@c Copyright (C) 1996-2012 John W. Eaton
@c
@c This file is part of Octave.
@c
@c Octave is free software; you can redistribute it and/or modify it
@c under the terms of the GNU General Public License as published by the
@c Free Software Foundation; either version 3 of the License, or (at
@c your option) any later version.
@c 
@c Octave is distributed in the hope that it will be useful, but WITHOUT
@c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
@c FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
@c for more details.
@c 
@c You should have received a copy of the GNU General Public License
@c along with Octave; see the file COPYING.  If not, see
@c <http://www.gnu.org/licenses/>.

@node Numerical Integration
@chapter Numerical Integration

Octave comes with several built-in functions for computing the integral
of a function numerically (termed quadrature).  These functions all solve
1-dimensional integration problems.

@menu
* Functions of One Variable::
* Orthogonal Collocation::
* Functions of Multiple Variables::
@end menu

@node Functions of One Variable
@section Functions of One Variable

Octave supports five different algorithms for computing the integral
@tex
$$
 \int_a^b f(x) d x
$$
@end tex
of a function @math{f} over the interval from @math{a} to @math{b}.
These are

@table @code
@item quad
Numerical integration based on Gaussian quadrature.

@item quadv
Numerical integration using an adaptive vectorized Simpson's rule.

@item quadl
Numerical integration using an adaptive Lobatto rule.

@item quadgk
Numerical integration using an adaptive Gauss-Konrod rule.

@item quadcc
Numerical integration using adaptive Clenshaw-Curtis rules.

@item trapz, cumtrapz
Numerical integration of data using the trapezoidal method.
@end table

@noindent
The best quadrature algorithm to use depends on the integrand.  If you have
empirical data, rather than a function, the choice is @code{trapz} or
@code{cumtrapz}.  If you are uncertain about the characteristics of the
integrand, @code{quadcc} will be the most robust as it can handle
discontinuities, singularities, oscillatory functions, and infinite intervals.
When the integrand is smooth @code{quadgk} may be the fastest of the
algorithms.

@multitable @columnfractions 0.05 0.15 .80
@headitem @tab Function @tab Characteristics
@item @tab quad   @tab Low accuracy with nonsmooth integrands
@item @tab quadv  @tab Medium accuracy with smooth integrands
@item @tab quadl  @tab Medium accuracy with smooth integrands.  Slower than quadgk.
@item @tab quadgk @tab Medium accuracy (@math{1e^{-6}}--@math{1e^{-9}}) with smooth integrands.
@item @tab        @tab Handles oscillatory functions and infinite bounds
@item @tab quadcc @tab Low to High accuracy with nonsmooth/smooth integrands
@item @tab        @tab Handles oscillatory functions, singularities, and infinite bounds
@end multitable


Here is an example of using @code{quad} to integrate the function
@tex
$$
 f(x) = x \sin (1/x) \sqrt {|1 - x|}
$$
from $x = 0$ to $x = 3$.
@end tex
@ifnottex

@example
  @var{f}(@var{x}) = @var{x} * sin (1/@var{x}) * sqrt (abs (1 - @var{x}))
@end example

@noindent
from @var{x} = 0 to @var{x} = 3.
@end ifnottex

This is a fairly difficult integration (plot the function over the range
of integration to see why).

The first step is to define the function:

@example
@group
function y = f (x)
  y = x .* sin (1./x) .* sqrt (abs (1 - x));
endfunction
@end group
@end example

Note the use of the `dot' forms of the operators.  This is not necessary for
the @code{quad} integrator, but is required by the other integrators.  In any
case, it makes it much easier to generate a set of points for plotting because
it is possible to call the function with a vector argument to produce a vector
result.

The second step is to call quad with the limits of integration:

@example
@group
[q, ier, nfun, err] = quad ("f", 0, 3)
     @result{} 1.9819
     @result{} 1
     @result{} 5061
     @result{} 1.1522e-07
@end group
@end example

Although @code{quad} returns a nonzero value for @var{ier}, the result
is reasonably accurate (to see why, examine what happens to the result
if you move the lower bound to 0.1, then 0.01, then 0.001, etc.).

The function @qcode{"f"} can be the string name of a function, a function
handle, or an inline function.  These options make it quite easy to do
integration without having to fully define a function in an m-file.  For
example:

@example
@group
# Verify integral (x^3) = x^4/4
f = inline ("x.^3");
quadgk (f, 0, 1)
     @result{} 0.25000

# Verify gamma function = (n-1)! for n = 4
f = @@(x) x.^3 .* exp (-x);
quadcc (f, 0, Inf)
     @result{} 6.0000
@end group
@end example

@DOCSTRING(quad)

@DOCSTRING(quad_options)

@DOCSTRING(quadv)

@DOCSTRING(quadl)

@DOCSTRING(quadgk)

@DOCSTRING(quadcc)

Sometimes one does not have the function, but only the raw (x, y) points from
which to perform an integration.  This can occur when collecting data in an
experiment.  The @code{trapz} function can integrate these values as shown in
the following example where "data" has been collected on the cosine function
over the range [0, pi/2). 

@example
@group
x = 0:0.1:pi/2;  # Uniformly spaced points
y = cos (x);
trapz (x, y)
     @result{} 0.99666
@end group
@end example

The answer is reasonably close to the exact value of 1.  Ordinary quadrature
is sensitive to the characteristics of the integrand.  Empirical integration 
depends not just on the integrand, but also on the particular points chosen to
represent the function.  Repeating the example above with the sine function
over the range [0, pi/2) produces far inferior results.

@example
@group
x = 0:0.1:pi/2;  # Uniformly spaced points
y = sin (x);
trapz (x, y)
     @result{} 0.92849
@end group
@end example

However, a slightly different choice of data points can change the result
significantly.  The same integration, with the same number of points, but
spaced differently produces a more accurate answer.

@example
@group
x = linspace (0, pi/2, 16);  # Uniformly spaced, but including endpoint
y = sin (x);
trapz (x, y)
     @result{} 0.99909
@end group
@end example

In general there may be no way of knowing the best distribution of points ahead
of time.  Or the points may come from an experiment where there is no freedom to
select the best distribution.  In any case, one must remain aware of this
issue when using @code{trapz}.

@DOCSTRING(trapz)

@DOCSTRING(cumtrapz)

@node Orthogonal Collocation
@section Orthogonal Collocation

@DOCSTRING(colloc)

Here is an example of using @code{colloc} to generate weight matrices
for solving the second order differential equation
@tex
$u^\prime - \alpha u^{\prime\prime} = 0$ with the boundary conditions
$u(0) = 0$ and $u(1) = 1$.
@end tex
@ifnottex
@var{u}' - @var{alpha} * @var{u}'' = 0 with the boundary conditions
@var{u}(0) = 0 and @var{u}(1) = 1.
@end ifnottex

First, we can generate the weight matrices for @var{n} points (including
the endpoints of the interval), and incorporate the boundary conditions
in the right hand side (for a specific value of
@tex
$\alpha$).
@end tex
@ifnottex
@var{alpha}).
@end ifnottex

@example
@group
n = 7;
alpha = 0.1;
[r, a, b] = colloc (n-2, "left", "right");
at = a(2:n-1,2:n-1);
bt = b(2:n-1,2:n-1);
rhs = alpha * b(2:n-1,n) - a(2:n-1,n);
@end group
@end example

Then the solution at the roots @var{r} is

@example
@group
u = [ 0; (at - alpha * bt) \ rhs; 1]
     @result{} [ 0.00; 0.004; 0.01 0.00; 0.12; 0.62; 1.00 ]
@end group
@end example

@node Functions of Multiple Variables
@section Functions of Multiple Variables

Octave does not have built-in functions for computing the integral of
functions of multiple variables directly.  It is possible, however, to
compute the integral of a function of multiple variables using the
existing functions for one-dimensional integrals.

To illustrate how the integration can be performed, we will integrate
the function
@tex
$$
  f(x, y) = \sin(\pi x y)\sqrt{x y}
$$
@end tex
@ifnottex

@example
f(x, y) = sin(pi*x*y)*sqrt(x*y)
@end example

@end ifnottex
for @math{x} and @math{y} between 0 and 1.

The first approach creates a function that integrates @math{f} with
respect to @math{x}, and then integrates that function with respect to
@math{y}.  Because @code{quad} is written in Fortran it cannot be called
recursively.  This means that @code{quad} cannot integrate a function
that calls @code{quad}, and hence cannot be used to perform the double
integration.  Any of the other integrators, however, can be used which is
what the following code demonstrates.

@example
@group
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)
      @result{} 0.30022
@end group
@end example

The above process can be simplified with the @code{dblquad} and
@code{triplequad} functions for integrals over two and three
variables.  For example:

@example
@group
I = dblquad (@@(x, y) sin (pi*x.*y) .* sqrt (x.*y), 0, 1, 0, 1)
      @result{} 0.30022
@end group
@end example

@DOCSTRING(dblquad)

@DOCSTRING(triplequad)

The above mentioned approach works, but is fairly slow, and that problem
increases exponentially with the dimensionality of the integral.  Another
possible solution is to use Orthogonal Collocation as described in the
previous section (@pxref{Orthogonal Collocation}).  The integral of a function
@math{f(x,y)} for @math{x} and @math{y} between 0 and 1 can be approximated
using @math{n} points by
@tex
$$
 \int_0^1 \int_0^1 f(x,y) d x d y \approx \sum_{i=1}^n \sum_{j=1}^n q_i q_j f(r_i, r_j),
$$
@end tex
@ifnottex
the sum over @code{i=1:n} and @code{j=1:n} of @code{q(i)*q(j)*f(r(i),r(j))},
@end ifnottex
where @math{q} and @math{r} is as returned by @code{colloc (n)}.  The
generalization to more than two variables is straight forward.  The
following code computes the studied integral using @math{n=8} points.

@example
@group
f = @@(x,y) sin (pi*x*y') .* sqrt (x*y');
n = 8;
[t, ~, ~, q] = colloc (n);
I = q'*f(t,t)*q;
      @result{} 0.30022
@end group
@end example

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