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%----------------------------------------------------------------------------------------------
% PROGRAM: overlap
% AUTHOR: Andri M. Gretarsson
% DATE: 7/10/04
%
% SYNTAX: coeff=overlap(z1,z2,domain <,metric,accuracy>)
% <...> indicates optional arguments
%
% Calculates the overlap integral of the two functions specified.
%
% INPUT ARGUMENTS:
% ----------------
% z1 = The values of the first function. Can be a 1D vector or a 2D matrix
% z2 = The values of the second function. z1 and z2 must be the same size.
% domain = the domain values at which z1 and z2 are specified. If z1 and z2 are 1D
% then domain is a nx2 matrix specifying one x,y pair for each value in z1
% and z2. If z1 and z2 are 2D then domain is a nxmx2 array where
% domain(:,:,1) and domain(:,:,2) are the x and y meshes corresponding
% to the values in z1 and z2. These meshes are often generated using meshgrid.m.
% metric = value, vector or matrix by which to multiply the elemental line or area dl or dS.
% For example, in 2D polar coordinates the elemental area is dS=r*dr*dtheta so
% metric should be specified as r.
% accuracy = round results to the nearest increment of accuracy. For example, if
% accuracy=0.3, then coeff 1.54 would be rounded to 1.5 while coeff=1.56 would be
% rounded to 1.8.
%
% OUTPUT ARGUMENTS:
% -----------------
% coeff = the numerical result of the overlap integral.
%
% EXAMPLE 1 (cartesian):
% [x,y]=meshgrid([0:0.01:2*pi],[0:0.01:2*pi]);
% clear domain; domain(:,:,1)=x; domain(:,:,2)=y;
% z1=sin(x+y); z2=sin(x+y);
% coeff=overlap(z1,z2,domain,1,0.0001)
% z1=sin(x+y); z2=cos(x+y);
% coeff=overlap(z1,z2,domain,1,0.0001)
%
% EXAMPLE 2 (polar):
% [r,theta]=meshgrid([0.01:0.01:1],[0:0.5:360]*pi/180);
% clear domain; domain(:,:,1)=r; domain(:,:,2)=theta;
% z1=r; z2=theta;
% subplot(1,3,1); [x,y]=pol2cart(theta,r); h=pcolor(x,y,z1); set(h,'EdgeColor','none'); axis square;
% subplot(1,3,2); [x,y]=pol2cart(theta,r); h=pcolor(x,y,z2); set(h,'EdgeColor','none'); axis square;
% subplot(1,3,3); [x,y]=pol2cart(theta,r); h=pcolor(x,y,z1.*z2); set(h,'EdgeColor','none'); axis square;
% coeff=overlap(z1,z2,domain,r,0.0001)
%
% Last updated: July 18, 2004 by AMG
%----------------------------------------------------------------------------------------------
% SYNTAX: coeff=overlap(z1,z2,domain <,metric,accuracy>)
%----------------------------------------------------------------------------------------------
function coeff=overlap(z1,z2,domain,varargin)
if nargin>=4;
metric=varargin{1};
if size(metric)==size(z1)
metric=metric(2:end,2:end);
end
else
metric=1;
end
if nargin>=5; accuracy=varargin{2}; else accuracy=0; end
if size(z1)~=size(z2), error('z1 and z2 must have same size'); end
if min(size(z1))==1 & min(size(domain))==1 % 1D plot along axis
dl=metric.*(domain(2:end)-domain(1:end-1));
coeff=sum(z1(2:end).*z2(2:end).*dl)/2;
end
if min(size(z1))==1 & size(domain,2)==2 % 1D plot along arbitrary axis
dl=metric.*sqrt((domain(2:end,1)-domain(1:end-1,1)).^2 + (domain(2:end,2)-domain(1:end-1,2)).^2);
coeff=sum(z1(2:end).*z2(2:end).*dl)/2;
end
if min(size(z1))>= 2 % 2D plot over xy plane
coord1=domain(:,:,1); coord2=domain(:,:,2);
dcoord1=diff(coord1,1,2); dcoord1=dcoord1(2:end,:);
dcoord2=diff(coord2,1,1); dcoord2=dcoord2(:,2:end);
dS = metric.*dcoord1.*dcoord2;
coeff=sum(sum(z1(2:end,2:end).*z2(2:end,2:end).*dS));
end
if accuracy~=0
coeff=round(coeff/accuracy)*accuracy;
end
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