Numerical integration of f(x,y) with flexible arguments for giving domain and method.


Updated 25 Jun 2009

View License

The intention is to make a more flexible and more general function than the Matlab function dblquad. Especially the domain may be given in a more general way, it may be a circle, a rectangle, or any convex polygon. Also different methods may be used.

FUN is (normally) a function handle. When method is 'dblquad' the
function Z=FUN(X,Y) should accept a vector X and a scalar Y and return
a vector Z of values as Matlab's 'dblquad'.
For other methods the function just need to handle single element
arguments. The function may accept matrix, and vector, arguments of the
same size and return a matrix also with this size, such that
Z(i) = FUN(X(i),Y(i)). The function will be used this way if parameter
'matrixarg' is also included in the PARAM struct and set to true.
Default is false, and used if 'matrixarg' is not a field in PARAM.
If FUN == 1, then area of domain is returned.

DOMAIN is a struct where the field 'type' gives how the domain is
given. Examples:
domain = struct('type','circle','xc',0,'yc',0,'radius',1);
domain = struct('type','sector','th1',-pi,'th2',pi,'r1',0,'r2',1,'xc',0,'yc',0);
domain = struct('type','box','x1',-1,'x2',1,'y1',-1,'y2',1);
domain = struct('type','area','x1',-1,'x2',1,'y1',@(x) sqrt(x),'y2',@(x) x.^2);
domain = struct('type','polygon','x',[0,2,2,1,0],'y',[0,0,1,2,1]);

PARAM is a struct where the field 'method' always must be included.
It gives the method used, often in both x and y direction.
Examples for Matlab quad, i.e. dblquad, fuction with argument tol,
Gauss Quadrature method and Clenshaw-Curtis method:
param = struct('method','dblquad','tol',1e-6);
param = struct('method','gauss','points',6);
param = struct('method','cc','points',10);
param.matrixarg = true; % use vector-arguments in fun

Q = DOUBLEINTEGRAL(FUN, DOMAIN, PARAM, verbose, makefigure)
verbose = 0 (display only errors), 1 (+ if arguments are overruled), 2
(+ results), 3 (+ some status information), 4 (+ some debugging info)
makefigure = 0 (default) for no figure, 1 for domain, 5 for contour of
function in domain.

fun = @(x,y) y*sin(x)+x*cos(y);
domain = struct('type','box','x1',pi,'x2',2*pi,'y1',0,'y2',pi);
param = struct('method','dblquad','tol',1e-6);
Q = doubleintegral(fun, domain, param); % -pi^2

fun = @(x,y) y.*sin(x)+x.*cos(y); % allows vectors in both x and y
domain = struct('type','box','x1',pi,'x2',2*pi,'y1',0,'y2',pi);
param = struct('method','gauss','points',6,'matrixarg',true);
Q = doubleintegral(fun, domain, param, 3, 5); % -pi^2

fun = 1; % Area is one dimensional integral of f(x) = abs(y2(x)-y1(x))
domain = struct('type','area','x1',0,'x2',1,'y1',@(x) x.^2,'y2',@(x) sqrt(x));
param = struct('method','cc','points',12);
Q = doubleintegral(fun, domain, param); % 1/3

fun = @(x,y) sqrt(max(zeros(size(x)), 1-x.^2-y.^2)); % unit sphere
domain = struct('type','circle','xc',0,'yc',0,'radius',1);
param = struct('method','gauss','points',25,'matrixarg',true);
Q = doubleintegral(fun, domain, param); % 2*pi/3
% an alternative is to use dblquad, domain may be a circle or a box.
domain = struct('type','box','x1',-1,'x2',1,'y1',-1,'y2',1);
param = struct('method','dblquad','tol',1e-6);
% or polygon to find just a sector
domain = struct('type','polygon','x',[0,1,1],'y',[0,0,1]);
Q = doubleintegral(fun, domain, param); % pi/12
% or grid of points in sector. Note still fun = @(x,y)
domain = struct('type','sector','th1',0,'th2',pi/4,'r1',0,'r2',1,'xc',0,'yc',0);
param = struct('method','dblquad','tol',1e-6);
Q = doubleintegral(fun, domain, param); % pi/12

Cite As

Karl Skretting (2023). doubleintegral (, MATLAB Central File Exchange. Retrieved .

MATLAB Release Compatibility
Created with R2007a
Compatible with any release
Platform Compatibility
Windows macOS Linux
Find more on Numerical Integration and Differential Equations in Help Center and MATLAB Answers

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!
Version Published Release Notes