## 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} =} quad2d (@var{f}, @var{xa}, @var{xb}, @var{ya}, @var{yb}) ## @deftypefnx {} {@var{q} =} quad2d (@var{f}, @var{xa}, @var{xb}, @var{ya}, @var{yb}, @var{prop}, @var{val}, @dots{}) ## @deftypefnx {} {[@var{q}, @var{err}, @var{iter}] =} quad2d (@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). ## ## @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 MaxFunEvals ## The maximum number of function calls to the vectorized function @var{f}. ## The default is 5000. ## ## @item Singular ## Use transforms to weaken singularities on the edge of the integration ## doamin. Valid options are @var{true} or @var{false}. The default is ## @var{true}. ## ## @item Vectorized ## Option to disable vectorized integration, forcing octave to use only scalar ## inputs when calling the integrand. ## ## @item FailurePlot ## If @code{quad2d} fails to converge to the desired error tolerance, a plot ## of the areas that still need refinement is plotted ## @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. @var{iter} is the number of vectorized function calls to the ## function @var{f} that were used. ## ## 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, quad2d, integral3, quad, quadgk, quadv, quadl, quadcc, ## trapz, dblquad, triplequad} ## @end deftypefn function [q, err, iter] = quad2d (f, xa, xb, ya, yb, varargin) if (nargin < 5 || (mod (nargin, 2) == 0)) print_usage (); endif ## Convert function given as a string to a function handle if (ischar (f)) f = @(x) feval (f, x); 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 = true; idx = 1; maxiter = 5000; failureplot = false; while (idx < nargin - 5) prop = varargin{idx++}; if (! ischar (prop)) error ("quad2d: property PROP must be a string"); endif switch (tolower (prop)) case "abstol" abstol = varargin{idx++}; if (! (isnumeric (abstol) && isscalar (abstol) && abstol >= 0)) error ("quad2d: AbsTol value must be a numeric scalar >= 0"); endif case "reltol" reltol = varargin{idx++}; if (! (isnumeric (reltol) && isscalar (reltol) && reltol >= 0)) error ("quad2d: RelTol value must be a numeric scalar >= 0"); endif case "vectorized" vectorized = varargin{idx++}; if !islogical (vectorized) error ("quad2d: 'vectorized' must be a logical value"); endif case "singular" singular = varargin{idx++}; if !islogical (singular) error ("quad2d: 'singular' must be a logical value"); endif case "maxfunevals" maxiter = varargin{idx++}; if (!(isnumeric (maxiter) && isscalar (maxiter) && floor(maxiter) == maxiter && maxiter >= 1)) error ("quad2d: MAxFunEvals value must be an integer scalar >= 1") endif case "failureplot" failureplot = varargin{idx++}; if !islogical (failureplot) error ("quad2d: 'failureplot' must be a logical value"); endif otherwise error ("quad2d: unknown property '%s'", prop); endswitch endwhile if ! vectorized f = @(x, y) arrayfun(f, x, y); endif # 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 ("quad2d: '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 ("quad2d: 'ya' must be a constant or a (vectorized) function.") endif endif iter = 0; qaccept = 0; qerraccept = 0; if singular # Shampine suggests using the singularity weakening transform # suggested by Havie # \int_a^b f(x) dx = \int_0^pi f (g(t)) (dx / dt) dt # where # g(t) = ((a - b) * cos(t) + (a + b)) / 2 # dx = - (a - b) * sin(t) / 2 dt # Now our integral is # \int_a^b \int_0^1 f(x,y) dydx # as we already subsitute for "y", so # gx(tx) = ((a - b) * cos(tx) + (a + b)) / 2 # gy(ty) = (1 - cos(ty)) / 2 # dydx = (b - a) * sin(tx) * sin(ty) / 4 dtydtx xtrans = @(tx) ((xa - xb) .* cos (tx) + (xa + xb)) ./ 2; ytrans = @(ty) (1 - cos (ty)) ./ 2; ztrans = @(tx, ty) (xb - xa) .* sin(tx) .* sin(ty) ./ 4; area = pi ^ 2; # Initialize tile list tilelist(1) = struct("xa", 0, "xb", pi, "ya", 0, "yb", pi, "q", 0, "qerr", Inf); else xtrans = @(tx) tx; ytrans = @(ty) ty; ztrans = @(tx, ty) 1; area = (xb - xa); # Initialize tile list tilelist(1) = struct("xa", xa, "xb", xb, "ya", 0, "yb", 1, "q", 0, "qerr", Inf); endif while (length(tilelist) > 0 && iter < maxiter) # Get tile with the largest error [~, indx] = max([tilelist.qerr]); tile = tilelist(indx); tilelist(indx) = []; # Subdivide the tile into 4 subtiles iter += 4; tiles(4) = struct("xa", tile.xa, "xb", tile.xa + (tile.xb - tile.xa) / 2, "ya", tile.ya, "yb", tile.ya + (tile.yb - tile.ya) / 2, "q", 0, "qerr", 0); tiles(3) = struct("xa", tile.xa, "xb", tile.xa + (tile.xb - tile.xa) / 2, "ya", tile.ya + (tile.yb - tile.ya) / 2, "yb", tile.yb, "q", 0, "qerr", 0); tiles(2) = struct("xa", tile.xa + (tile.xb - tile.xa) / 2, "xb", tile.xb, "ya", tile.ya, "yb", tile.ya + (tile.yb - tile.ya)/ 2, "q", 0, "qerr", 0); tiles(1) = struct("xa", tile.xa + (tile.xb - tile.xa) / 2, "xb", tile.xb, "ya", tile.ya + (tile.yb - tile.ya) / 2, "yb", tile.yb, "q", 0, "qerr", 0); # Perform the quadrature of 4 subtiles for i = 1:4 [tiles(i).q, tiles(i).qerr] = tensorproduct (f, ya, yb, tiles(i), xtrans, ytrans, ztrans, singular); endfor q = qaccept + sum([[tilelist.q], tiles.q]); err = qerraccept + sum([[tilelist.qerr], tiles.qerr]); tol = max (abstol, reltol .* abs (q)) / 4; # Shampine suggests taking a margin of a factor of 8 for # the local tolerance localtol = tol * ([tile.xb] - [tile.xa]) * ([tile.yb] - [tile.ya]) ... / area / 8; # If global tolerance is met, return if (err < tol) break; endif # Accept the tiles meeting the tolerance, and add the others back to # the list of tiles to treat indx = find([tiles.qerr] < localtol); qaccept += sum([tiles(indx).q]); qerraccept += sum([tiles(indx).qerr]); tiles(indx) = []; tilelist = [tilelist, tiles]; endwhile if (iter >= maxiter) if (err > max (abstol, reltol .* abs (q))) warning ("quad2d : Maximum number of sub-tiles without convergence.") else warning("quad2d : Maximum number of sub-tiles, maybe low accuracy.") endif if failureplot newplot title("QUAD2D : Areas needing refinement") for tile = tilelist xaa = xtrans(tile.xa); xbb = xtrans(tile.xb); y1 = ya(xaa) + ytrans(tile.ya) * (yb(xaa) - ya(xaa)); y2 = ya(xaa) + ytrans(tile.yb) * (yb(xaa) - ya(xaa)); y3 = ya(xbb) + ytrans(tile.yb) * (yb(xbb) - ya(xbb)); y4 = ya(xbb) + ytrans(tile.ya) * (yb(xbb) - ya(xbb)); patch ([xaa, xaa, xbb, xbb, xaa],[y1, y2, y3, y4, y1], 'b'); endfor endif endif endfunction function [q, qerr] = tensorproduct (f, ya, yb, tile, xtrans, ytrans, ztrans, singular) # The Shampine TwoD paper proposes using a G3,K7 rule in a tensor product # I couldn't find a tabulated abscissas et weighrs of a G3,K7 rule publically # available, so I'm using a G7,G15 rule from Octave's implementation of quadgk persistent abscissa = [-0.9914553711208126e+00, -0.9491079123427585e+00, ... -0.8648644233597691e+00, -0.7415311855993944e+00, ... -0.5860872354676911e+00, -0.4058451513773972e+00, ... -0.2077849550078985e+00, 0.0000000000000000e+00, ... 0.2077849550078985e+00, 0.4058451513773972e+00, ... 0.5860872354676911e+00, 0.7415311855993944e+00, ... 0.8648644233597691e+00, 0.9491079123427585e+00, ... 0.9914553711208126e+00]; persistent weights15 = ... [0.2293532201052922e-01, 0.6309209262997855e-01, ... 0.1047900103222502e+00, 0.1406532597155259e+00, ... 0.1690047266392679e+00, 0.1903505780647854e+00, ... 0.2044329400752989e+00, 0.2094821410847278e+00, ... 0.2044329400752989e+00, 0.1903505780647854e+00, ... 0.1690047266392679e+00, 0.1406532597155259e+00, ... 0.1047900103222502e+00, 0.6309209262997855e-01, ... 0.2293532201052922e-01]; persistent weights7 = [0.0, ... 0.1294849661688697e+00, 0.0, ... 0.2797053914892767e+00, 0.0, ... 0.3818300505051889e+00, 0.0, ... 0.4179591836734694e+00, 0.0, ... 0.3818300505051889e+00, 0.0, ... 0.2797053914892767e+00, 0.0, ... 0.1294849661688697e+00, 0.0]; xaa = tile.xa; xbb = tile.xb; yaa = tile.ya; ybb = tile.yb; tx = ((xbb - xaa) * abscissa + xaa + xbb) / 2; x = xtrans(tx); ty = (abscissa' * (ybb - yaa) + yaa + ybb) / 2; y = ones (15, 1) * ya(x) + ytrans(ty) * (yb(x) - ya(x)); xhalfwidth = (xbb - xaa ) / 2; yhalfwidth = ones (15, 1) * (yb(x) - ya(x)) .* (ybb - yaa) ./ 2; x = ones (15, 1) * x; tx = ones (15,1) * tx; ty = ty * ones (1, 15); z = yhalfwidth .* f (x, y) .* ztrans(tx, ty) .* xhalfwidth; q = weights15 * (weights15 * z)'; qerr = abs(weights7 * (weights7 * z)' - q); endfunction ## Test input validation %!error quad2d %!error quad2d (0, 1 ,2 ,3 ,4) %!error quad2d (@plus) %!error quad2d (@plus, 1) %!error quad2d (@plus, 1, 2) %!error quad2d (@plus, 1, 2, 3) %!error quad2d (@plus, 1, 2, 3, [4 5]) %!error quad2d (@plus, 1, 2, 3, "test") %!error quad2d (@plus, 1, 2, 3, 4, "foo") %!error quad2d (@plus, 1, 2, 3, 4, "foo", "bar") %!error quad2d (@plus, 1, 2, 3, 4, 99, "bar") %!error quad2d (@plus, 1, 2, 3, 4, "AbsTol", "foo") %!error quad2d (@plus, 1, 2, 3, 4, "AbsTol", [1, 2]) %!error quad2d (@plus, 1, 2, 3, 4, "AbsTol", -1) %!error quad2d (@plus, 1, 2, 3, 4, "RelTol", "foo") %!error quad2d (@plus, 1, 2, 3, 4, "RelTol", [1, 2]) %!error quad2d (@plus, 1, 2, 3, 4, "RelTol", -1) %!test %! f = @(x, y) x .* y; %! assert (quad2d (f, 0, 1, 0, 1), 0.25, 1e-10); %!test %! f = @(x, y) 9 * x.^2 + 15 * y.^2; %! assert (quad2d (f, 0, 5, -5, 0, "AbsTol", 1e-9), 5000, 1e-9); %! assert (quad2d (f, 0, 5, -5, 0, "RelTol", 1e-6), 5000, -1e-6); %! assert (quad2d (f, 0, 5, -5, 0, "RelTol", 1e-6, "AbsTol", 1e-9), %! 5000, 1e-9); ## tests from dblquad %!assert (quad2d (@(x, y) 1 ./ (x+y), 0, 1, 0, 1, "AbsTol", 1e-7), %! 2*log (2), 1e-6) %!assert (quad2d (@(x, y) 1 ./ (x+y), 0, 1, 0, 1, "RelTol", 1e-6), %! 2*log (2), -1e-6) %!assert (quad2d (@(x, y) 1 ./ (x+y), 0, 1, 0, 1, "AbsTol", 1e-8, %! "RelTol", 1e-6), 2*log (2), -1e-6) %!assert (quad2d (@(x, y) exp (-x.^2 - y.^2) , -1, 1, -1, 1), %! pi * erf (1).^2, 1e-10) %!assert (quad2d (@plus, 1, 2, 3, 4), 5, 1e-10) %!assert (quad2d (@(x,y) 1 ./ (x + y), 0, 1, 0, @(x) 1 - x), 1, -1e-6) %!assert (quad2d (@(x,y) 1 ./ (x + y), 0, 1, 0, @(x) 1 - x, "Singular", true), 1, -1e-6)