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########################################################################
##
## Copyright (C) 2013-2024 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or <https://octave.org/copyright/>.
##
## 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
## <https://www.gnu.org/licenses/>.
##
########################################################################
## -*- texinfo -*-
## @deftypefn {} {@var{h} =} starting_stepsize (@var{order}, @var{fcn}, @var{t0}, @var{x0}, @var{AbsTol}, @var{RelTol}, @var{normcontrol}, @var{args})
##
## Determine a good initial timestep for an ODE solver of order @var{order}
## using the algorithm described in reference [1].
##
## The input argument @var{fcn}, is the function describing the differential
## equations, @var{t0} is the initial time, and @var{x0} is the initial
## condition. @var{AbsTol} and @var{RelTol} are the absolute and relative
## tolerance on the ODE integration taken from an ode options structure.
##
## Reference:
## E. Hairer, S.P. Norsett and G. Wanner,
## @cite{Solving Ordinary Differential Equations I: Nonstiff Problems},
## Springer.
## @end deftypefn
##
## @seealso{odepkg}
function h = starting_stepsize (order, fcn, t0, x0,
AbsTol, RelTol, normcontrol,
args = {})
## compute norm of initial conditions
d0 = AbsRel_norm (x0, x0, AbsTol, RelTol, normcontrol);
## compute norm of the function evaluated at initial conditions
y = fcn (t0, x0, args{:});
if (iscell (y))
y = y{1};
endif
d1 = AbsRel_norm (y, y, AbsTol, RelTol, normcontrol);
if (d0 < 1e-5 || d1 < 1e-5)
h0 = 1e-6;
else
h0 = .01 * (d0 / d1);
endif
## compute one step of Explicit-Euler
x1 = x0 + h0 * y;
## approximate the derivative norm
yh = fcn (t0+h0, x1, args{:});
if (iscell (yh))
yh = yh{1};
endif
d2 = (1 / h0) * ...
AbsRel_norm (yh - y, yh - y, AbsTol, RelTol, normcontrol);
if (max (d1, d2) <= 1e-15)
h1 = max (1e-6, h0 * 1e-3);
else
h1 = (1e-2 / max (d1, d2)) ^(1 / (order+1));
endif
h = min (100 * h0, h1);
endfunction
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