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¡°Simpson ·¨ÇóÊýֵ΢·Ö¡± Ë㷨˵Ã÷

ÐÁÆÕÉ­Êýֵ΢·ÖÊÇÓÃÀ´ÇóµÈ¾à½ÚµãÔڽڵ㴦µÄµ¼ÊýµÄ£¬ÐÁÆÕÉ­Êýֵ΢·Ö¹«Ê½ÈçÏ£º 4 1 f¡¯(x1) 3(y2-y0)/h-f¡¯(x0) 1 4 1 f¡¯(x2) 3(y3-y1)/h 1 4 . 3(y4-y2)/h f¡¯(x3)

. . . . . . . . . . . . . . . . 1 . .

1 4 3(yn-yn-2)/h-f¡¯(xn)

f¡¯(x=n-1)

ÆäÖУ¬yn=f(xn),xn=xn+nh

Èç¹û¶Ëµãµ¼ÊýÖµ£­f(x0)ºÍ-f¡¯(xn)δ֪£¬Ôò½«ËüÃǵÄÖеã΢·Ö¹«Ê½½üËÆ£¬ÕâʱµÄÐÁÆÕÉ­Êýֵ΢·Ö¹«Ê½Îª£º

2 0 f¡¯(x1) y2-y0/h

1 4 1 f¡¯(x2) 3(y3-y1)/h 1 4 . 3(y4-y2)/h f¡¯(x3) . . . . . . . . . . . .

. . . . 1 . . 0 .2 f¡¯(x=n-1) (yn-yn-2)/h

ÔÚMATLABÖУ¬±à³ÌʵÏÖµÄÐÁÆÕÉ­ÊýÖµ·¨£¨ÓÃÓÚÒÑÖªº¯Êý±í´ïʽ£©º¯ÊýΪ£ºCISimpson¡£ ¹¦ÄÜ£ºÐÁÆÕÉ­ÊýÖµ·¨ÇóÒÑÖªº¯ÊýÔÚijµãµÄµ¼ÊýÖµ¡£ µ÷Óøñʽ£ºdf=CISimpson(func,x0,n,h)¡£ ÆäÖУ¬funcΪº¯ÊýÃû£» xoΪÇ󵼵㣻

nΪ½«ÒÑÖªº¯ÊýÀëÉ¢µÄÊý¾ÝµãÊý£»

hΪÀëÉ¢²½³¤£» dfΪµ¼ÊýÖµ¡£

ÐÁÆÕÉ­Êýֵ΢·Ö·¨£¨ÊÊÓÃÓÚÒÑÖªº¯Êý±í´ïʽ£©µÄMATLAB³ÌÐò´úÂëÈçÏ£º function df=CISimpson(func,x0,n,h)

%ÐÁÆÕÉ­ÊýÖµ·¨ÇóÒÑÖªº¯ÊýfuncÔÚx0µãµÄµ¼ÊýÖµ %º¯ÊýÃû£ºfunc %Ç󵼵㣺x0

%½«ÒÑÖªº¯ÊýÀëÉ¢µÄÊý¾ÝµãÊý£ºn %ÀëÉ¢²½³¤£ºh %µ¼ÊýÖµ£ºdf

= = if nargin ==2 %ÒÔÏÂÊDzÎÊýµÄÅжϹý³Ì h=0.1; n=5; else

if(nargin ==3) if (n<5)

disp('n²»ÄÜСÓÚ5!'); return; else

h=0.1; end

else (nargin ==4 && h ==0.0) disp('h²»ÄÜΪ0!'); return; end end

for(i=1:n) %ÕâÊǸöÑ­»·¼ÆËã½ÚµãµÄº¯ÊýÖµ if (mod(n,2) ==0)

y(i)=subs(sym(func),findsym(sym(func)),x0+(i-n/2)*h); else

y(i)=subs(sym(func),findsym(sym(func)),x0+(i-(n+1)/2)*h); end end

f(1)=(y(3)-y(1))/(2*h);

f(2)=(y(n)-y(n-2))/(2*h); %ÕâÁ½ÐÐÓÃÖÐÐÄ΢·Ö·¨¸ø³ö¶ËµãµÄµ¼Êý b(1:n-2,1)=zeros(n-2,1); b(1,1)=3*(y(3)-y(1))/h-f(1); b(n-2,1)=3*(y(n)-y(n-2))/h-f(2); for (i=2:(n-3))

b(i,1)=3*(y(i+2)-y(i))/h;

end %ÕâÒ»¿éÊÇÐÁÆÕÉ­¹«Ê½µÄÓұߵÄÁÐÏòÁ¿ for(i=1:n-2) for(j=1:n-2)

if((i ==j+1) ||(j ==i+1)) A(i,j)=1; else if(i ==j) A(i,j)=4; end end end

end %ÕâÒ»¿éÊÇϵÊý¾ØÕó [Q,R]=qr(A);

DF = R\\(Q\\b); %ÓÃQR·Ö½â·¨Çó½â if(mod(n,2) ==0) df = DF(n/2); else

df=DF((n+1)/2);

end %ÕâÀïÊÇÇó³öx0´¦µÄµ¼ÊýÖµ

ÐÁÆÕÉ­Êýֵ΢·Ö·¨£¨ÊÊÓÃÓÚÊý¾ÝÏòÁ¿£©µÄÁíÒ»¸öMATLAB³ÌÐò´úÂëÈçÏ£º function df=DISimpson(X,Y,n,p)

%ÐÁÆÕÉ­ÊýÖµ·¨Çón¸öÊý¾ÝµãÔÚµÚp¸öµã´¦µÄµ¼ÊýÖµ %ÀëÉ¢Êý¾ÝµÄx×ø±êÏòÁ¿£ºX %ÀëÉ¢Êý¾ÝµÄy×ø±êÏòÁ¿£ºY %Êý¾ÝµãÊý£ºn %ÒªÇóµ¼ÊýµÄµã:p %µ¼ÊýÖµ:df if n<5

disp('n²»ÄÜСÓÚ5!'); return; end

if p==0

disp('n²»ÄÜСÓÚ0!'); return; end

h=X(2)-X(1);

xx=linspace(X(1),X(n),h); if(xx~=X)

disp('½ÚµãÖ®¼ä²»ÊǵȾàµÄ£¡'); return; end

f(1)=(Y(3)-Y(1))/(2*h); f(2)=(Y(n)-Y(n-2))/(2*h); b(1,1)=3*(Y(3)-Y(1))/h-f(1); b(n-2,1)=3*(Y(n)-Y(n-2))/h-f(2); for(i=2:(n-3))

b(i,1)=3*(Y(i+2)-Y(i))/h; end

for(i=1:n-2) for(j=1:n-2)

if((i==j+1)||(j==i+1)) A(i,j)=1; else if(i==j) A(i,j)=4; end end end end

[Q,R]=qr(A); DF=R\\(Q\\b); if(p==1) df=f(1); else

df=DF(p-1);

end

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