--- /dev/null
+## Copyright (C) 2008, 2009 VZLU Prague, a.s.
+## Copyright (C) 2010 Olaf Till <olaf.till@uni-jena.de>
+##
+## This program 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.
+##
+## This program 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
+## this program; if not, see <http://www.gnu.org/licenses/>.
+
+## -*- texinfo -*-
+## @deftypefn {Function File} {} vfzero (@var{fun}, @var{x0})
+## @deftypefnx {Function File} {} vfzero (@var{fun}, @var{x0}, @var{options})
+## @deftypefnx {Function File} {[@var{x}, @var{fval}, @var{info}, @var{output}] =} vfzero (@dots{})
+## A variant of @code{fzero}. Finds a zero of a vector-valued
+## multivariate function where each output element only depends on the
+## input element with the same index (so the Jacobian is diagonal).
+##
+## @var{fun} should be a handle or name of a function returning a column
+## vector. @var{x0} should be a two-column matrix, each row specifying
+## two points which bracket a zero of the respective output element of
+## @var{fun}.
+##
+## If @var{x0} is a single-column matrix then several nearby and distant
+## values are probed in an attempt to obtain a valid bracketing. If
+## this is not successful, the function fails. @var{options} is a
+## structure specifying additional options. Currently, @code{vfzero}
+## recognizes these options: @code{"FunValCheck"}, @code{"OutputFcn"},
+## @code{"TolX"}, @code{"MaxIter"}, @code{"MaxFunEvals"}. For a
+## description of these options, see @ref{doc-optimset,,optimset}.
+##
+## On exit, the function returns @var{x}, the approximate zero and
+## @var{fval}, the function value thereof. @var{info} is a column vector
+## of exit flags that can have these values:
+##
+## @itemize
+## @item 1 The algorithm converged to a solution.
+##
+## @item 0 Maximum number of iterations or function evaluations has been
+## reached.
+##
+## @item -1 The algorithm has been terminated from user output function.
+##
+## @item -5 The algorithm may have converged to a singular point.
+## @end itemize
+##
+## @var{output} is a structure containing runtime information about the
+## @code{fzero} algorithm. Fields in the structure are:
+##
+## @itemize
+## @item iterations Number of iterations through loop.
+##
+## @item nfev Number of function evaluations.
+##
+## @item bracketx A two-column matrix with the final bracketing of the
+## zero along the x-axis.
+##
+## @item brackety A two-column matrix with the final bracketing of the
+## zero along the y-axis.
+## @end itemize
+## @seealso{optimset, fsolve}
+## @end deftypefn
+
+## This is essentially the ACM algorithm 748: Enclosing Zeros of
+## Continuous Functions due to Alefeld, Potra and Shi, ACM Transactions
+## on Mathematical Software, Vol. 21, No. 3, September 1995. Although
+## the workflow should be the same, the structure of the algorithm has
+## been transformed non-trivially; instead of the authors' approach of
+## sequentially calling building blocks subprograms we implement here a
+## FSM version using one interior point determination and one bracketing
+## per iteration, thus reducing the number of temporary variables and
+## simplifying the algorithm structure. Further, this approach reduces
+## the need for external functions and error handling. The algorithm has
+## also been slightly modified.
+
+## Author: Jaroslav Hajek <highegg@gmail.com>
+
+## PKG_ADD: __all_opts__ ("vfzero");
+
+function [x, fval, info, output] = vfzero (fun, x0, options = struct ())
+
+ ## Get default options if requested.
+ if (nargin == 1 && ischar (fun) && strcmp (fun, 'defaults'))
+ x = optimset ("MaxIter", Inf, "MaxFunEvals", Inf, "TolX", 1e-8, \
+ "OutputFcn", [], "FunValCheck", "off");
+ return;
+ endif
+
+ if (nargin < 2 || nargin > 3)
+ print_usage ();
+ endif
+
+ if (ischar (fun))
+ fun = str2func (fun, "global");
+ endif
+
+ ## TODO
+ ## displev = optimget (options, "Display", "notify");
+ funvalchk = strcmpi (optimget (options, "FunValCheck", "off"), "on");
+ outfcn = optimget (options, "OutputFcn");
+ tolx = optimget (options, "TolX", 1e-8);
+ maxiter = optimget (options, "MaxIter", Inf);
+ maxfev = optimget (options, "MaxFunEvals", Inf);
+ nx = rows (x0);
+ ## fun may assume a certain length of x, so we will always call it
+ ## with the full-length x, even if only some elements are needed
+
+ persistent mu = 0.5;
+
+ if (funvalchk)
+ ## Replace fun with a guarded version.
+ fun = @(x) guarded_eval (fun, x);
+ endif
+
+ ## The default exit flag if exceeded number of iterations.
+ info = zeros (nx, 1);
+ niter = 0;
+ nfev = 0;
+
+ x = fval = fc = a = fa = b = fb = aa = c = u = fu = NaN (nx, 1);
+ bracket_ready = false (nx, 1);
+ eps = eps (class (x0));
+
+ ## Prepare...
+ a = x0(:, 1);
+ fa = fun (a)(:);
+ nfev = 1;
+ if (columns (x0) > 1)
+ b = x0(:, 2);
+ fb = fun (b)(:);
+ nfev += 1;
+ else
+ ## Try to get b.
+ aa(idx = a == 0) = 1;
+ aa(! idx) = a(! idx);
+ for tb = [0.9*aa, 1.1*aa, aa-1, aa+1, 0.5*aa 1.5*aa, -aa, 2*aa, -10*aa, 10*aa]
+ tfb = fun (tb)(:); nfev += 1;
+ idx = ! bracket_ready & sign (fa) .* sign (tfb) <= 0;
+ bracket_ready |= idx;
+ b(idx) = tb(idx);
+ fb(idx) = tfb(idx);
+ if (all (bracket_ready))
+ break;
+ endif
+ endfor
+ endif
+
+ tp = a(idx = b < a);
+ a(idx) = b(idx);
+ b(idx) = tp;
+
+ tp = fa(idx);
+ fa(idx) = fb(idx);
+ fb(idx) = tp;
+
+ if (! all (sign (fa) .* sign (fb) <= 0))
+ error ("fzero:bracket", "vfzero: not a valid initial bracketing");
+ endif
+
+ slope0 = (fb - fa) ./ (b - a);
+
+ idx = fa == 0;
+ b(idx) = a(idx);
+ fb(idx) = fa(idx);
+
+ idx = (! idx & fb == 0);
+ a(idx) = b(idx);
+ fa(idx) = fb(idx);
+
+ itype = ones (nx, 1);
+
+ idx = abs (fa) < abs (fb);
+ u(idx) = a(idx); fu(idx) = fa(idx);
+ u(! idx) = b(! idx); fu(! idx) = fb(! idx);
+
+ d = e = u;
+ fd = fe = fu;
+ mba = mu * (b - a);
+ not_ready = true (nx, 1);
+ while (niter < maxiter && nfev < maxfev && any (not_ready))
+
+ ## itype == 1
+ type1idx = not_ready & itype == 1;
+ ## The initial test.
+ idx = b - a <= 2*(2 * eps * abs (u) + tolx) & type1idx;
+ x(idx) = u(idx); fval(idx) = fu(idx);
+ info(idx) = 1;
+ not_ready(idx) = false;
+ type1idx &= not_ready;
+ exclidx = type1idx;
+ ## Secant step.
+ idx = type1idx & \
+ (tidx = abs (fa) <= 1e3*abs (fb) & abs (fb) <= 1e3*abs (fa));
+ c(idx) = u(idx) - (a(idx) - b(idx)) ./ (fa(idx) - fb(idx)) .* fu(idx);
+ ## Bisection step.
+ idx = type1idx & ! tidx;
+ c(idx) = 0.5*(a(idx) + b(idx));
+ d(type1idx) = u(type1idx); fd(type1idx) = fu(type1idx);
+ itype(type1idx) = 5;
+
+ ## itype == 2 or 3
+ type23idx = not_ready & ! exclidx & (itype == 2 | itype == 3);
+ exclidx |= type23idx;
+ uidx = cellfun (@ (x) length (unique (x)), \
+ num2cell ([fa, fb, fd, fe], 2)) == 4;
+ oidx = sign (c - a) .* sign (c - b) > 0;
+ ## Inverse cubic interpolation.
+ idx = type23idx & (uidx & ! oidx);
+ q11 = (d(idx) - e(idx)) .* fd(idx) ./ (fe(idx) - fd(idx));
+ q21 = (b(idx) - d(idx)) .* fb(idx) ./ (fd(idx) - fb(idx));
+ q31 = (a(idx) - b(idx)) .* fa(idx) ./ (fb(idx) - fa(idx));
+ d21 = (b(idx) - d(idx)) .* fd(idx) ./ (fd(idx) - fb(idx));
+ d31 = (a(idx) - b(idx)) .* fb(idx) ./ (fb(idx) - fa(idx));
+ q22 = (d21 - q11) .* fb(idx) ./ (fe(idx) - fb(idx));
+ q32 = (d31 - q21) .* fa(idx) ./ (fd(idx) - fa(idx));
+ d32 = (d31 - q21) .* fd(idx) ./ (fd(idx) - fa(idx));
+ q33 = (d32 - q22) .* fa(idx) ./ (fe(idx) - fa(idx));
+ c(idx) = a(idx) + q31 + q32 + q33;
+ ## Quadratic interpolation + newton.
+ idx = type23idx & (oidx | ! uidx);
+ a0 = fa(idx);
+ a1 = (fb(idx) - fa(idx))./(b(idx) - a(idx));
+ a2 = ((fd(idx) - fb(idx))./(d(idx) - b(idx)) - a1) ./ (d(idx) - a(idx));
+ ## Modification 1: this is simpler and does not seem to be worse.
+ c(idx) = a(idx) - a0./a1;
+ taidx = a2 != 0;
+ tidx = idx;
+ tidx(tidx) = taidx;
+ c(tidx) = a(tidx)(:) - (a0(taidx)./a1(taidx))(:);
+ for i = 1:3
+ tidx &= i <= itype;
+ taidx = tidx(idx);
+ pc = a0(taidx)(:) + (a1(taidx)(:) + \
+ a2(taidx)(:).*(c(tidx) - b(tidx))(:)) \
+ .*(c(tidx) - a(tidx))(:);
+ pdc = a1(taidx)(:) + a2(taidx)(:).*(2*c(tidx) - a(tidx) - b(tidx))(:);
+ tidx0 = tidx;
+ tidx0(tidx0, 1) &= (p0idx = pdc == 0);
+ taidx0 = tidx0(idx);
+ tidx(tidx, 1) &= ! p0idx;
+ c(tidx0) = a(tidx0)(:) - (a0(taidx0)./a1(taidx0))(:);
+ c(tidx) = c(tidx)(:) - (pc(! p0idx)./pdc(! p0idx))(:);
+ endfor
+ itype(type23idx) += 1;
+
+ ## itype == 4
+ type4idx = not_ready & ! exclidx & itype == 4;
+ exclidx |= type4idx;
+ ## Double secant step.
+ idx = type4idx;
+ c(idx) = u(idx) - 2*(b(idx) - a(idx))./(fb(idx) - fa(idx)).*fu(idx);
+ ## Bisect if too far.
+ idx = type4idx & abs (c - u) > 0.5*(b - a);
+ c(idx) = 0.5 * (b(idx) + a(idx));
+ itype(type4idx) = 5;
+
+ ## itype == 5
+ type5idx = not_ready & ! exclidx & itype == 5;
+ ## Bisection step.
+ idx = type5idx;
+ c(idx) = 0.5 * (b(idx) + a(idx));
+ itype(type5idx) = 2;
+
+ ## Don't let c come too close to a or b.
+ delta = 2*0.7*(2 * eps * abs (u) + tolx);
+ nidx = not_ready & ! (idx = b - a <= 2*delta);
+ idx &= not_ready;
+ c(idx) = (a(idx) + b(idx))/2;
+ c(nidx) = max (a(nidx) + delta(nidx), \
+ min (b(nidx) - delta(nidx), c(nidx)));
+
+ ## Calculate new point.
+ idx = not_ready;
+ x(idx, 1) = c(idx, 1);
+ if (any (idx))
+ c(! idx) = u(! idx); # to have some working place-holders since
+ # fun() might expect full-length
+ # argument
+ fval(idx, 1) = fc(idx, 1) = fun (c)(:)(idx, 1);
+ niter ++; nfev ++;
+ endif
+
+ ## Modification 2: skip inverse cubic interpolation if
+ ## nonmonotonicity is detected.
+ nidx = not_ready & ! (idx = sign (fc - fa) .* sign (fc - fb) >= 0);
+ idx &= not_ready;
+ ## The new point broke monotonicity.
+ ## Disable inverse cubic.
+ fe(idx) = fc(idx);
+ ##
+ e(nidx) = d(nidx); fe(nidx) = fd(nidx);
+
+ ## Bracketing.
+ idx1 = not_ready & sign (fa) .* sign (fc) < 0;
+ idx2 = not_ready & ! idx1 & sign (fb) .* sign (fc) < 0;
+ idx3 = not_ready & ! (idx1 | idx2) & fc == 0;
+ d(idx1) = b(idx1); fd(idx1) = fb(idx1);
+ b(idx1) = c(idx1); fb(idx1) = fc(idx1);
+ d(idx2) = a(idx2); fd(idx2) = fa(idx2);
+ a(idx2) = c(idx2); fa(idx2) = fc(idx2);
+ a(idx3) = b(idx3) = c(idx3); fa(idx3) = fb(idx3) = fc(idx3);
+ info(idx3) = 1;
+ not_ready(idx3) = false;
+ if (any (not_ready & ! (idx1 | idx2 | idx3)))
+ ## This should never happen.
+ error ("fzero:bracket", "vfzero: zero point is not bracketed");
+ endif
+
+ ## If there's an output function, use it now.
+ if (! isempty (outfcn))
+ optv.funccount = nfev;
+ optv.fval = fval;
+ optv.iteration = niter;
+ idx = not_ready & outfcn (x, optv, "iter");
+ info(idx) = -1;
+ not_ready(idx) = false;
+ endif
+
+ nidx = not_ready & ! (idx = abs (fa) < abs (fb));
+ idx &= not_ready;
+ u(idx) = a(idx); fu(idx) = fa(idx);
+ u(nidx) = b(nidx); fu(nidx) = fb(nidx);
+ idx = not_ready & b - a <= 2*(2 * eps * abs (u) + tolx);
+ info(idx) = 1;
+ not_ready(idx) = false;
+
+ ## Skip bisection step if successful reduction.
+ itype(not_ready & itype == 5 & (b - a) <= mba) = 2;
+ idx = not_ready & itype == 2;
+ mba(idx) = mu * (b(idx) - a(idx));
+ endwhile
+
+ ## Check solution for a singularity by examining slope
+ idx = not_ready & info == 1 & (b - a) != 0;
+ idx(idx, 1) &= \
+ abs ((fb(idx, 1) - fa(idx, 1))./(b(idx, 1) - a(idx, 1)) \
+ ./ slope0(idx, 1)) > max (1e6, 0.5/(eps+tolx));
+ info(idx) = - 5;
+
+ output.iterations = niter;
+ output.funcCount = nfev;
+ output.bracketx = [a, b];
+ output.brackety = [fa, fb];
+
+endfunction
+
+## An assistant function that evaluates a function handle and checks for
+## bad results.
+function fx = guarded_eval (fun, x)
+ fx = fun (x);
+ if (! isreal (fx))
+ error ("fzero:notreal", "vfzero: non-real value encountered");
+ elseif (any (isnan (fx)))
+ error ("fzero:isnan", "vfzero: NaN value encountered");
+ endif
+endfunction
+
+%!shared opt0
+%! opt0 = optimset ("tolx", 0);
+%!assert(vfzero(@cos, [0, 3], opt0), pi/2, 10*eps)
+%!assert(vfzero(@(x) x^(1/3) - 1e-8, [0,1], opt0), 1e-24, 1e-22*eps)
git://git.creatis.insa-lyon.fr / CreaPhase.git / blobdiff