######################################################################## ## ## Copyright (C) 2017-2020 The Octave Project Developers ## ## See the file COPYRIGHT.md in the top-level directory of this ## distribution or . ## ## 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} =} integral (@var{f}, @var{a}, @var{b}) ## @deftypefnx {} {@var{q} =} integral (@var{f}, @var{a}, @var{b}, @var{prop}, @var{val}, @dots{}) ## ## Numerically evaluate the integral of @var{f} from @var{a} to @var{b} using ## adaptive quadrature. ## ## @code{integral} is a wrapper for @code{quadcc} (general real, scalar ## integrands and limits), @code{quadgk} (integrals with specified integration ## paths), and @code{quadv} (array-valued integrands) that is intended to ## provide @sc{matlab} compatibility. More control of the numerical integration ## may be achievable by calling the various quadrature functions directly. ## ## @var{f} is a function handle, inline function, or string containing the name ## of the function to evaluate. The function @var{f} must be vectorized and ## return a vector of output values when given a vector of input values. ## ## @var{a} and @var{b} are the lower and upper limits of integration. Either ## or both limits may be infinite or contain weak end singularities. If either ## or both limits are complex, @code{integral} will perform a straight line ## path integral. Alternatively, a complex domain path can be specified using ## the @qcode{"Waypoints"} option (see below). ## ## Additional optional parameters can be specified using ## @qcode{"@var{property}", @var{value}} pairs. Valid properties are: ## ## @table @code ## @item Waypoints ## Specifies points to be used in defining subintervals of the quadrature ## algorithm, or if @var{a}, @var{b}, or @var{waypoints} are complex then ## the quadrature is calculated as a contour integral along a piecewise ## continuous path. For more detail see @code{quadgk}. ## ## @item ArrayValued ## @code{integral} expects @var{f} to return a scalar value unless ## @var{arrayvalued} is specified as true. This option will cause ## @code{integral} to perform the integration over the entire array and return ## @var{q} with the same dimensions as returned by @var{f}. For more detail ## see @code{quadv}. ## ## @item AbsTol ## Define the absolute error tolerance for the quadrature. The default ## absolute tolerance is 1e-10 (1e-5 for single). ## ## @item RelTol ## Define the relative error tolerance for the quadrature. The default ## relative tolerance 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 ## ## 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 ## As a consequence of using @code{quadcc}, @code{quadgk}, and @code{quadv}, ## certain option combinations are not supported. Currently, ## @qcode{"ArrayValued"} cannot be combined with @qcode{"RelTol"} or ## @qcode{"Waypoints"}. ## @end enumerate ## ## @seealso{integral2, integral3, quad, quadgk, quadv, quadl, quadcc, trapz, ## dblquad, triplequad} ## @end deftypefn function q = integral (f, a, b, varargin) if (nargin < 3 || (mod (nargin, 2) == 0)) print_usage (); endif ## quadcc can't handle complex limits & integrands, but quadgk & quadv can ## check for simple cases of complex limits and integrand eval to complex ## more general integrand / return check performed later with quadcc call complex_check = false; if (iscomplex(a) || iscomplex(b)) complex_check = true; elseif (iscomplex (feval (f, a)) || iscomplex (feval (f, b))) complex_check = true; endif if (nargin == 3) ## Pass the simplest case directly to general integrator. ## Let quadcc function handle input checks on function and limits. if (!complex_check) try q = quadcc (f, a, b); catch quaderror if strcmpi (quaderror.message, "quadcc: integrand F must return a single, real-valued vector") q = quadgk (f, a, b); else error(quaderror.message) endif end_try_catch else q = quadgk (f, a, b); endif else ## Parse options to determine how to call integrator. abstol = []; reltol = []; waypoints = []; arrayvalued = false; idx = 1; while (idx < nargin - 3) prop = varargin{idx++}; if (! ischar (prop)) error ("integral: property PROP must be a string"); endif switch (tolower (prop)) case "reltol" reltol = varargin{idx++}; case "abstol" abstol = varargin{idx++}; case "waypoints" waypoints = varargin{idx++}(:); case "arrayvalued" arrayvalued = varargin{idx++}; otherwise error ("integral: unknown property '%s'", prop); endswitch endwhile issingle = (isa (a, "single") || isa (b, "single") || isa (waypoints, "single")); if (arrayvalued) ## Pass vector-valued function to quadv, checking for conflicting params ## FIXME: Replace warning when have array compatible call with waypoints if (! isempty (waypoints)) warning(["integral: array-valued quadrature routine currently ", ... "unable to handle WayPoints. WayPoints are ignored."]); endif ## FIXME: Remove warning once we have reltol compatible arrayval'd quadfn if (! isempty (reltol)) warning(["integral: array-valued quadrature only accepts AbsTol.", ... " RelTol ignored."]); endif if (isempty (abstol)) abstol = ifelse (issingle, 1e-5, 1e-10); endif q = quadv (f, a, b, abstol); else if (isempty (abstol)) abstol = ifelse (issingle, 1e-5, 1e-10); endif if (isempty (reltol)) reltol = ifelse (issingle, 1e-4, 1e-6); endif if (! isempty (waypoints)) q = quadgk (f, a, b, "AbsTol", abstol, "RelTol", reltol, "WayPoints", waypoints); else if (!complex_check) try q = quadcc (f, a, b, [abstol, reltol]); catch quaderror if strcmpi (quaderror.message, "quadcc: integrand F must return a single, real-valued vector") q = quadgk (f, a, b, "AbsTol", abstol, "RelTol", reltol); else error(quaderror.message) endif end_try_catch else q = quadgk (f, a, b, "AbsTol", abstol, "RelTol", reltol); endif endif endif endif endfunction ## Matlab compatibility tests %!test %! f = @(x) exp (-x.^2) .* log (x).^2; %! emgamma = 0.57721566490153286; %! exact = (sqrt (pi)*(8*log (2)^2+8*emgamma*log (2)+pi^2+2*emgamma^2))/16; %! assert (integral (f, 0, Inf), exact, -1e-6); %! assert (integral (f, 0, Inf, "RelTol", 1e-12), exact, -1e-12); %!test # with parameter %! f = @(x, c) 1 ./ (x.^3 - 2*x - c); %! assert (integral (@(x) f(x,5), 0, 2), -0.4605015338467329, 1e-10); %!test # with tolerances %! f = @(x) log(x); %! assert (integral (@(x) f(x), 0, 1, "AbsTol", 1e-6), -1, 1e-6); %!test # waypoints %! f = @(x) 1./(2.*x-1); %! assert (integral (f, 0, 0, "Waypoints", [1+1i, 1-1i]), -pi*1i, 1e-10); %!test # an array-valued function %! f = @(x) sin ((1:5)*x); %! assert (integral (f, 0, 1, "ArrayValued", true), 1./[1:5]-cos(1:5)./[1:5], %! 1e-10); %!test # test single input/output %! assert (integral (@sin, 0, 1), cos(0)-cos(1), 1e-10); %! assert (class (integral (@sin, single (0), 1)), "single"); %! assert (class (integral (@sin, 0, single (1))), "single"); %! assert (class (integral (@sin, single (0), single (1))), "single"); %! assert (integral (@sin, 0, 1, "Waypoints", 0.5), cos(0)-cos(1), 1e-10); %! assert (class (integral (@sin, 0, 1, "Waypoints", single (0.5))), "single"); %! assert (class (integral (@sin, single (0), 1, "Waypoints", 0.5)), "single"); %! assert (class (integral (@sin, 0, single (1), "Waypoints", 0.5)), "single"); %!test # test complex argument handling %! f = @(x) round (exp (i*x)); %! assert (integral(f, 0, pi), quadgk(f, 0, pi), eps); %! assert (integral(f, -1, 1), 2, 5*eps); %! assert (integral (@sin, -i, i), 0, eps); %! assert (1.5 * integral (@sqrt, -1, 0), i, eps); %!test %! f = @(x) x.^5 .* exp (-x) .* sin (x); %! assert (integral (f, 0, inf, "RelTol", 1e-8, "AbsTol", 1e-12), -15, -1e-8); ## tests from quadcc %!assert (integral (@sin, -pi, pi), 0, 1e-10) %!assert (integral (inline ("sin"), -pi, pi), 0, 1e-10) %!assert (integral ("sin", -pi, pi), 0, 1e-10) %!assert (integral (@sin, -pi, 0), -2, 1e-10) %!assert (integral (@sin, 0, pi), 2, 1e-10) %!assert (integral (@(x) 1./(sqrt (x).*(x+1)), 0, Inf), pi, -1e-6) %!assert (integral (@(x) 1./(sqrt (x).*(x+1)), 0, Inf, %! "AbsTol", 0, "RelTol", 1e-8), %! pi, -1e-8) %!assert (integral (@(x) exp (-x .^ 2), -Inf, Inf), sqrt (pi), 1e-10) %!assert (integral (@(x) exp (-x .^ 2), -Inf, 0), sqrt (pi)/2, 1e-10) ## tests from quadgk %!assert (integral (@sin,-pi,pi, "WayPoints",0, "AbsTol",1e-6, "RelTol",1e-3), %! 0, 1e-6) %!assert (integral (@(x) abs (1 - x.^2), 0, 2, "Waypoints", 1), 2, 1e-10) %!assert (integral (@(z) log (z),1+1i,1+1i, "WayPoints", [1-1i, -1,-1i, -1+1i]), %! complex (0, pi), 1e-10) ## tests from quadv ## Test vector-valued functions %!assert (integral (@(x) [(sin (x)), (sin (2*x))], 0, pi, "ArrayValued", 1), %! [2, 0], 1e-10) ## Test matrix-valued functions %!test %! assert (integral (@(x) [x,x,x; x,exp(x),x; x,x,x], 0, 1, "ArrayValued", 1), %! [0.5,0.5,0.5; 0.5,(exp (1) - 1),0.5; 0.5,0.5,0.5], 1e-10); ## Test input validation %!error integral (@sin) %!error integral (@sin, 0) %!error integral (@sin, 0, 1, 1e-6, true, 4) %!error integral (@sin, 0, 1, "DummyArg") %!error integral (@sin, 0, 1, 2, 3) %!error integral (@sin, 0, 1, "foo", 3) %!error integral (@sin, 0, 1, "AbsTol", ones (2,2)) %!error integral (@sin, 0, 1, "AbsTol", -1) %!error integral (@sin, 0, 1, "RelTol", ones (2,2)) %!error integral (@sin, 0, 1, "RelTol", -1) %%TO DO ADD COMPLEX TESTS