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Use Octave coding conventions in all m-file %!test blocks
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Use Octave coding conventions in all m-file %!test blocks
author | Rik <octave@nomad.inbox5.com> |
---|---|
date | Mon, 13 Feb 2012 07:29:44 -0800 |
parents | 7277fe922e99 |
children | 5d3a684236b0 |
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## Copyright (C) 2008-2012 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 ## <http://www.gnu.org/licenses/>. ## -*- texinfo -*- ## @deftypefn {Function File} {@var{q} =} quadgk (@var{f}, @var{a}, @var{b}) ## @deftypefnx {Function File} {@var{q} =} quadgk (@var{f}, @var{a}, @var{b}, @var{abstol}) ## @deftypefnx {Function File} {@var{q} =} quadgk (@var{f}, @var{a}, @var{b}, @var{abstol}, @var{trace}) ## @deftypefnx {Function File} {@var{q} =} quadgk (@var{f}, @var{a}, @var{b}, @var{prop}, @var{val}, @dots{}) ## @deftypefnx {Function File} {[@var{q}, @var{err}] =} quadgk (@dots{}) ## ## Numerically evaluate the integral of @var{f} from @var{a} to @var{b} ## using adaptive Gauss-Konrod quadrature. ## @var{f} is a function handle, inline function, or string ## containing the name of the function to evaluate. ## The formulation is based on a proposal by L.F. Shampine, ## @cite{"Vectorized adaptive quadrature in @sc{matlab}", Journal of ## Computational and Applied Mathematics, pp131-140, Vol 211, Issue 2, ## Feb 2008} where all function evaluations at an iteration are ## calculated with a single call to @var{f}. Therefore, the function ## @var{f} must be vectorized and must accept a vector of input values @var{x} ## and return an output vector representing the function evaluations at the ## given values of @var{x}. ## ## @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. ## Variable transformation will be used to treat any infinite intervals and ## weaken the singularities. For example: ## ## @example ## quadgk (@@(x) 1 ./ (sqrt (x) .* (x + 1)), 0, Inf) ## @end example ## ## @noindent ## Note that the formulation of the integrand uses the ## element-by-element operator @code{./} and all user functions to ## @code{quadgk} should do the same. ## ## The optional argument @var{tol} defines the absolute tolerance used to stop ## the integration procedure. The default value is @math{1e^{-10}}. ## ## The algorithm used by @code{quadgk} involves subdividing the ## integration interval and evaluating each subinterval. ## If @var{trace} is true then after computing each of these partial ## integrals display: (1) the number of subintervals at this step, ## (2) the current estimate of the error @var{err}, (3) the current estimate ## for the integral @var{q}. ## ## Alternatively, properties of @code{quadgk} can be passed to the function as ## pairs @code{"@var{prop}", @var{val}}. Valid properties are ## ## @table @code ## @item AbsTol ## Define the absolute error tolerance for the quadrature. The default ## absolute tolerance is 1e-10. ## ## @item RelTol ## Define the relative error tolerance for the quadrature. The default ## relative tolerance is 1e-5. ## ## @item MaxIntervalCount ## @code{quadgk} initially subdivides the interval on which to perform ## the quadrature into 10 intervals. Subintervals that have an ## unacceptable error are subdivided and re-evaluated. If the number of ## subintervals exceeds 650 subintervals at any point then a poor ## convergence is signaled and the current estimate of the integral is ## returned. The property "MaxIntervalCount" can be used to alter the ## number of subintervals that can exist before exiting. ## ## @item WayPoints ## Discontinuities in the first derivative of the function to integrate can be ## flagged with the @code{"WayPoints"} property. This forces the ends of ## a subinterval to fall on the breakpoints of the function and can result in ## significantly improved estimation of the error in the integral, faster ## computation, or both. For example, ## ## @example ## quadgk (@@(x) abs (1 - x.^2), 0, 2, "Waypoints", 1) ## @end example ## ## @noindent ## signals the breakpoint in the integrand at @code{@var{x} = 1}. ## ## @item Trace ## If logically true @code{quadgk} prints information on the ## convergence of the quadrature at each iteration. ## @end table ## ## If any of @var{a}, @var{b}, or @var{waypoints} is complex then the ## quadrature is treated as a contour integral along a piecewise ## continuous path defined by the above. In this case the integral is ## assumed to have no edge singularities. For example, ## ## @example ## @group ## quadgk (@@(z) log (z), 1+1i, 1+1i, "WayPoints", ## [1-1i, -1,-1i, -1+1i]) ## @end group ## @end example ## ## @noindent ## integrates @code{log (z)} along the square defined by @code{[1+1i, ## 1-1i, -1-1i, -1+1i]} ## ## The result of the integration is returned in @var{q}. ## @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. ## ## @seealso{quad, quadv, quadl, quadcc, trapz, dblquad, triplequad} ## @end deftypefn function [q, err] = quadgk (f, a, b, varargin) if (nargin < 3) print_usage (); endif if (b < a) [q, err] = quadgk (f, b, a, varargin{:}); q = -q; else abstol = 1e-10; reltol = 1e-5; waypoints = []; maxint = 650; trace = false; if (nargin > 3) if (! ischar (varargin{1})) if (!isempty (varargin{1})) abstol = varargin{1}; reltol = 0; endif if (nargin > 4) trace = varargin{2}; endif if (nargin > 5) error ("quadgk: can not pass additional arguments to user function"); endif else idx = 1; while (idx < nargin - 3) if (ischar (varargin{idx})) str = varargin{idx++}; if (strcmpi (str, "reltol")) reltol = varargin{idx++}; elseif (strcmpi (str, "abstol")) abstol = varargin{idx++}; elseif (strcmpi (str, "waypoints")) waypoints = varargin{idx++} (:); if (isreal(waypoints)) waypoints (waypoints < a | waypoints > b) = []; endif elseif (strcmpi (str, "maxintervalcount")) maxint = varargin{idx++}; elseif (strcmpi (str, "trace")) trace = varargin{idx++}; else error ("quadgk: unknown property %s", str); endif else error ("quadgk: expecting property to be a string"); endif endwhile if (idx != nargin - 2) error ("quadgk: expecting properties in pairs"); endif endif endif ## Convert function given as a string to a function handle if (ischar (f)) f = @(x) feval (f, x); endif ## Use variable subsitution to weaken endpoint singularities and to ## perform integration with endpoints at infinity. No transform for ## contour integrals if (iscomplex (a) || iscomplex (b) || iscomplex(waypoints)) ## contour integral, no transform subs = [a; waypoints; b]; h = sum (abs (diff (subs))); h0 = h; trans = @(t) t; elseif (isinf (a) && isinf(b)) ## Standard Infinite to finite integral transformation. ## \int_{-\infinity_^\infinity f(x) dx = \int_-1^1 f (g(t)) g'(t) dt ## where ## g(t) = t / (1 - t^2) ## g'(t) = (1 + t^2) / (1 - t^2) ^ 2 ## waypoint transform is then ## t = (2 * g(t)) ./ (1 + sqrt(1 + 4 * g(t) .^ 2)) if (!isempty (waypoints)) trans = @(x) (2 * x) ./ (1 + sqrt(1 + 4 * x .^ 2)); subs = [-1; trans(waypoints); 1]; else subs = linspace (-1, 1, 11)'; endif h = 2; h0 = b - a; trans = @(t) t ./ (1 - t.^2); f = @(t) f (t ./ (1 - t .^ 2)) .* (1 + t .^ 2) ./ ((1 - t .^ 2) .^ 2); elseif (isinf(a)) ## Formula defined in Shampine paper as two separate steps. One to ## weaken singularity at finite end, then a second to transform to ## a finite interval. The singularity weakening transform is ## \int_{-\infinity}^b f(x) dx = ## - \int_{-\infinity}^0 f (b - t^2) 2 t dt ## (note minus sign) and the finite interval transform is ## \int_{-\infinity}^0 f(b - t^2) 2 t dt = ## \int_{-1}^0 f (b - g(s) ^ 2) 2 g(s) g'(s) ds ## where ## g(s) = s / (1 + s) ## g'(s) = 1 / (1 + s) ^ 2 ## waypoint transform is then ## t = sqrt (b - x) ## s = - t / (t + 1) if (!isempty (waypoints)) tmp = sqrt (b - waypoints); trans = @(x) - x ./ (x + 1); subs = [-1; trans(tmp); 0]; else subs = linspace (-1, 0, 11)'; endif h = 1; h0 = b - a; trans = @(t) b - (t ./ (1 + t)).^2; f = @(s) - 2 * s .* f (b - (s ./ (1 + s)) .^ 2) ./ ((1 + s) .^ 3); elseif (isinf(b)) ## Formula defined in Shampine paper as two separate steps. One to ## weaken singularity at finite end, then a second to transform to ## a finite interval. The singularity weakening transform is ## \int_a^\infinity f(x) dx = \int_0^\infinity f (a + t^2) 2 t dt ## and the finite interval transform is ## \int_0^\infinity f(a + t^2) 2 t dt = ## \int_0^1 f (a + g(s) ^ 2) 2 g(s) g'(s) ds ## where ## g(s) = s / (1 - s) ## g'(s) = 1 / (1 - s) ^ 2 ## waypoint transform is then ## t = sqrt (x - a) ## s = t / (t + 1) if (!isempty (waypoints)) tmp = sqrt (waypoints - a); trans = @(x) x ./ (x + 1); subs = [0; trans(tmp); 1]; else subs = linspace (0, 1, 11)'; endif h = 1; h0 = b - a; trans = @(t) a + (t ./ (1 - t)).^2; f = @(s) 2 * s .* f (a + (s ./ (1 - s)) .^ 2) ./ ((1 - s) .^ 3); else ## Davis, Rabinowitz, "Methods of Numerical Integration" p441 2ed. ## Presented in section 5 of the Shampine paper as ## g(t) = ((b - a) / 2) * (t / 2 * (3 - t^2)) + (b + a) / 2 ## g'(t) = ((b-a)/4) * (3 - 3t^2); ## waypoint transform can then be found by solving for t with ## Maxima (solve (c + 3*t - 3^3, t);). This gives 3 roots, two of ## which are complex for values between a and b and so can be ## ignored. The third is ## c = (-4*x + 2*(b+a)) / (b-a); ## k = ((sqrt(c^2 - 4) + c)/2)^(1/3); ## t = (sqrt(3)* 1i * (1 - k^2) - (1 + k^2)) / 2 / k; if (! isempty (waypoints)) trans = @__quadgk_finite_waypoint__; subs = [-1; trans(waypoints, a, b); 1]; else subs = linspace(-1, 1, 11)'; endif h = 2; h0 = b - a; trans = @(t) ((b - a) ./ 4) * t .* (3 - t.^2) + (b + a) ./ 2; f = @(t) f((b - a) ./ 4 .* t .* (3 - t.^2) + (b + a) ./ 2) .* ... 3 .* (b - a) ./ 4 .* (1 - t.^2); endif ## Split interval into at least 10 subinterval with a 15 point ## Gauss-Kronrod rule giving a minimum of 150 function evaluations while (length (subs) < 11) subs = [subs' ; subs(1:end-1)' + diff(subs') ./ 2, NaN](:)(1 : end - 1); endwhile subs = [subs(1:end-1), subs(2:end)]; warn_state = warning ("query", "Octave:divide-by-zero"); unwind_protect ## Singularity will cause divide by zero warnings warning ("off", "Octave:divide-by-zero"); ## Initial evaluation of the integrand on the subintervals [q_subs, q_errs] = __quadgk_eval__ (f, subs); q0 = sum (q_subs); err0 = sum (q_errs); if (isa (a, "single") || isa (b, "single") || isa (waypoints, "single")) myeps = eps ("single"); else myeps = eps; endif first = true; while (true) ## Check for subintervals that are too small. Test must be ## performed in untransformed subintervals. What is a good ## value for this test. Shampine suggests 100*eps if (any (abs (diff (trans (subs), [], 2) / h0) < 100 * myeps)) q = q0; err = err0; break; endif ## Quit if any evaluations are not finite (Inf or NaN) if (any (! isfinite (q_subs))) warning ("quadgk: non finite integrand encountered"); q = q0; err = err0; break; endif tol = max (abstol, reltol .* abs (q0)); ## If the global error estimate is meet exit if (err0 < tol) q = q0; err = err0; break; endif ## Accept the subintervals that meet the convergence criteria idx = find (abs (q_errs) < tol .* abs(diff (subs, [], 2)) ./ h); if (first) q = sum (q_subs (idx)); err = sum (q_errs(idx)); first = false; else q0 = q + sum (q_subs); err0 = err + sum (q_errs); q += sum (q_subs (idx)); err += sum (q_errs(idx)); endif subs(idx,:) = []; ## If no remaining subintervals exit if (rows (subs) == 0) break; endif if (trace) disp([rows(subs), err, q0]); endif ## Split remaining subintervals in two mid = (subs(:,2) + subs(:,1)) ./ 2; subs = [subs(:,1), mid; mid, subs(:,2)]; ## If the maximum subinterval count is met accept remaining ## subinterval and exit if (rows (subs) > maxint) warning ("quadgk: maximum interval count (%d) met", maxint); q += sum (q_subs); err += sum (q_errs); break; endif ## Evaluation of the integrand on the remaining subintervals [q_subs, q_errs] = __quadgk_eval__ (f, subs); endwhile if (err > max (abstol, reltol * abs(q))) warning ("quadgk: Error tolerance not met. Estimated error %g", err); endif unwind_protect_cleanup if (strcmp (warn_state.state, "on")) warning ("on", "Octave:divide-by-zero"); endif end_unwind_protect endif endfunction function [q, err] = __quadgk_eval__ (f, subs) ## A (15,7) point pair of Gauss-Konrod quadrature rules. The abscissa ## and weights are copied directly from dqk15w.f from quadpack 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 = ... diag ([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 = ... diag ([0.1294849661688697e+00, 0.2797053914892767e+00, ... 0.3818300505051889e+00, 0.4179591836734694e+00, ... 0.3818300505051889e+00, 0.2797053914892767e+00, ... 0.1294849661688697e+00]); halfwidth = diff (subs, [], 2) ./ 2; center = sum (subs, 2) ./ 2;; x = bsxfun (@plus, halfwidth * abscissa, center); y = reshape (f (x(:)), size(x)); ## This is faster than using bsxfun as the * operator can use a ## single BLAS call, rather than rows(sub) calls to the @times ## function. q = sum (y * weights15, 2) .* halfwidth; err = abs (sum (y(:,2:2:end) * weights7, 2) .* halfwidth - q); endfunction function t = __quadgk_finite_waypoint__ (x, a, b) c = (-4 .* x + 2.* (b + a)) ./ (b - a); k = ((sqrt(c .^ 2 - 4) + c) ./ 2) .^ (1/3); t = real ((sqrt(3) .* 1i * (1 - k .^ 2) - (1 + k .^ 2)) ./ 2 ./ k); endfunction %!assert (quadgk (@sin,-pi,pi), 0, 1e-6) %!assert (quadgk (inline ("sin"),-pi,pi), 0, 1e-6) %!assert (quadgk ("sin",-pi,pi), 0, 1e-6) %!assert (quadgk (@sin,-pi,pi, "waypoints", 0, "MaxIntervalCount", 100, "reltol", 1e-3, "abstol", 1e-6, "trace", false), 0, 1e-6) %!assert (quadgk (@sin,-pi,pi, 1e-6,false), 0, 1e-6) %!assert (quadgk (@sin,-pi,0), -2, 1e-6) %!assert (quadgk (@sin,0,pi), 2, 1e-6) %!assert (quadgk (@(x) 1./sqrt (x),0,1), 2, 1e-6) %!assert (quadgk (@(x) abs (1 - x.^2),0,2, "Waypoints", 1), 2, 1e-6) %!assert (quadgk (@(x) 1./(sqrt (x) .* (x+1)),0,Inf), pi, 1e-6) %!assert (quadgk (@(z) log (z),1+1i,1+1i, "WayPoints", [1-1i, -1,-1i, -1+1i]), -pi * 1i, 1e-6) %!assert (quadgk (@(x) exp (-x .^ 2),-Inf,Inf), sqrt (pi), 1e-6) %!assert (quadgk (@(x) exp (-x .^ 2),-Inf,0), sqrt (pi)/2, 1e-6) %error (quadgk (@sin)) %error (quadgk (@sin, -pi)) %error (quadgk (@sin, -pi, pi, "DummyArg"))