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## Copyright (C) 2008-2013 David Bateman
## Copyright (C) 2012 Alexander Klein
##
## This file is part of Octave.
##
## Octave is free software; you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 3 of the License, or (at
## your option) any later version.
##
## Octave is distributed in the hope that it will be useful, but
## WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
## General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING. If not, see
## <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} {@var{q} =} quadv (@var{f}, @var{a}, @var{b})
## @deftypefnx {Function File} {@var{q} =} quadv (@var{f}, @var{a}, @var{b}, @var{tol})
## @deftypefnx {Function File} {@var{q} =} quadv (@var{f}, @var{a}, @var{b}, @var{tol}, @var{trace})
## @deftypefnx {Function File} {@var{q} =} quadv (@var{f}, @var{a}, @var{b}, @var{tol}, @var{trace}, @var{p1}, @var{p2}, @dots{})
## @deftypefnx {Function File} {[@var{q}, @var{nfun}] =} quadv (@dots{})
##
## Numerically evaluate the integral of @var{f} from @var{a} to @var{b}
## using an adaptive Simpson's rule.
## @var{f} is a function handle, inline function, or string
## containing the name of the function to evaluate.
## @code{quadv} is a vectorized version of @code{quad} and the function
## defined by @var{f} must accept a scalar or vector as input and return a
## scalar, vector, or array as output.
##
## @var{a} and @var{b} are the lower and upper limits of integration. Both
## limits must be finite.
##
## The optional argument @var{tol} defines the tolerance used to stop
## the adaptation procedure. The default value is @math{1e^{-6}}.
##
## The algorithm used by @code{quadv} involves recursively subdividing the
## integration interval and applying Simpson's rule on each subinterval.
## If @var{trace} is true then after computing each of these partial
## integrals display: (1) the total number of function evaluations,
## (2) the left end of the subinterval, (3) the length of the subinterval,
## (4) the approximation of the integral over the subinterval.
##
## Additional arguments @var{p1}, etc., are passed directly to the function
## @var{f}. To use default values for @var{tol} and @var{trace}, one may pass
## empty matrices ([]).
##
## The result of the integration is returned in @var{q}. @var{nfun} indicates
## the number of function evaluations that were made.
##
## Note: @code{quadv} is written in Octave's scripting language and can be
## used recursively in @code{dblquad} and @code{triplequad}, unlike the
## similar @code{quad} function.
## @seealso{quad, quadl, quadgk, quadcc, trapz, dblquad, triplequad}
## @end deftypefn
function [q, nfun] = quadv (f, a, b, tol, trace, varargin)
## TODO: Make norm for convergence testing configurable
if (nargin < 3)
print_usage ();
endif
if (nargin < 4)
tol = [];
endif
if (nargin < 5)
trace = [];
endif
if (isa (a, "single") || isa (b, "single"))
myeps = eps ("single");
else
myeps = eps;
endif
if (isempty (tol))
tol = 1e-6;
endif
if (isempty (trace))
trace = 0;
endif
## Split the interval into 3 abscissa, and apply a 3 point Simpson's rule
c = (a + b) / 2;
fa = feval (f, a, varargin{:});
fc = feval (f, c, varargin{:});
fb = feval (f, b, varargin{:});
nfun = 3;
## If have edge singularities, move edge point by eps*(b-a) as
## discussed in Shampine paper used to implement quadgk
if (any (isinf (fa(:))))
fa = feval (f, a + myeps * (b-a), varargin{:});
endif
if (any (isinf (fb(:))))
fb = feval (f, b - myeps * (b-a), varargin{:});
endif
h = (b - a);
q = (b - a) / 6 * (fa + 4 * fc + fb);
[q, nfun, hmin] = simpsonstp (f, a, b, c, fa, fb, fc, q, nfun, abs (h),
tol, trace, varargin{:});
if (nfun > 10000)
warning ("maximum iteration count reached");
elseif (any (! isfinite (q(:))))
warning ("infinite or NaN function evaluations were returned");
elseif (hmin < (b - a) * myeps)
warning ("minimum step size reached -- possibly singular integral");
endif
endfunction
function [q, nfun, hmin] = simpsonstp (f, a, b, c, fa, fb, fc, q0,
nfun, hmin, tol, trace, varargin)
if (nfun > 10000)
q = q0;
else
d = (a + c) / 2;
e = (c + b) / 2;
fd = feval (f, d, varargin{:});
fe = feval (f, e, varargin{:});
nfun += 2;
q1 = (c - a) / 6 * (fa + 4 * fd + fc);
q2 = (b - c) / 6 * (fc + 4 * fe + fb);
q = q1 + q2;
if (abs(a - c) < hmin)
hmin = abs (a - c);
endif
if (trace)
disp ([nfun, a, b-a, q]);
endif
## Force at least one adpative step.
## Not vectorizing q-q0 in the norm provides a more rigid criterion for
## matrix-valued functions.
if (nfun == 5 || norm (q - q0, Inf) > tol)
[q1, nfun, hmin] = simpsonstp (f, a, c, d, fa, fc, fd, q1, nfun, hmin,
tol, trace, varargin{:});
[q2, nfun, hmin] = simpsonstp (f, c, b, e, fc, fb, fe, q2, nfun, hmin,
tol, trace, varargin{:});
q = q1 + q2;
endif
endif
endfunction
%!assert (quadv (@sin, 0, 2 * pi), 0, 1e-5)
%!assert (quadv (@sin, 0, pi), 2, 1e-5)
%% Handles weak singularities at the edge
%!assert (quadv (@(x) 1 ./ sqrt (x), 0, 1), 2, 1e-5)
%% Handles vector-valued functions
%!assert (quadv (@(x) [(sin (x)), (sin (2 * x))], 0, pi), [2, 0], 1e-5)
%% Handles matrix-valued functions
%!assert (quadv (@(x) [ x, x, x; x, 1./sqrt(x), x; x, x, x ], 0, 1 ), [0.5, 0.5, 0.5; 0.5, 2, 0.5; 0.5, 0.5, 0.5], 1e-5)
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