta*(0.1713245*subs(sym(f),t,ta*0.9324695+tb)+...

0.1713245*subs(sym(f),t,-ta*0.9324695+tb)+...

0.3607616*subs(sym(f),t,ta*0.6612094+tb)+...

0.3607616*subs(sym(f),t,-ta*0.6612094+tb)+...

0.4679139*subs(sym(f),t,ta*0.2386292+tb)+...

0.4679139*subs(sym(f),t,-ta*0.2386292+tb));

end

I=simplify(I);

I=vpa(I,6);

y =

2.04798

N=4:

y =

3.08862

2. 分别用三点和四点Gauss-Lagurre公式计算积分

function I = GaussLagurre(f,n)

syms t;

t= findsym(sym(f));

switch n

case 2

I=0.7110930*subs(sym(f),t,0.4157746)+...

0.2785177*subs(sym(f),t,2.2942804)+...

0.0103893*subs(sym(f),t,6.2899451);

case 3

I=0.6031541*subs(sym(f),t,0.3225477)+...

0.3574187*subs(sym(f),t,1.7457611)+...

0.0388879*subs(sym(f),t,4.5366203) +...

0.0005393*subs(sym(f),t,9.3950710);

end

I=simplify(I);

I=vpa(I,6);

syms x

f=exp(-10*x)*sin(x);

f=f./exp(-x);

a=0;b=inf;

n=2;

y= GaussLagurre(f,n)

y =

0.00680897

N=4:

y =

0.0104892

3. 设，分别取，，用以下三个公式计算，

列表比较三个公式的计算误差，从误差可以得出什么结论？

function [df1,df2,df3,w1,w2,w3]=MidPoint(func,a)

if (nargin == 3 && h == 0.0)

disp('h不能为0');

return;

end

for k=1:6

h=1/10^k;

y0=subs(sym(func), findsym(sym(func)),a);

y1 = subs(sym(func), findsym(sym(func)),a+h);

y2 = subs(sym(func), findsym(sym(func)),a-h);

df1(k) = (y1-y0)/h;

df2(k) = (y1-y2)/(2*h);

y3=subs(sym(func), findsym(sym(func)),a+2*h);

y4=subs(sym(func), findsym(sym(func)),a-2*h);

df3(k)=(y4-8*y2+8*y1-y3)/(12*h);

w1(k)=1/a-df1(k);

w2(k)=1/a-df2(k);

w3(k)=1/a-df3(k);

end

df1=simplify(df1); df1=vpa(df1,6);

df2=simplify(df2); df2=vpa(df2,6);

df3=simplify(df3); df3=vpa(df3,6);

w1=simplify(w1); w1=vpa(w1,6);

w2=simplify(w2); w2=vpa(w2,6);

w3=simplify(w3); w3=vpa(w3,6);

syms x

f=log(x);

a=0.7;

[y1,y2,y3,w1,w2,w3]=MidPoint(f,a)

y1 =

[ 1.33531, 1.41846, 1.42755, 1.42847, 1.42856, 1.42857]

y2 =

[ 1.43841, 1.42867, 1.42857, 1.42857, 1.42857, 1.42857]

y3 =

[ 1.42806, 1.42857, 1.42857, 1.42857, 1.42857, 1.42857]

w1 =

[ 0.0932575, 0.0101079, 0.00101944, 0.000102031, 0.0000102043, 0.00000101738]

w2 =

[ -0.00983893, -0.0000971936, -9.71814e-7, -9.72193e-9, 1.92131e-10, -3.02526e-9]

w3 =

[ 0.000513166, 4.76342e-8, 8.04334e-12, -1.10191e-10, 3.37991e-10, -4.5859e-9]