/usr/share/octave/packages/optim-1.4.0/d2_min.m is in octave-optim 1.4.0-1.
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##
## 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/>.
## [x,v,nev,h,args] = d2_min(f,d2f,args,ctl,code) - Newton-like minimization
##
## Minimize f(x) using 1st and 2nd derivatives. Any function w/ second
## derivatives can be minimized, as in Newton. f(x) decreases at each
## iteration, as in Levenberg-Marquardt. This function is inspired from the
## Levenberg-Marquardt algorithm found in the book "Numerical Recipes".
##
## ARGUMENTS :
## f : string : Cost function's name
##
## d2f : string : Name of function returning the cost (1x1), its
## differential (1xN) and its second differential or it's
## pseudo-inverse (NxN) (see ctl(5) below) :
##
## [v,dv,d2v] = d2f (x).
##
## args : list : f and d2f's arguments. By default, minimize the 1st
## or matrix : argument.
##
## ctl : vector : Control arguments (see below)
## or struct
##
## code : string : code will be evaluated after each outer loop that
## produced some (any) improvement. Variables visible from
## "code" include "x", the best parameter found, "v" the
## best value and "args", the list of all arguments. All can
## be modified. This option can be used to re-parameterize
## the argument space during optimization
##
## CONTROL VARIABLE ctl : (optional). May be a struct or a vector of length
## ---------------------- 5 or less where NaNs are ignored. Default values
## are written <value>.
## FIELD VECTOR
## NAME POS
##
## ftol, f N/A : Stop search when value doesn't improve, as tested by
##
## f > Deltaf/max(|f(x)|,1)
##
## where Deltaf is the decrease in f observed in the last
## iteration. <10*sqrt(eps)>
##
## utol, u N/A : Stop search when updates are small, as tested by
##
## u > max { dx(i)/max(|x(i)|,1) | i in 1..N }
##
## where dx is the change in the x that occured in the last
## iteration. <NaN>
##
## dtol, d N/A : Stop search when derivative is small, as tested by
##
## d > norm (dv) <eps>
##
## crit, c ctl(1) : Set one stopping criterion, 'ftol' (c=1), 'utol' (c=2)
## or 'dtol' (c=3) to the value of by the 'tol' option. <1>
##
## tol, t ctl(2) : Threshold in termination test chosen by 'crit' <10*eps>
##
## narg, n ctl(3) : Position of the minimized argument in args <1>
## maxev,m ctl(4) : Maximum number of function evaluations <inf>
## maxout,m : Maximum number of outer loops <inf>
## id2f, i ctl(5) : 0 if d2f returns the 2nd derivatives, 1 if <0>
## it returns its pseudo-inverse.
##
## verbose, v N/A : Be more or less verbose (quiet=0) <0>
function [xbest,vbest,nev,hbest,args] = d2_min (f,d2f,args,ctl,code)
persistent warned = false;
if (! warned)
warned = true;
warning ("Octave:deprecated-function",
"The standalone function `d2_min' has been deprecated, and will be removed in the future. The algorithm can now be used with the frontend `nonlin_min'.");
endif
maxout = inf;
maxinner = 30 ;
tcoeff = 0.5 ; # Discount on total weight
ncoeff = 0.5 ; # Discount on weight of newton
ocoeff = 1.5 ; # Factor for outwards searching
report = 0 ; # Never report
verbose = 0 ; # Be quiet
prudent = 1 ; # Check coherence of d2f and f?
niter = 0 ;
crit = 0; # Default control variables
ftol = 10 * sqrt (eps);
dtol = eps;
utol = tol = nan;
narg = 1;
maxev = inf;
id2f = 0;
if nargin >= 4 # Read arguments
if isnumeric (ctl)
if length (ctl)>=1 && !isnan (ctl(1)), crit = ctl(1); end
if length (ctl)>=2 && !isnan (ctl(2)), tol = ctl(2); end
if length (ctl)>=3 && !isnan (ctl(3)), narg = ctl(3); end
if length (ctl)>=4 && !isnan (ctl(4)), maxev = ctl(4); end
if length (ctl)>=5 && !isnan (ctl(5)), id2f = ctl(5); end
elseif isstruct (ctl)
if isfield (ctl, "crit") , crit = ctl.crit ; end
if isfield (ctl, "tol") , tol = ctl.tol ; end
if isfield (ctl, "narg") , narg = ctl.narg ; end
if isfield (ctl, "maxev") , maxev = ctl.maxev ; end
if isfield (ctl, "maxout") , maxout = ctl.maxout ; end
if isfield (ctl, "id2f") , id2f = ctl.id2f ; end
if isfield (ctl, "verbose"), verbose = ctl.verbose; end
if isfield (ctl, "code") , code = ctl.code ; end
else
error ("The 'ctl' argument should be either a vector or a struct");
end
end
if crit == 1, ftol = tol;
elseif crit == 2, utol = tol;
elseif crit == 3, dtol = tol;
elseif crit, error ("crit is %i. Should be 1,2 or 3.\n");
end
if nargin < 5, code = "" ; end
if iscell (args) # List of arguments
x = args{narg};
else # Single argument
x = args;
args = {args};
end
############################## Checking ##############################
if narg > length (args)
error ("d2_min : narg==%i, length (args)==%i\n",
narg, length (args));
end
if tol <= 0
printf ("d2_min : tol=%8.3g <= 0\n",tol) ;
end
if !ischar (d2f) || !ischar (f)
printf ("d2_min : f and d2f should be strings!\n");
end
sz = size (x); N = prod (sz);
v = feval (f, args{:});
nev = [1,0];
if prudent && (! isnumeric (v) || isnan (v) || any (size (v)>1))
error ("Function '%s' returns inadequate output", f);
end
xbest = x = x(:);
vold = vbest = nan ; # Values of f
hbest = nan ; # Inv. Hessian
if verbose
printf ( "d2_min : Initially, v=%8.3g\n",v);
end
while niter <= maxout
niter += 1;
if nev(1) > maxev, break; end;
[v,d,h] = feval (d2f, args{1:narg-1},reshape(x,sz),args{narg+1:end});
nev(2)++;
if prudent && niter <= 1 && ...
(! isnumeric (v) || isnan (v) || any (size (v)>1) || ...
! isnumeric (d) || length (d(:)) != N || ...
! isnumeric (h) || any (size (h) != N))
error ("Function '%s' returns inadequate output", d2f);
end
if ! id2f, h = pinv (h); end
d = d(:);
if prudent
v2 = feval (f, args{1:narg-1},reshape(x,sz),args{narg+1:end});
nev(1)++;
if abs(v2-v) > 0.001 * sqrt(eps) * max (abs(v2), 1)
printf ("d2_min : f and d2f disagree %8.3g\n",abs(v2-v));
end
end
xbest = x ;
if ! isnan (vbest) # Check that v ==vbest
if abs (vbest - v) > 1000*eps * max (vbest, 1)
printf ("d2_min : vbest changed at beginning of outer loop\n");
end
end
vold = vbest = v ;
hbest = h ;
if length (code), abest = args; end # Eventually stash all args
if verbose || (report && rem(niter,max(report,1)) == 1)
printf ("d2_min : niter=%d, v=%8.3g\n",niter,v );
end
if norm (d) < dtol # Check for small derivative
if verbose || report
printf ("d2_min : Exiting because of low gradient\n");
end
break; # Exit outer loop
end
dnewton = -h*d ; # Newton step
# Heuristic for negative hessian
if dnewton'*d > 0, dnewton = -100*d; end
wn = 1 ; # Weight of Newton step
wt = 1 ; # Total weight
ninner = 0;
done_inner = 0; # 0=not found. 1=Ready to quit inner.
# ##########################################
while ninner < maxinner, # Inner loop ###############################
ninner += 1;
# Proposed step
dx = wt*(wn*dnewton - (1-wn)*d) ;
xnew = x+dx ;
if verbose
printf (["Weight : total=%8.3g, newtons's=%8.3g vbest=%8.3g ",...
"Norm:Newton=%8.3g, deriv=%8.3g\n"],...
wt,wn,vbest,norm(wt*wn*dnewton),norm(wt*(1-wn)*d));
end
if any(isnan(xnew))
printf ("d2_min : Whoa!! any(isnan(xnew)) (1)\n");
end
vnew = feval (f, args{1:narg-1},reshape(xnew,sz),args{narg+1:end});
nev(1)++;
if vnew<vbest # Stash best values
dbest = dx ;
vbest = vnew;
xbest = xnew;
done_inner = 1 ; # Will go out at next increase
if verbose
printf ( "d2_min : Found better value\n");
end
elseif done_inner == 1 # Time to go out
if verbose
printf ( "d2_min : Quitting %d th inner loop\n",niter);
end
break; # out of inner loop
end
wt = wt*tcoeff ; # Reduce norm of proposed step
wn = wn*ncoeff ; # And bring it closer to derivative
end # End of inner loop ########################
# ##########################################
wbest = 0; # Best coeff for dbest
if ninner >= maxinner # There was a problem
if verbose
printf ( "d2_min : Too many inner loops (vnew=%8.3g)\n",vnew);
end
# ##########################################
else # Look for improvement along dbest
wn = ocoeff ;
xnew = x+wn*dbest;
if any(isnan(xnew)),
printf ("d2_min : Whoa!! any(isnan(xnew)) (2)\n");
end
vnew = feval (f, args{1:narg-1},reshape(xnew,sz),args{narg+1:end});
nev(1)++;
while vnew < vbest,
vbest = vnew; # Stash best values
wbest = wn;
xbest = xnew;
wn = wn*ocoeff ;
xnew = x+wn*dbest;
vnew = feval (f, args{1:narg-1},reshape(xnew,sz),args{narg+1:length(args)});
if verbose
printf ( "Looking farther : v = %8.3g\n",vnew);
end
nev(1)++;
end
end # End of improving along dbest
# ##########################################
if verbose || rem(niter,max(report,1)) == 1
if vold,
if verbose
printf ("d2_min : Inner loop : vbest=%8.5g, vbest/vold=%8.5g\n",...
vbest,vbest/vold);
end
else
if verbose
printf ( "d2_min : Inner loop : vbest=%8.5g, vold=0\n", vbest);
end
end
end
if vbest < vold
## "improvement found"
if prudent
tmpv = feval (f, args{1:narg-1},reshape(xbest,sz),args{2:end});
nev(1)++;
if abs (tmpv-vbest) > eps
printf ("d2_min : Whoa! Value at xbest changed by %g\n",...
abs(tmpv-vbest));
end
end
v = vbest; x = xbest;
if ! isempty (code)
if verbose
printf ("d2_min : Going to eval (\"%s\")\n",code);
end
xstash = xbest;
astash = abest;
args = abest; # Here : added 2001/11/07. Is that right?
x = xbest;
eval (code, "printf (\"code fails\\n\");");
xbest = x;
abest = args;
# Check whether eval (code) changes value
if prudent
tmpv = feval (f, args{1:narg-1},reshape(x,sz),args{2:end});
nev(1)++;
if abs (tmpv-vbest) > max (min (100*eps,0.00001*abs(vbest)), eps) ,
printf ("d2_min : Whoa! Value changes by %g after eval (code)\n",...
abs (tmpv-vbest));
end
end
end
end
if ! isnan (ftol) && ftol > (vold-vbest)/max(vold,1),
if verbose || report
printf ("d2_min : Quitting, niter=%-3d v=%8.3g, ",niter,v);
if vold, printf ("v/vold=%8.3g \n",v/vold);
else printf ("vold =0 \n",v);
end
end
break ; # out of outer loop
end
if ! isnan (utol) && utol > max (abs (wbest*dbest)) / max(abs (xbest),1)
if verbose || report
printf ("d2_min : Quitting, niter=%-3d v=%8.3g, ",niter,v);
if vold, printf ("v/vold=%8.3g \n",v/vold);
else printf ("vold =0 \n",v);
end
end
break ; # out of outer loop
end
end # End of outer loop ##################
xbest = reshape (xbest, sz);
if length (code)
args = abest;
args(narg) = xbest;
end
if niter > maxout
if verbose
printf ( "d2_min : Outer loop lasts forever\n");
end
end
# One last check
if prudent
err = feval (f, args{1:narg-1},reshape(xbest,sz),args{2:end});
nev(1)++;
if abs (err-vbest) > eps,
printf ("d2_min : Whoa!! xbest does not eval to vbest\n");
printf (" : %8.3e - %8.3e = %8.3e != 0\n",err,vbest,err-vbest);
end
end
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