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improvements to fsolve & family
author Jaroslav Hajek <highegg@gmail.com>
date Tue, 17 Mar 2009 08:49:08 +0100
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1 ## Copyright (C) 2008, 2009 Jaroslav Hajek
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2 ##
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3 ## This file is part of Octave.
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4 ##
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5 ## Octave is free software; you can redistribute it and/or modify it
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6 ## under the terms of the GNU General Public License as published by
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7 ## the Free Software Foundation; either version 3 of the License, or (at
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8 ## your option) any later version.
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9 ##
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10 ## Octave is distributed in the hope that it will be useful, but
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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13 ## General Public License for more details.
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14 ##
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15 ## You should have received a copy of the GNU General Public License
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16 ## along with Octave; see the file COPYING. If not, see
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17 ## <http://www.gnu.org/licenses/>.
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18
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19 ## -*- texinfo -*-
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20 ## @deftypefn{Function File} {@var{x}} = __dogleg__ (@var{r}, @var{b}, @var{x}, @var{d}, @var{delta})
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21 ## Undocumented internal function.
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22 ## @end deftypefn
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23
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24 ## Solve the double dogleg trust-region problem:
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25 ## Minimize norm(r*x-b) subject to the constraint norm(d.*x) <= delta,
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26 ## x being a convex combination of the gauss-newton and scaled gradient.
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27
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28 ## TODO: error checks
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29 ## TODO: handle singularity, or leave it up to mldivide?
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30
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31 function x = __dogleg__ (r, b, d, delta)
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32 ## Get Gauss-Newton direction.
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33 x = r \ b;
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34 xn = norm (d .* x);
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35 if (xn > delta)
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36 ## GN is too big, get scaled gradient.
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37 s = (r' * b) ./ d;
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38 sn = norm (s);
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39 if (sn > 0)
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40 ## Normalize and rescale.
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41 s = (s / sn) ./ d;
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42 ## Get the line minimizer in s direction.
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43 tn = norm (r*s);
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44 snm = (sn / tn) / tn;
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45 if (snm < delta)
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46 ## Get the dogleg path minimizer.
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47 bn = norm (b);
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48 dxn = delta/xn; snmd = snm/delta;
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49 t = (bn/sn) * (bn/xn) * snmd;
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50 t -= dxn * snmd^2 - sqrt ((t-dxn)^2 + (1-dxn^2)*(1-snmd^2));
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51 alpha = dxn*(1-snmd^2) / t;
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52 else
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53 alpha = 0;
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54 endif
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55 else
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56 alpha = delta / xn;
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57 snm = 0;
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58 endif
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59 ## Form the appropriate convex combination.
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60 x = alpha * x + ((1-alpha) * min (snm, delta)) * s;
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61 endif
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62 endfunction
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