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23.3 Functions of Multiple Variables

Octave includes several functions for computing the integral of functions of multiple variables. This procedure can generally be performed by creating a function that integrates f with respect to x, and then integrates that function with respect to y. This procedure can be performed manually using the following example which integrates the function: 

f(x, y) = sin(pi*x*y) * sqrt(x*y)

for x and y between 0 and 1.

Using quadgk in the example below, a double integration can be performed. (Note that any of the 1-D quadrature functions can be used in this fashion except for quad since it is written in Fortran and cannot be called recursively.) 

function q = g(y)
  q = ones (size (y));
  for i = 1:length (y)
    f = @(x) sin (pi*x.*y(i)) .* sqrt (x.*y(i));
    q(i) = quadgk (f, 0, 1);
  endfor
endfunction

I = quadgk ("g", 0, 1)
      ⇒ 0.30022

The algorithm above is implemented in the function dblquad for integrals over two variables. The 3-D equivalent of this process is implemented in triplequad for integrals over three variables. As an example, the result above can be replicated with a call to dblquad as shown below. 

I = dblquad (@(x, y) sin (pi*x.*y) .* sqrt (x.*y), 0, 1, 0, 1)
      ⇒ 0.30022
: q = dblquad (f, xa, xb, ya, yb)
: q = dblquad (f, xa, xb, ya, yb, tol)
: q = dblquad (f, xa, xb, ya, yb, tol, quadf)
: q = dblquad (f, xa, xb, ya, yb, tol, quadf, …)

Numerically evaluate the double integral of f.

f is a function handle, inline function, or string containing the name of the function to evaluate. The function f must have the form z = f(x,y) where x is a vector and y is a scalar. It should return a vector of the same length and orientation as x.

xa, ya and xb, yb are the lower and upper limits of integration for x and y respectively. The underlying integrator determines whether infinite bounds are accepted.

The optional argument tol defines the absolute tolerance used to integrate each sub-integral. The default value is 1e-6.

The optional argument quadf specifies which underlying integrator function to use. Any choice but quad is available and the default is quadcc.

Additional arguments, are passed directly to f. To use the default value for tol or quadf one may pass ':' or an empty matrix ([]).

See also: integral2, integral3, triplequad, quad, quadv, quadl, quadgk, quadcc, trapz.

: q = triplequad (f, xa, xb, ya, yb, za, zb)
: q = triplequad (f, xa, xb, ya, yb, za, zb, tol)
: q = triplequad (f, xa, xb, ya, yb, za, zb, tol, quadf)
: q = triplequad (f, xa, xb, ya, yb, za, zb, tol, quadf, …)

Numerically evaluate the triple integral of f.

f is a function handle, inline function, or string containing the name of the function to evaluate. The function f must have the form w = f(x,y,z) where either x or y is a vector and the remaining inputs are scalars. It should return a vector of the same length and orientation as x or y.

xa, ya, za and xb, yb, zb are the lower and upper limits of integration for x, y, and z respectively. The underlying integrator determines whether infinite bounds are accepted.

The optional argument tol defines the absolute tolerance used to integrate each sub-integral. The default value is 1e-6.

The optional argument quadf specifies which underlying integrator function to use. Any choice but quad is available and the default is quadcc.

Additional arguments, are passed directly to f. To use the default value for tol or quadf one may pass ':' or an empty matrix ([]).

See also: integral3, integral2, dblquad, quad, quadv, quadl, quadgk, quadcc, trapz.

The recursive algorithm for quadrature presented above is referred to as "iterated". A separate 2-D integration method is implemented in the function quad2d. This function performs a "tiled" integration by subdividing the integration domain into rectangular regions and performing separate integrations over those domains. The domains are further subdivided in areas requiring refinement to reach the desired numerical accuracy. For certain functions this method can be faster than the 2-D iteration used in the other functions above.

: q = quad2d (f, xa, xb, ya, yb)
: q = quad2d (f, xa, xb, ya, yb, prop, val, …)
: [q, err, iter] = quad2d (…)

Numerically evaluate the two-dimensional integral of f using adaptive quadrature over the two-dimensional domain defined by xa, xb, ya, yb using tiled integration. Additionally, ya and yb may be scalar functions of x, allowing for the integration over non-rectangular domains.

f is a function handle, inline function, or string containing the name of the function to evaluate. The function f must be of the form z = f(x,y) where x is a vector and y is a scalar. It should return a vector of the same length and orientation as x.

Additional optional parameters can be specified using "property", value pairs. Valid properties are:

AbsTol

Define the absolute error tolerance for the quadrature. The default value is 1e-10 (1e-5 for single).

RelTol

Define the relative error tolerance for the quadrature. The default value is 1e-6 (1e-4 for single).

MaxFunEvals

The maximum number of function calls to the vectorized function f. The default value is 5000.

Singular

Enable/disable transforms to weaken singularities on the edge of the integration domain. The default value is true.

Vectorized

Option to disable vectorized integration, forcing Octave to use only scalar inputs when calling the integrand. The default value is false.

FailurePlot

If quad2d fails to converge to the desired error tolerance before MaxFunEvals is reached, a plot of the areas that still need refinement is created. The default value is false.

Adaptive quadrature is used to minimize the estimate of error until the following is satisfied: 

        error <= max (AbsTol, RelTol*|q|)

The optional output err is an approximate bound on the error in the integral abs (q - I), where I is the exact value of the integral. The optional output iter is the number of vectorized function calls to the function f that were used.

Example 1 : integrate a rectangular region in x-y plane 

f = @(x,y) 2*ones (size (x));
q = quad2d (f, 0, 1, 0, 1)
  ⇒ q =  2

The result is a volume, which for this constant-value integrand, is just Length * Width * Height.

Example 2 : integrate a triangular region in x-y plane 

f = @(x,y) 2*ones (size (x));
ymax = @(x) 1 - x;
q = quad2d (f, 0, 1, 0, ymax)
  ⇒ q =  1

The result is a volume, which for this constant-value integrand, is the Triangle Area x Height or 1/2 * Base * Width * Height.

Programming Notes: If there are singularities within the integration region it is best to split the integral and place the singularities on the boundary.

Known MATLAB incompatibility: If tolerances are left unspecified, and any integration limits are of type single, then Octave’s integral functions automatically reduce the default absolute and relative error tolerances as specified above. If tighter tolerances are desired they must be specified. MATLAB leaves the tighter tolerances appropriate for double inputs in place regardless of the class of the integration limits.

Reference: L.F. Shampine, MATLAB program for quadrature in 2D, Applied Mathematics and Computation, pp. 266–274, Vol 1, 2008.

See also: integral2, dblquad, integral, quad, quadgk, quadv, quadl, quadcc, trapz, integral3, triplequad.

Finally, the functions integral2 and integral3 are provided as general 2-D and 3-D integration functions. They will auto-select between iterated and tiled integration methods and, unlike dblquad and triplequad, will work with non-rectangular integration domains.

: q = integral2 (f, xa, xb, ya, yb)
: q = integral2 (f, xa, xb, ya, yb, prop, val, …)
: [q, err] = integral2 (…)

Numerically evaluate the two-dimensional integral of f using adaptive quadrature over the two-dimensional domain defined by xa, xb, ya, yb (scalars may be finite or infinite). Additionally, ya and yb may be scalar functions of x, allowing for integration over non-rectangular domains.

f is a function handle, inline function, or string containing the name of the function to evaluate. The function f must be of the form z = f(x,y) where x is a vector and y is a scalar. It should return a vector of the same length and orientation as x.

Additional optional parameters can be specified using "property", value pairs. Valid properties are:

AbsTol

Define the absolute error tolerance for the quadrature. The default value is 1e-10 (1e-5 for single).

RelTol

Define the relative error tolerance for the quadrature. The default value is 1e-6 (1e-4 for single).

Method

Specify the two-dimensional integration method to be used, with valid options being "auto" (default), "tiled", or "iterated". When using "auto", Octave will choose the "tiled" method unless any of the integration limits are infinite.

Vectorized

Enable or disable vectorized integration. A value of false forces Octave to use only scalar inputs when calling the integrand, which enables integrands f(x,y) that have not been vectorized and only accept x and y as scalars to be used. The default value is true.

Adaptive quadrature is used to minimize the estimate of error until the following is satisfied: 

        error <= max (AbsTol, RelTol*|q|)

err is an approximate bound on the error in the integral abs (q - I), where I is the exact value of the integral.

Example 1 : integrate a rectangular region in x-y plane 

f = @(x,y) 2*ones (size (x));
q = integral2 (f, 0, 1, 0, 1)
  ⇒ q =  2

The result is a volume, which for this constant-value integrand, is just Length * Width * Height.

Example 2 : integrate a triangular region in x-y plane 

f = @(x,y) 2*ones (size (x));
ymax = @(x) 1 - x;
q = integral2 (f, 0, 1, 0, ymax)
  ⇒ q =  1

The result is a volume, which for this constant-value integrand, is the Triangle Area x Height or 1/2 * Base * Width * Height.

Programming Notes: If there are singularities within the integration region it is best to split the integral and place the singularities on the boundary.

Known MATLAB incompatibility: If tolerances are left unspecified, and any integration limits are of type single, then Octave’s integral functions automatically reduce the default absolute and relative error tolerances as specified above. If tighter tolerances are desired they must be specified. MATLAB leaves the tighter tolerances appropriate for double inputs in place regardless of the class of the integration limits.

Reference: L.F. Shampine, MATLAB program for quadrature in 2D, Applied Mathematics and Computation, pp. 266–274, Vol 1, 2008.

See also: quad2d, dblquad, integral, quad, quadgk, quadv, quadl, quadcc, trapz, integral3, triplequad.

: q = integral3 (f, xa, xb, ya, yb, za, zb)
: q = integral3 (f, xa, xb, ya, yb, za, zb, prop, val, …)

Numerically evaluate the three-dimensional integral of f using adaptive quadrature over the three-dimensional domain defined by xa, xb, ya, yb, za, zb (scalars may be finite or infinite). Additionally, ya and yb may be scalar functions of x and za, and zb maybe be scalar functions of x and y, allowing for integration over non-rectangular domains.

f is a function handle, inline function, or string containing the name of the function to evaluate. The function f must be of the form z = f(x,y) where x is a vector and y is a scalar. It should return a vector of the same length and orientation as x.

Additional optional parameters can be specified using "property", value pairs. Valid properties are:

AbsTol

Define the absolute error tolerance for the quadrature. The default value is 1e-10 (1e-5 for single).

RelTol

Define the relative error tolerance for the quadrature. The default value is 1e-6 (1e-4 for single).

Method

Specify the two-dimensional integration method to be used, with valid options being "auto" (default), "tiled", or "iterated". When using "auto", Octave will choose the "tiled" method unless any of the integration limits are infinite.

Vectorized

Enable or disable vectorized integration. A value of false forces Octave to use only scalar inputs when calling the integrand, which enables integrands f(x,y) that have not been vectorized and only accept x and y as scalars to be used. The default value is true.

Adaptive quadrature is used to minimize the estimate of error until the following is satisfied: 

        error <= max (AbsTol, RelTol*|q|)

err is an approximate bound on the error in the integral abs (q - I), where I is the exact value of the integral.

Example 1 : integrate over a rectangular volume 

f = @(x,y,z) ones (size (x));
q = integral3 (f, 0, 1, 0, 1, 0, 1)
  ⇒ q =  1.00000

For this constant-value integrand, the result is a volume which is just Length * Width * Height.

Example 2 : integrate over a spherical volume 

f = @(x,y) ones (size (x));
ymax = @(x) sqrt (1 - x.^2);
zmax = @(x,y) sqrt (1 - x.^2 - y.^2);
q = integral3 (f, 0, 1, 0, ymax, 0, zmax)
  ⇒ q =  0.52360

For this constant-value integrand, the result is a volume which is 1/8th of a unit sphere or 1/8 * 4/3 * pi.

Programming Notes: If there are singularities within the integration region it is best to split the integral and place the singularities on the boundary.

Known MATLAB incompatibility: If tolerances are left unspecified, and any integration limits are of type single, then Octave’s integral functions automatically reduce the default absolute and relative error tolerances as specified above. If tighter tolerances are desired they must be specified. MATLAB leaves the tighter tolerances appropriate for double inputs in place regardless of the class of the integration limits.

Reference: L.F. Shampine, MATLAB program for quadrature in 2D, Applied Mathematics and Computation, pp. 266–274, Vol 1, 2008.

See also: triplequad, integral, quad, quadgk, quadv, quadl, quadcc, trapz, integral2, quad2d, dblquad.

The above integrations can be fairly slow, and that problem increases exponentially with the dimensionality of the integral. Another possible solution for 2-D integration is to use Orthogonal Collocation as described in the previous section (see Orthogonal Collocation). The integral of a function f(x,y) for x and y between 0 and 1 can be approximated using n points by the sum over i=1:n and j=1:n of q(i)*q(j)*f(r(i),r(j)), where q and r is as returned by colloc (n). The generalization to more than two variables is straight forward. The following code computes the studied integral using n=8 points. 

f = @(x,y) sin (pi*x*y') .* sqrt (x*y');
n = 8;
[t, ~, ~, q] = colloc (n);
I = q'*f(t,t)*q;
      ⇒ 0.30022

It should be noted that the number of points determines the quality of the approximation. If the integration needs to be performed between a and b, instead of 0 and 1, then a change of variables is needed.

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