/usr/share/octave/packages/3.2/optim-1.0.17/jacobs.m is in octave-optim 1.0.17-1.
This file is owned by root:root, with mode 0o644.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 | ## Copyright (C) 2011 Fotios Kasolis <fotios.kasolis@gmail.com>
##
## 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} {Df =} jacobs (@var{x}, @var{f})
## @deftypefnx {Function File} {Df =} jacobs (@var{x}, @var{f}, @var{hook})
## Calculate the jacobian of a function using the complex step method.
##
## Let @var{f} be a user-supplied function. Given a point @var{x} at
## which we seek for the Jacobian, the function @command{jacobs} returns
## the Jacobian matrix @code{d(f(1), @dots{}, df(end))/d(x(1), @dots{},
## x(n))}. The function uses the complex step method and thus can be
## applied to real analytic functions.
##
## The optional argument @var{hook} is a structure with additional options. @var{hook}
## can have the following fields:
## @itemize @bullet
## @item
## @code{h} - can be used to define the magnitude of the complex step and defaults
## to 1e-20; steps larger than 1e-3 are not allowed.
## @item
## @code{fixed} - is a logical vector internally usable by some optimization
## functions; it indicates for which elements of @var{x} no gradient should be
## computed, but zero should be returned.
## @end itemize
##
## For example:
##
## @example
## @group
## f = @@(x) [x(1)^2 + x(2); x(2)*exp(x(1))];
## Df = jacobs ([1, 2], f)
## @end group
## @end example
## @end deftypefn
function Df = jacobs (x, f, hook)
if ( (nargin < 2) || (nargin > 3) )
print_usage ();
endif
if (ischar (f))
f = str2func (f, "global");
endif
n = numel (x);
default_h = 1e-20;
max_h = 1e-3; # must be positive
if (nargin > 2)
if (isfield (hook, "h"))
if (! (isscalar (hook.h)))
error ("complex step magnitude must be a scalar");
endif
if (abs (hook.h) > max_h)
warning ("complex step magnitude larger than allowed, set to %e", ...
max_h)
h = max_h;
else
h = hook.h;
endif
else
h = default_h;
endif
if (isfield (hook, "fixed"))
if (numel (hook.fixed) != n)
error ("index of fixed parameters has wrong dimensions");
endif
fixed = hook.fixed;
else
fixed = false (n, 1);
endif
else
h = default_h;
fixed = false (n, 1);
endif
if (all (fixed))
error ("all elements of 'x' are fixed");
endif
x = repmat (x(:), 1, n) + h * 1i * eye (n);
idx = find (! fixed);
## after first evaluation, dimensionness of 'f' is known
t_Df = imag (f (x(:, idx(1)))(:));
dim = numel (t_Df);
Df = zeros (dim, n);
Df(:, idx(1)) = t_Df;
for count = idx(2:end)
Df(:, count) = imag (f (x(:, count))(:));
endfor
Df /= h;
endfunction
%!assert (jacobs (1, @(x) x), 1)
%!assert (jacobs (6, @(x) x^2), 12)
%!assert (jacobs ([1; 1], @(x) [x(1)^2; x(1)*x(2)]), [2, 0; 1, 1])
%!assert (jacobs ([1; 2], @(x) [x(1)^2 + x(2); x(2)*exp(x(1))]), [2, 1; 2*exp(1), exp(1)])
%% Test input validation
%!error jacobs ()
%!error jacobs (1)
%!error jacobs (1, 2, 3, 4)
%!error jacobs (@sin, 1, [1, 1])
%!error jacobs (@sin, 1, ones(2, 2))
%!demo
%! # Relative error against several h-values
%! k = 3:20; h = 10 .^ (-k); x = 0.3*pi;
%! err = zeros (1, numel (k));
%! for count = 1 : numel (k)
%! err(count) = abs (jacobs (x, @sin, struct ("h", h(count))) - cos (x)) / abs (cos (x)) + eps;
%! endfor
%! loglog (h, err); grid minor;
%! xlabel ("h"); ylabel ("|Df(x) - cos(x)| / |cos(x)|")
%! title ("f(x)=sin(x), f'(x)=cos(x) at x = 0.3pi")
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