Previous: Orthogonal Collocation, Up: Numerical Integration [Contents][Index]
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
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.
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.
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.
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.
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.
Previous: Orthogonal Collocation, Up: Numerical Integration [Contents][Index]