## Copyright (C) 2017 Nicholas Jankowski, David Bateman ## ## 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 ## . ## -*- texinfo -*- ## @deftypefn {} {@var{q} =} integral2 (@var{f}, @var{xa}, @var{xb}, @var{ya}, @var{yb}) ## @deftypefnx {} {@var{q} =} integral2 (@var{f}, @var{xa}, @var{xb}, @var{ya}, @var{yb}, @var{prop}, @var{val}, @dots{}) ## @deftypefnx {} {[@var{q}, @var{err}] =} integral2 (@dots{}) ## ## Numerically evaluate the two-dimensional integral of @var{f} using adaptive ## quadrature over the two-dimensional domain defined by @var{xa}, @var{xb}, ## @var{ya}, @var{yb} (scalars may be finite or infinite). Additionally, ## @var{ya} and @var{yb} may be scalar functions of @var{x}, allowing for the ## integration over non-rectangular domains. ## ## @var{f} is a function handle, inline function, or string containing the name ## of the function to evaluate. The function @var{f} must be of the form ## @math{z = f(x,y)} where @var{x} is a vector and @var{y} is a scalar. It ## should return a vector of the same length and orientation as @var{x}. ## ## Additional optional parameters can be specified using ## @qcode{"@var{property}", @var{value}} pairs. Valid properties are: ## ## @table @code ## @item AbsTol ## Define the absolute error tolerance for the quadrature. The default ## value is 1e-10 (1e-5 for single). ## ## @item RelTol ## Define the relative error tolerance for the quadrature. The default ## value is 1e-6 (1e-4 for single). ## ## @item Vectorized ## Option to disable vectorized integration, forcing octave to use only scalar ## inputs when calling the integrand. ## @end table ## ## Adaptive quadrature is used to minimize the estimate of error until the ## following is satisfied: ## @tex ## $$error \leq \max \left( AbsTol, RelTol\cdot\vert q\vert \right)$$ ## @end tex ## @ifnottex ## ## @example ## @group ## @var{error} <= max (@var{AbsTol}, @var{RelTol}*|@var{q}|). ## @end group ## @end example ## ## @end ifnottex ## ## @var{err} is an approximate bound on the error in the integral ## @code{abs (@var{q} - @var{I})}, where @var{I} is the exact value of the ## integral. ## ## Known @sc{matlab} incompatibilities: ## ## @enumerate ## @item ## If tolerances are left unspecified, and any integration limits or waypoints ## are of type @code{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. ## @sc{matlab} leaves the tighter tolerances appropriate for @code{double} ## inputs in place regardless of the class of the integration limits. ## ## @item ## @code{integral2} currently does not have an implemented 2d 'tiled' ## integration method. As such the @qcode{Method} property is ignored. ## @end enumerate ## ## @seealso{integral, integral3, quad, quadgk, quadv, quadl, quadcc, trapz, ## dblquad, triplequad} ## @end deftypefn function [q, err] = integral2 (f, xa, xb, ya, yb, varargin) if (nargin < 5 || (mod (nargin, 2) == 0)) print_usage (); endif if (! is_function_handle (f)) print_usage (); endif if (! (isscalar (xa) && isscalar (xb))) print_usage (); endif ## Check for single or double limits to set appropriate default tolerance. issingle = isa ([xa, xb], "single"); issingle = issingle || (!is_function_handle(ya) && isa(ya, "single")); issingle = issingle || (!is_function_handle(yb) && isa(yb, "single")); ## Set defaults, update with any specified parameters. if issingle abstol = 1e-5; reltol = 1e-4; else abstol = 1e-10; reltol = 1e-6; endif vectorized = true; singular = false; # leave hooks in place for tiled method, but default all to iterated for now. method = "iterated"; idx = 1; while (idx < nargin - 5) prop = varargin{idx++}; if (! ischar (prop)) error ("integral2: property PROP must be a string"); endif switch (tolower (prop)) case "abstol" abstol = varargin{idx++}; if (! (isnumeric (abstol) && isscalar (abstol) && abstol >= 0)) error ("integral2: AbsTol value must be a numeric scalar >= 0"); endif case "reltol" reltol = varargin{idx++}; if (! (isnumeric (reltol) && isscalar (reltol) && reltol >= 0)) error ("integral2: RelTol value must be a numeric scalar >= 0"); endif case "method" # FIXME: after having a tiled method, remove warning method = tolower (varargin{idx++}); if strcmp(method, "tiled") warning (["integral2: tiled method not implemented yet." ... "Defaulting to iterated method."]); endif case "singular" # Undocumented option to force weakening of edge singularities, but # only supported by the 'tiled' method singular = varargin{idx++}; if !islogical (singular) error ("integral2: 'singular' must be a logical value"); elseif !(strcmp(method, "iterated") || strcmp(method, "auto")) error ("integral2: method '%s' unrecognized", method) endif case "vectorized" # option to allow unvectorized functions to be treated vectorized = varargin{idx++}; if !islogical (vectorized) error ("integral2: 'vectorized' must be a logical value"); endif otherwise error ("integral2: unknown property '%s'", prop); endswitch endwhile if ! vectorized f = @(x, y) arrayfun(f, x, y); endif if strcmp(method, "iterated") if (! (isscalar (ya) && isscalar (yb))) ## FIXME: should this also check whether a ya(x) or yb(x) returns a ## scalar? As it is, a ya = @(x) [x x] returs true to isscalar despite ## having a vector return value error (["integral2: Non scalar limits currently unsupported with the" ... "'iterated' method"]); endif q = outer_iterated (f, xa, xb, ya, yb, abstol, reltol); if nargout == 2 warning("integral2: 'iterated' method can not return estimated error"); err = 0; endif else ## FIXME: replace with 'tiled' method block. Add the following back into ## the help text: ## Method ## Specifies the two dimensional integration method to be used, with valid ## options being @var{"auto"}, @var{"tiled"}, or @var{"iterated"}. ## @code{integral} will use @var{"auto"} by default, where it will usually ## choose @var{"tiled"} unless any of the integration limits are infinite. ## ## Also add documentaino for 'singular' parameter that currently has no ## effect. error("integral2: method tiled not implemented yet"); endif endfunction function q = outer_iterated (f, xa, xb, ya, yb, abstol, reltol) # check upper and lower bounds of y if !is_function_handle(ya) if isscalar (ya) ya = @(x) ya * ones(rows(x), columns(x)); else error ("integral2: 'ya' must be a constant or a (vectorized) function"); endif endif if !is_function_handle(yb) if isscalar (yb) yb = @(x) yb * ones(rows(x), columns(x)); else error ("integral2: 'ya' must be a constant or a (vectorized) function"); endif endif inner = @inner_iterated; q = feval (@quadcc, @(x) inner (x, f, ya, yb, abstol, reltol), xa, xb, ... [abstol, reltol]); endfunction function q = inner_iterated (x, f, ya, yb, abstol, reltol) q = zeros (size (x)); for i = 1 : length (x) q(i) = feval (@quadcc, @(y) f(x(i), y), ya(x(i)), yb(x(i)), ... [abstol, reltol]); endfor endfunction % method="auto" which will default to 'tiled' %!test %! f = @(x, y) x .* y; %! assert (integral2 (f, 0, 1, 0, 1), 0.25, 1e-10); %!test %! f = @(x, y) 9 * x.^2 + 15 * y.^2; %! assert (integral2 (f, 0, 5, -5, 0, "AbsTol", 1e-9), 5000, 1e-9); %! assert (integral2 (f, 0, 5, -5, 0, "RelTol", 1e-6), 5000, -1e-6); %! assert (integral2 (f, 0, 5, -5, 0, "RelTol", 1e-6, "AbsTol", 1e-9), %! 5000, 1e-9); ## tests from dblquad %!assert (integral2 (@(x, y) 1 ./ (x+y), 0, 1, 0, 1, "AbsTol", 1e-7), %! 2*log (2), 1e-7); %!assert (integral2 (@(x, y) 1 ./ (x+y), 0, 1, 0, 1, "RelTol", 1e-6), %! 2*log (2), -1e-6); %!assert (integral2 (@(x, y) 1 ./ (x+y), 0, 1, 0, 1, "AbsTol", 1e-8, %! "RelTol", 1e-6), 2*log (2), -1e-6); %!assert (integral2 (@(x, y) exp (-x.^2 - y.^2) , -1, 1, -1, 1), %! pi * erf (1).^2, 1e-10); %!assert (integral2 (@plus, 1, 2, 3, 4), 5, 1e-10); %!assert (integral2 (@(x,y) 1 ./ (x + y), 0, 1, 0, @(x) 1 - x), 1, -1e-6); % method="iterated" %!test %! f = @(x, y) x .* y; %! assert (integral2 (f, 0, 1, 0, 1, "method", "iterated"), 0.25, 1e-10); %!test %! f = @(x, y) 9 * x.^2 + 15 * y.^2; %! assert (integral2 (f, 0, 5, -5, 0, "AbsTol", 1e-9, "method", "iterated"), %! 5000, 1e-9); %! assert (integral2 (f, 0, 5, -5, 0, "RelTol", 1e-6, "method", "iterated"), %! 5000, -1e-6); %! assert (integral2 (f, 0, 5, -5, 0, "RelTol", 1e-6, "method", "iterated", %! "AbsTol", 1e-9),5000, 1e-9); ## tests from dblquad %!assert (integral2 (@(x, y) 1 ./ (x+y), 0, 1, 0, 1, "AbsTol", 1e-7, "method", %! "iterated"), 2*log (2), 1e-7); %!assert (integral2 (@(x, y) 1 ./ (x+y), 0, 1, 0, 1, "RelTol", 1e-6, "method", %! "iterated"), 2*log (2), -1e-6); %!assert (integral2 (@(x, y) 1 ./ (x+y), 0, 1, 0, 1, "AbsTol", 1e-8, %! "RelTol", 1e-6, "method", "iterated"), 2*log (2), -1e-6); %!assert (integral2 (@(x, y) exp (-x.^2 - y.^2) , -1, 1, -1, 1, "method", %! "iterated"), pi * erf (1).^2, 1e-10); %!assert (integral2 (@plus, 1, 2, 3, 4, "method", "iterated"), 5, 1e-10); %!assert (integral2 (@(x,y) 1 ./ (x + y), 0, 1, 0, @(x) 1 - x, "method", %! "iterated"), 1, -1e-6); ## Test input validation %!error integral2 %!error integral2 (0, 1 ,2 ,3 ,4) %!error integral2 (@plus) %!error integral2 (@plus, 1) %!error integral2 (@plus, 1, 2) %!error integral2 (@plus, 1, 2, 3) %!error integral2 (@plus, 1, 2, 3, [4 5]) %!error integral2 (@plus, 1, 2, 3, "test") %!error integral2 (@plus, 1, 2, 3, 4, "foo") %!error integral2 (@plus, 1, 2, 3, 4, "foo", "bar") %!error integral2 (@plus, 1, 2, 3, 4, 99, "bar") %!error integral2 (@plus, 1, 2, 3, 4, "AbsTol", "foo") %!error integral2 (@plus, 1, 2, 3, 4, "AbsTol", [1, 2]) %!error integral2 (@plus, 1, 2, 3, 4, "AbsTol", -1) %!error integral2 (@plus, 1, 2, 3, 4, "RelTol", "foo") %!error integral2 (@plus, 1, 2, 3, 4, "RelTol", [1, 2]) %!error integral2 (@plus, 1, 2, 3, 4, "RelTol", -1)