quad2dg.m

工具箱 (数值积分工具箱) matlab 中使用

function [int, tol1,k] = quad2dg(fun,xlow,xhigh,ylow,yhigh,tol,p1,p2,p3,p4,p5,p6,p7,p8,p9)
%usage:  [int tol] = quad2dg('Fun',xlow,xhigh,ylow,yhigh)
%or
%        [int tol] = quad2dg('Fun',xlow,xhigh,ylow,yhigh,reltol,p1,p2,...)
%
%This function is similar to QUAD or QUAD8 for 2-dimensional integration,
%but it uses a Gaussian quadrature integration scheme.  
%       int     -- value of the integral
%       tol     -- absolute tolerance abs( int-intold)
%       Fun     -- Fun(x,y) (function to be integrated)
%       xlow    -- lower x limit of integration
%       xhigh   -- upper x limit of integration
%       ylow    -- lower y limit of integration
%       yhigh   -- upper y limit of integration
%       reltol     -- relative tolerance parameter (optional)
%
%Note that if there are discontinuities the region of integration 
%should be broken up into separate pieces.  And if there are singularities,
%a more appropriate integration quadrature should be used 
%(such as the Gauss-Chebyshev for a specific type of singularity).
%
%modified by Per A. Brodtkorb 17.11.98 : 
% -accept multiple integrations limits
% -optimized by saving the weights as global constants and only  
%  computing the integrals which did not converge 
% -enabled the integration of directly given functions enclosed in
%  parenthesis 
%
%Example:  integration from 0 to 2 and from 2 to 4 for both x and y is done by:
%
%     quad2dg('(x.^2.*y)',[0 2],[2 4],[0 2],[2 4])
%

global bpx2 bpy2 wfxy2;


if isempty(bpx2)| isempty(bpy2)| isempty(wfxy2)
  [bpx2,bpy2,wfxy2] = grule2d(2,2);
end


if exist('tol')~=1,
  tol=1e-3;
elseif isempty(tol),
  tol=1e-3;
end
[errorcode ,xlow,xhigh,ylow,yhigh]=distchck(4,xlow,xhigh,ylow,yhigh);
if errorcode > 0
  error('Requires non-scalar arguments to match in size.');
end
%size(xlow),size(xhigh),  size(ylow),size(yhigh)  
%if prod(size(xlow))==1,
%  xlow=xlow(ones(size(xhigh)));;
%elseif prod(size(xhigh))==1,
%  xhigh=xhigh(ones(size(xlow)));;
%elseif any( size(xhigh)~=size(xlow) )
%  error('The input must have equal size!')
%end


[N M]=size(xlow);
%setup mapping parameters &  make sure they all are column vectors
xlow=xlow(:); xhigh=xhigh(:);ylow=ylow(:);yhigh=yhigh(:); 
nk=N*M;


%Map to x
qx=(xhigh-xlow)/2;
px=(xhigh+xlow)/2;
x=permute(qx(:,ones(1,2), ones(1,2) ),[ 2 3 1]).*bpx2(:,:,ones(1,nk)) ...
    +permute(px(:,ones(1,2), ones(1,2) ),[2 3 1]);
%Map to y
qy=(yhigh-ylow)/2;
py=(yhigh+ylow)/2;
y=permute(qy(:,ones(1,2), ones(1,2) ),[ 2 3 1]).*bpy2(:,:,ones(1,nk)) ...
    +permute(py(:,ones(1,2), ones(1,2) ),[2 3 1]);


if any(fun=='(') & any(fun=='y')& any(fun=='x'),
  exec_string=['fv=', fun, ';']; %the call function is already setup
else%set up the call function 
  exec_string=['fv=', fun, ' (x,y'];
  num_parameters=nargin-6;
  for i=1:num_parameters,
    exec_string=[exec_string,',p',int2str(i)];
  end
  exec_string=[exec_string,');'];
end
eval(exec_string);
int_old = squeeze(sum(sum(wfxy2(:,:,ones(1,nk)).*fv))).*qx.*qy;
int=zeros(size(int_old));tol1=int;
k=1:nk;


% Break out of the iteration loop for three reasons:
%  1) the last update is very small (compared to int)
%  2) the last update is very small (compared to tol)
%  3) There are more than 8 iterations. This should NEVER happen. 


converge='n';
for i=1:8,
  gnum=int2str(2^(i+1));
  eval(['global bpx',gnum,' wfxy',gnum,'  bpy',gnum,';']);
  if isempty(eval(['bpx',gnum]))| isempty(eval(['bpy',gnum])) ,  
    eval(['[bpx',gnum,',bpy',gnum,',wfxy',gnum,']=grule2d(',gnum,',', gnum,');']);
  end
  eval(['x=permute(qx(k,ones(1,',gnum,'), ones(1,',gnum,') ),[ 2 3 1]).*bpx',gnum,'(:,:,ones(1,nk)) +permute(px(k,ones(1,',gnum,'), ones(1,',gnum,') ),[2 3 1]);']);


  eval(['y=permute(qy(k,ones(1,',gnum,'), ones(1,',gnum,') ),[ 2 3 1]).*bpy',gnum,'(:,:,ones(1,nk)) +permute(py(k,ones(1,',gnum,'), ones(1,',gnum,') ),[2 3 1]);']);
    eval(exec_string);
    eval(['int(k) = squeeze(sum(sum((wfxy',gnum,'(:,:,ones(1,nk)).*fv)))).*qx(k).*qy(k);']);
    tol1(k)=abs(int_old(k)-int(k)); % absolute tolerance
    k=find(tol1 > abs(tol*int)|tol1 > abs(tol));%indices to integrals which did not converge
  
    if any(k), % compute integrals again
      nk=length(k);%# of integrals we have to compute again
    else
      converge='y';
      break;
    end
    
     int_old(k)=int(k);
end
int=reshape(int,[N M]); % reshape int to the same size as input arguments
if converge=='n',
  disp('Integral did not converge--singularity likely')
end