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. . . . . . . . . . . . . . . . 1 . .
1 4 3(yn-yn-2)/h-f¡¯(xn)
f¡¯(x=n-1)
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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
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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»¿éÊÇÐÁÆÕɹ«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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