## Copyright (C) 2017 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} =} integral3 (@var{f}, @var{xa}, @var{xb}, @var{ya}, @var{yb}, @var{za}, @var{zb}) ## @deftypefnx {} {@var{q} =} integral3 (@var{f}, @var{xa}, @var{xb}, @var{ya}, @var{yb}, @var{za}, @var{zb}, @var{prop}, @var{val}, @dots{}) ## ## Numerically evaluate the three-dimensional integral of @var{f} using adaptive ## quadrature over the three-dimensional domain defined by @var{xa}, @var{xb}, ## @var{ya}, @var{yb}, @var{za}, @var{zb} (scalars may be finite or infinite). Additionally, ## @var{ya} and @var{yb} may be scalar functions of @var{x} and @var{za} and @var{zb} ## maybe be scalar functions of @var{x} and @var{y}, 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 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. ## ## @item Vectorized ## Option to enable/disable vectorized integration. False forces octave to use ## only scalar inputs when calling the integrand, which enables integrands ## @math{f(x,y)} that have not been vectorized and only accept @var{x} and ## @var{y} as scalars. Default value is @code{true}. ## ## @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). ## @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 ## 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. ## @end enumerate ## ## ## Reference: @nospell{L.F. Shampine}, ## @cite{"@sc{matlab} program for quadrature in 2D"}, Applied Mathematics and ## Computation, pp. 266--274, Vol 1, 2008. ## ## @seealso{integral, integral2, quad, quadgk, quadv, quadl, quadcc, trapz, ## quad2d, dblquad, triplequad} ## @end deftypefn function q = integral3 (f, xa, xb, ya, yb, za, zb, varargin) if (nargin < 7 || (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") || ... ((! is_function_handle (ya)) && isa (ya, "single")) || ... ((! is_function_handle (yb)) && isa (yb, "single")) || ... ((! is_function_handle (za)) && isa (za, "single")) || ... ((! is_function_handle (zb)) && isa (zb, "single")); ## Set defaults, update with any specified parameters. if issingle abstol = 1e-5; reltol = 1e-4; else abstol = 1e-10; reltol = 1e-6; endif method = "auto"; vectorized = true; idx = 1; while (idx < nargin - 7) prop = varargin{idx++}; if (! ischar (prop)) error ("integral3: property PROP must be a string"); endif switch (tolower (prop)) case "abstol" abstol = varargin{idx++}; if (! ((isnumeric (abstol)) && (isscalar (abstol)) && (abstol >= 0))) error ("integral3: AbsTol value must be a numeric scalar >= 0"); endif case "reltol" reltol = varargin{idx++}; if (! ((isnumeric (reltol)) && (isscalar (reltol)) && (reltol >= 0))) error ("integral3: RelTol value must be a numeric scalar >= 0"); endif case "method" method = tolower (varargin{idx++}); if (! any (strcmp (method, {"auto", "iterated", "tiled"}))) error ("integral3 : method '%s' unrecognized", method); endif case "vectorized" # option to allow unvectorized functions to be used vectorized = varargin{idx++}; if (! islogical (vectorized)) error ("integral3: 'vectorized' must be a logical value"); endif otherwise error ("integral3: unknown property '%s'", prop); endswitch endwhile if strcmp (method, "auto") if ((isinf (xa)) || (isinf (xb)) || ... ((! is_function_handle(ya)) && (isinf (ya))) || ... ((! is_function_handle(yb)) && (isinf (yb))) || ... ((! is_function_handle(za)) && (isinf (za))) || ... ((! is_function_handle(zb)) && (isinf (zb)))) method = "iterated"; else method = "tiled"; endif endif # check upper and lower bounds of y if (! is_function_handle (ya)) if isscalar (ya) ya = @(x) ya * ones (size (x)); else error ("integral3: 'ya' must be a constant or a (vectorized) function."); endif endif if (! is_function_handle (yb)) if isscalar (yb) yb = @(x) yb * ones (size (x)); else error ("integral3: 'ya' must be a constant or a (vectorized) function."); endif endif # check upper and lower bounds of z if (! is_function_handle (za)) if isscalar (za) za = @(x, y) za * ones (size(y)); else error ("integral3: 'za' must be a constant or a (vectorized) function."); endif endif if (! is_function_handle (zb)) if isscalar (zb) zb = @(x, y) zb * ones (size (y)); else error ("integral3: 'za' must be a constant or a (vectorized) function.") endif endif inner = @inner; q = feval (@quadcc, @(x) inner (x, f, ya, yb, za, zb, vectorized, method, ... abstol, reltol), xa, xb, [abstol, reltol]); endfunction function q = inner (x, f, ya, yb, za, zb, vectorized, method, abstol, reltol) q = zeros (size (x)); for i = 1 : length (x) za2 = @(y) za(x(i), y); zb2 = @(y) zb(x(i), y); f2 = @(y, z) f(x(i), y, z); if (! vectorized) f2 = @(y, z) arrayfun (f2, y, z); endif if strcmp (method, "iterated") inner = @inner_iterated; q(i) = feval (@quadcc, @(y) inner (y, f2, za2, zb2, abstol, reltol), ... ya(x(i)), yb(x(i)), [abstol, reltol]); else q(i) = quad2d (f2, ya(x(i)), yb(x(i)), za2, zb2, 'AbsTol', abstol, ... 'RelTol', reltol); endif endfor endfunction function q = inner_iterated (y, f2, za2, zb2, abstol, reltol) q = zeros (size (y)); for i = 1 : length (y) q(i) = feval (@quadcc, @(z) f2(y(i), z), za2(y(i)), zb2(y(i)), ... [abstol, reltol]); endfor endfunction ## method tests %!test %! f = @(x, y, z) x .* y .* z; %! assert (integral3 (f, 0, 1, 0, 1, 0, 1), 0.125, 1e-10); %! assert (integral3 (f, 0, 1, 0, 1, 0, 1, "method", "tiled"), 0.125, 1e-10); %! assert (integral3 (f, 0, 1, 0, 1, 0, 1, "method", "iterated"), 0.125, 1e-10); %! assert (integral3 (f, 0, 1, 0, 1, 0, 1, "method", "auto"), 0.125, 1e-10); ## vectorized = false test %!test %! f = @(x, y, z) x * y * z; %! assert (integral3 (f, 0, 1, 0, 1, 0, 1, "vectorized", false), 0.125, 1e-10); ## tolerance tests %!test %! f = @(x, y, z) 2 * x.^2 + 3 * y.^2 + 4 * z.^2; %! assert (integral3 (f, 0, 5, -5, 0, 0, 5, "AbsTol", 1e-9), 9375, 1e-9); %! assert (integral3 (f, 0, 5, -5, 0, 0, 5, "RelTol", 1e-6), 9375, -1e-6); %! assert (integral3 (f, 0, 5, -5, 0, 0, 5, "RelTol", 1e-6, "AbsTol", 1e-9), %! 9375, 1e-9); ## non rectangular region ## This test is too slow with "iterated" method %!assert (integral3 (@(x,y,z) 1 ./ (x + y + z), 0, 1, 0, @(x) 1 - x, 0, %! @(x, y) 1 - x - y, "method", "tiled"), 0.25, 1e-6) ## Test input validation %!error integral3 %!error integral3 (0, 1 ,2 ,3 ,4, 5, 6) %!error integral3 (@plus) %!error integral3 (@plus, 1) %!error integral3 (@plus, 1, 2) %!error integral3 (@plus, 1, 2, 3) %!error integral3 (@plus, 1, 2, 3, 4, 5, [6 7]) %!error integral3 (@plus, 1, 2, 3, 4, 5, "test") %!error integral3 (@plus, 1, 2, 3, 4, 5, 6, "foo") %!error integral3 (@plus, 1, 2, 3, 4, 6, 6, "foo", "bar") %!error integral3 (@plus, 1, 2, 3, 4, 5, 6, 99, "bar") %!error integral3 (@plus, 1, 2, 3, 4, 5, 6, "AbsTol", "foo") %!error integral3 (@plus, 1, 2, 3, 4, 5, 6, "AbsTol", [1, 2]) %!error integral3 (@plus, 1, 2, 3, 4, 5, 6, "AbsTol", -1) %!error integral3 (@plus, 1, 2, 3, 4, 5, 6, "RelTol", "foo") %!error integral3 (@plus, 1, 2, 3, 4, 5, 6, "RelTol", [1, 2]) %!error integral3 (@plus, 1, 2, 3, 4, 5, 6, "RelTol", -1) %!error integral3 (@plus, 1, 2, 3, 4, 5, 6, "method", "bad")