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quad2d - Numerically evaluate double integral over planar region

Syntax

q = quad2d(fun,a,b,c,d) [q,errbnd] = quad2d(...) q = quad2d(fun,a,b,c,d,param1,val1,param2,val2,...)

Description

q = quad2d(fun,a,b,c,d) approximates the integral of fun(x,y) over the planar region and . fun is a function handle, c and d may each be a scalar or a function handle.

All input functions must be vectorized. The function Z=fun(X,Y) must accept 2-D matrices X and Y of the same size and return a matrix Z of corresponding values. The functions ymin=c(X) and ymax=d(X) must accept matrices and return matrices of the same size with corresponding values.

[q,errbnd] = quad2d(...). errbnd is an approximate upper bound on the absolute error, |Q - I|, where I denotes the exact value of the integral.

q = quad2d(fun,a,b,c,d,param1,val1,param2,val2,...) performs the integration as above with specified values of optional parameters:

`AbsTol`	absolute error tolerance
`RelTol`	relative error tolerance

quad2d attempts to satisfy ERRBND <= max(AbsTol,RelTol*|Q|). This is absolute error control when |Q| is sufficiently small and relative error control when |Q| is larger. A default tolerance value is used when a tolerance is not specified. The default value of AbsTol is 1e-5. The default value of RelTol is 100*eps(class(Q)). This is also the minimum value of RelTol. Smaller RelTol values are automatically increased to the default value.

MaxFunEvals Maximum allowed number of evaluations of fun reached.

The MaxFunEvals parameter limits the number of vectorized calls to fun. The default is 2000.

FailurePlot Generate a plot if MaxFunEvals is reached.

Setting FailurePlot to true generates a graphical representation of the regions needing further refinement when MaxFunEvals is reached. No plot is generated if the integration succeeds before reaching MaxFunEvals. These (generally) 4-sided regions are mapped to rectangles internally. Clusters of small regions indicate the areas of difficulty. The default is false.

Singular Problem may have boundary singularities

With Singular set to true, quad2d will employ transformations to weaken boundary singularities for better performance. The default is true. Setting Singular to false will turn these transformations off, which may provide a performance benefit on some smooth problems.

Examples

Example 1

Integrate over , . The true value of the integral is .

Q = quad2d(@(x,y) y.*sin(x)+x.*cos(y),pi,2*pi,0,pi)

Example 2

Integrate over the triangle and . The integrand is infinite at (0,0). The true value of the integral is .

fun = @(x,y) 1./(sqrt(x + y) .* (1 + x + y).^2 )

In Cartesian coordinates:

ymax = @(x) 1 - x;
Q = quad2d(fun,0,1,0,ymax)

In polar coordinates:

polarfun = @(theta,r) fun(r.*cos(theta),r.*sin(theta)).*r;
rmax = @(theta) 1./(sin(theta) + cos(theta));
Q = quad2d(polarfun,0,pi/2,0,rmax)

Limitations

quad2d begins by mapping the region of integration to a rectangle. Consequently, it may have trouble integrating over a region that does not have four sides or has a side that cannot be mapped smoothly to a straight line. If the integration is unsuccessful, some helpful tactics are leaving Singular set to its default value of true, changing between Cartesian and polar coordinates, or breaking the region of integration into pieces and adding the results of integration over the pieces.

For example:

fun = @(x,y)abs(x.^2 + y.^2 - 0.25);
c = @(x)-sqrt(1 - x.^2);
d = @(x)sqrt(1 - x.^2);
quad2d(fun,-1,1,c,d,'AbsTol',1e-8,...
    'FailurePlot',true,'Singular',false)
Warning: Reached the maximum number of function ... 
     evaluations (2000). The result fails the  ... 
     global error test.

The failure plot shows two areas of difficulty, near the points (-1,0) and (1,0) and near the circle :

Changing the value of Singular to true will cope with the geometric singularities at (-1,0) and (1,0). The larger shaded areas may need refinement but are probably not areas of difficulty.

Q = quad2d(fun,-1,1,c,d,'AbsTol',1e-8, ... 
     'FailurePlot',true,'Singular',true)
Warning: Reached the maximum number of function ...
     evaluations (2000). The result passes the ... 
     global error test.

From here you can take advantage of symmetry:

Q = 4*quad2d(fun,0,1,0,d,'Abstol',1e-8,...
     'Singular',true, 'FailurePlot',true)

However, the code is still working very hard near the singularity. It may not be able to provide higher accuracy:

Q = 4*quad2d(fun,0,1,0,d,'Abstol',1e-10,...
     'Singular',true,'FailurePlot',true)
Warning: Reached the maximum number of function ...
     evaluations (2000). The result passes the ...
     global error test.

At higher accuracy, a change in coordinates may work better.

polarfun = @(theta,r) fun(r.*cos(theta),r.*sin(theta)).*r;
Q = 4*quad2d(polarfun,0,pi/2,0,1,'AbsTol',1e-10)

It is best to put the singularity on the boundary by splitting the region of integration into two parts:

Q1 = 4*quad2d(polarfun,0,pi/2,0,0.5,'AbsTol',5e-11);
Q2 = 4*quad2d(polarfun,0,pi/2,0.5,1,'AbsTol',5e-11);
Q = Q1 + Q2

References

[1] L.F. Shampine "Vectorized Adaptive Quadrature in MATLAB," Journal of Computational and Applied Mathematics, 211, 2008, pp.131–140.