/usr/share/octave/packages/optim-1.4.0/private/__lm_svd__.m is in octave-optim 1.4.0-1.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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## Copyright (C) 1992-1994 Arthur Jutan
## Copyright (C) 1992-1994 Ray Muzic
## Copyright (C) 2010-2013 Olaf Till <i7tiol@t-online.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/>.
function [p, resid, cvg, outp] = __lm_svd__ (F, pin, hook)
## This is a backend for optimization. This code was originally
## contained in leasqr.m, which is now a frontend.
## some backend specific defaults
fract_prec_default = 0;
max_fract_step_default = Inf;
n = length (pin);
## passed constraints
mc = hook.mc; # matrix of linear constraints
vc = hook.vc; # vector of linear constraints
f_cstr = hook.f_cstr; # function of all constraints
df_cstr = hook.df_cstr; # function of derivatives of all constraints
n_gencstr = hook.n_gencstr; # number of non-linear constraints
eq_idx = hook.eq_idx; # logical index of equality constraints in all
# constraints
lbound = hook.lbound; # bounds, subset of linear inequality
ubound = hook.ubound; # constraints in mc and vc
## passed values of constraints for initial parameters
pin_cstr = hook.pin_cstr;
## passed return value of F for initial parameters
f_pin = hook.f_pin;
## passed derivative of residual function
dfdp = hook.dfdp;
## passed function for complementary pivoting
cpiv = hook.cpiv;
## passed options
maxstep = hook.max_fract_change;
maxstep(isna (maxstep)) = max_fract_step_default;
pprec = hook.fract_prec;
pprec(isna (pprec)) = fract_prec_default;
## keep absolute precision positive for non-null relative precision;
## arbitrary value, added to parameters before multiplying with
## relative precision
add_pprec = zeros (n, 1);
add_pprec(pprec > 0) = sqrt (eps);
##
stol = hook.TolFun;
niter = hook.MaxIter;
if (isempty (niter)) niter = 20; endif
wt = hook.weights;
fixed = hook.fixed;
verbose = strcmp (hook.Display, "iter");
user_interaction = hook.user_interaction;
## only preliminary, for testing
if (isfield (hook, "testing"))
testing = hook.testing;
else
testing = false;
endif
if (isfield (hook, "new_s"))
new_s = hook.new_s;
else
new_s = false;
endif
## some useful variables derived from passed variables
n_lcstr = size (vc, 1);
have_constraints_except_bounds = ...
n_lcstr + n_gencstr > ...
sum (lbound ~= -Inf) + sum (ubound ~= Inf);
wtl = wt(:);
nz = 20 * eps; # This is arbitrary. Constraint function will be
# regarded as <= zero if less than nz.
## backend-specific checking of options and constraints
if (have_constraints_except_bounds)
if (any (pin_cstr.inequ.lin_except_bounds < 0) || ...
(n_gencstr > 0 && any (pin_cstr.inequ.gen < 0)))
warning ("initial parameters violate inequality constraints");
endif
if (any (abs (pin_cstr.equ.lin) >= nz) || ...
(n_gencstr > 0 && any (abs (pin_cstr.equ.gen) >= nz)))
warning ("initial parameters violate equality constraints");
endif
endif
idx = lbound == ubound;
if (any (idx))
warning ("lower and upper bounds identical for some parameters, fixing the respective parameters");
fixed(idx) = true;
endif
if (all (fixed))
error ("no free parameters");
endif
lidx = pin < lbound;
uidx = pin > ubound;
if (any (lidx | uidx) && have_constraints_except_bounds)
warning ("initial parameters outside bounds, not corrected since other constraints are given");
else
if (any (lidx))
warning ("some initial parameters set to lower bound");
pin(lidx, 1) = lbound(lidx, 1);
endif
if (any (uidx))
warning ("some initial parameters set to upper bound");
pin(uidx, 1) = ubound(uidx, 1);
endif
endif
if (n_gencstr > 0 && any (~isinf (maxstep)))
warning ("setting both a maximum fractional step change of parameters and general constraints may result in inefficiency and failure");
endif
## fill constant fields of hook for derivative-functions; some fields
## may be backend-specific
dfdp_hook.fixed = fixed; # this may be handled by the frontend, but
# the backend still may add to it
## set up for iterations
##
p = pin;
f = f_pin; fbest=f; pbest=p;
m = prod (size (f));
r = wt .* f;
r = r(:);
if (~isreal (r)) error ("weighted residuals are not real"); endif
ss = r.' * r;
sbest=ss;
chgprev=Inf*ones(n,1);
cvg=0;
epsLlast=1;
epstab=[.1, 1, 1e2, 1e4, 1e6];
ac_idx = true (n_lcstr + n_gencstr, 1); # all constraints
nc_idx = false (n_lcstr + n_gencstr, 1); # none of all constraints
gc_idx = cat (1, false (n_lcstr, 1), true (n_gencstr, 1)); # gen. constr.
lc_idx = ~gc_idx;
if (([stop, outp.user_interaction] = ...
__do_user_interaction__ (user_interaction, p,
struct ("iteration", 0,
"residual", f),
"init")))
outp.niter = 0;
resid = f;
cvg = -1;
return;
endif
## do iterations
##
for iter = 1:niter
deb_printf (testing, "\nstart outer iteration\n");
v_cstr = f_cstr (p, ac_idx);
## index of active constraints
c_act = v_cstr < nz | eq_idx; # equality constraints might be
# violated at start
if (any (c_act))
if (n_gencstr > 0)
## full gradient is needed later
dct = df_cstr (p, ac_idx, ...
setfield (dfdp_hook, "f", v_cstr));
dct(:, fixed) = 0; # for user supplied dfdp; necessary?
dcat = dct(c_act, :);
else
dcat = df_cstr (p, c_act, ...
setfield (dfdp_hook, "f", v_cstr));
dcat(:, fixed) = 0; # for user supplied dfdp; necessary?
endif
dca = dcat.';
endif
nrm = zeros (1, n);
pprev=pbest;
prt = dfdp (p, setfield (dfdp_hook, "f", fbest(:)));
prt(:, fixed) = 0; # for user supplied dfdp; necessary?
r = wt .* -fbest;
r = r(:);
if (~isreal (r)) error ("weighted residuals are not real"); endif
sprev=sbest;
sgoal=(1-stol)*sprev;
msk = ~fixed;
prt(:, msk) = prt(:, msk) .* wtl(:, ones (1, sum (msk)));
nrm(msk) = sumsq (prt(:, msk), 1);
msk = nrm > 0;
nrm(msk) = 1 ./ sqrt (nrm(msk));
prt = prt .* nrm(ones (1, m), :);
nrm = nrm.';
[prt,s,v]=svd(prt,0);
s=diag(s);
g = prt.' * r;
for jjj=1:length(epstab)
deb_printf (testing, "\nstart inner iteration\n");
epsL = max(epsLlast*epstab(jjj),1e-7);
## printf ("epsL: %e\n", epsL); # for testing
## Usage of this "ser" later is equivalent to pre-multiplying the
## gradient with a positive-definit matrix, but not with a
## diagonal matrix, at epsL -> Inf; so there is a fallback to
## gradient descent, but not in general to descent for each
## gradient component. Using the commented-out "ser" ((1 / (1 +
## epsL^2)) * (1 ./ se + epsL * s)) would be equivalent to using
## Marquardts diagonal of the Hessian-approximation for epsL ->
## Inf, but currently this gives no advantages in tests, even with
## constraints.
### ser = 1 ./ sqrt((s.*s)+epsL);
se = sqrt ((s.*s) + epsL);
if (new_s)
## for testing
ser = (1 / (1 + epsL^2)) * (1 ./ se + epsL * s);
else
ser = 1 ./ se;
endif
tp1 = (v * (g .* ser)) .* nrm;
if (any (c_act))
deb_printf (testing, "constraints are active:\n");
deb_printf (testing, "%i\n", c_act);
## calculate chg by "quadratic programming"
nrme= diag (nrm);
ser2 = diag (ser .* ser);
mfc1 = nrme * v * ser2 * v.' * nrme;
tp2 = mfc1 * dca;
a_eq_idx = eq_idx(c_act);
[lb, bidx, ridx, tbl] = cpiv (dcat * tp1, dcat * tp2, a_eq_idx);
chg = tp1 + tp2(:, bidx) * lb; # if a parameter is "fixed",
# the respective component of chg should
# be zero too, even here (with active
# constraints)
deb_printf (testing, "change:\n");
deb_printf (testing, "%e\n", chg);
deb_printf (testing, "\n");
## indices for different types of constraints
c_inact = ~c_act; # inactive constraints
c_binding = nc_idx;
c_binding(c_act) = bidx; # constraints selected binding
c_unbinding = nc_idx;
c_unbinding(c_act) = ridx; # constraints unselected binding
c_nonbinding = c_act & ~(c_binding | c_unbinding); # constraints
# selected non-binding
else
## chg is the Levenberg/Marquardt step
chg = tp1;
## indices for different types of constraints
c_inact = ac_idx; # inactive constraints consist of all
# constraints
c_binding = nc_idx;
c_unbinding = nc_idx;
c_nonbinding = nc_idx;
endif
## apply constraints to step width (since this is a
## Levenberg/Marquardt algorithm, no line-search is performed
## here)
k = 1;
c_tp = c_inact(1:n_lcstr);
mcit = mc(:, c_tp).';
vci = vc(c_tp);
hstep = mcit * chg;
idx = hstep < 0;
if (any (idx))
k = min (1, min (- (vci(idx) + mcit(idx, :) * pprev) ./ ...
hstep(idx)));
endif
if (k < 1)
deb_printf (testing, "stepwidth: linear constraints\n");
endif
if (n_gencstr > 0)
c_tp = gc_idx & (c_nonbinding | c_inact);
if (any (c_tp) && any (f_cstr (pprev + k * chg, c_tp) < 0))
[k, fval, info] = ...
fzero (@ (x) min (cat (1, ...
f_cstr (pprev + x * chg, c_tp), ...
k - x, ...
ifelse (x < 0, -Inf, Inf))), ...
0);
if (info ~= 1 || abs (fval) >= nz)
error ("could not find stepwidth to satisfy inactive and non-binding general inequality constraints");
endif
deb_printf (testing, "general constraints limit stepwidth\n");
endif
endif
chg = k * chg;
if (any (gc_idx & c_binding)) # none selected binding =>
# none unselected binding
deb_printf (testing, "general binding constraints must be regained:\n");
## regain binding constraints and one of the possibly active
## previously inactive or non-binding constraints
ptp1 = pprev + chg;
tp = true;
nt_nosuc = true;
lim = 20;
while (nt_nosuc && lim >= 0)
deb_printf (testing, "starting from new value of p in regaining:\n");
deb_printf (testing, "%e\n", ptp1);
## we keep d_p.' * inv (mfc1) * d_p minimal in each step of
## the inner loop; this is both sensible (this metric
## considers a guess of curvature of sum of squared residuals)
## and convenient (we have useful matrices available for it)
c_tp0 = c_inact | c_nonbinding;
c_tp1 = c_inact | (gc_idx & c_nonbinding);
btbl = tbl(bidx, bidx);
c_tp2 = c_binding;
if (any (tp)) # if none before, does not get true again
tp = f_cstr (ptp1, c_tp1) < nz;
if (any (tp)) # could be less clumsy, but ml-compatibility..
## keep only the first true entry in tp
tp(tp) = logical (cat (1, 1, zeros (sum (tp) - 1, 1)));
## supplement binding index with one (the first) getting
## binding in c_tp1
c_tp2(c_tp1) = tp;
## gradient of this added constraint
caddt = dct(c_tp2 & ~c_binding, :);
cadd = caddt.';
C = dct(c_binding, :) * mfc1 * cadd;
Ct = C.';
G = [btbl, btbl * C; ...
-Ct * btbl, caddt * mfc1 * cadd - Ct * btbl * C];
btbl = gjp (G, size (G, 1));
endif
endif
dcbt = dct(c_tp2, :);
mfc = - mfc1 * dcbt.' * btbl;
deb_printf (testing, "constraints to regain:\n");
deb_printf (testing, "%i\n", c_tp2);
ptp2 = ptp1;
nt_niter_start = 100;
nt_niter = nt_niter_start;
while (nt_nosuc && nt_niter >= 0)
hv = f_cstr (ptp2, c_tp2);
if (all (abs (hv) < nz))
nt_nosuc = false;
chg = ptp2 - pprev;
else
ptp2 = ptp2 + mfc * hv; # step should be zero for each
# component for which the parameter is
# "fixed"
endif
nt_niter = nt_niter - 1;
endwhile
deb_printf (testing, "constraints after regaining:\n");
deb_printf (testing, "%e\n", hv);
if (nt_nosuc || ...
any (abs (chg) > abs (pprev .* maxstep)) || ...
any (f_cstr (ptp2, c_tp0) < -nz))
if (nt_nosuc)
deb_printf (testing, "regaining did not converge\n");
else
deb_printf (testing, "regaining violated type 3 and 4\n");
endif
nt_nosuc = true;
ptp1 = (pprev + ptp1) / 2;
endif
if (~nt_nosuc)
tp = f_cstr (ptp2, c_unbinding);
if (any (tp) < 0) # again ml-compatibility clumsyness..
[discarded, id] = min(tp);
tid = find (ridx);
id = tid(id); # index within active constraints
unsuccessful_exchange = false;
if (abs (tbl(id, id)) < nz) # Bard: not absolute value
## exchange this unselected binding constraint against a
## binding constraint, but not against an equality
## constraint
tbidx = bidx & ~a_eq_idx;
if (~any (tbidx))
unsuccessful_exchange = true;
else
[discarded, idm] = max (abs (tbl(tbidx, id)));
tid = find (tbidx);
idm = tid(idm); # -> index within active constraints
tbl = gjp (tbl, idm);
bidx(idm) = false;
ridx(idm) = true;
endif
endif
if (unsuccessful_exchange)
## It probably doesn't look good now; this desperate
## last attempt is not in the original algortithm, since
## that didn't account for equality constraints.
ptp1 = (pprev + ptp1) / 2;
else
tbl = gjp (tbl, id);
bidx(id) = true;
ridx(id) = false;
c_binding = nc_idx;
c_binding(c_act) = bidx;
c_unbinding = nc_idx;
c_unbinding(c_act) = ridx;
endif
nt_nosuc = true;
deb_printf (testing, "regaining violated type 2\n");
endif
endif
if (~nt_nosuc)
deb_printf (testing, "regaining successful, converged with %i iterations:\n", ...
nt_niter_start - nt_niter);
deb_printf (testing, "%e\n", ptp2);
endif
lim = lim - 1;
endwhile
if (nt_nosuc)
error ("could not regain binding constraints");
endif
else
## check the maximal stepwidth and apply as necessary
ochg=chg;
idx = ~isinf(maxstep);
limit = abs(maxstep(idx).*pprev(idx));
chg(idx) = min(max(chg(idx),-limit),limit);
if (verbose && any(ochg ~= chg))
disp(['Change in parameter(s): ', ...
sprintf("%d ",find(ochg ~= chg)), "maximal fractional stepwidth enforced"]);
endif
endif
aprec = pprec .* (abs (pbest) + add_pprec);
## ss=scalar sum of squares=sum((wt.*f)^2).
if (any(abs(chg) > 0.1*aprec))#--- # only worth evaluating
# function if there is some non-miniscule
# change
## In the code of the outer loop before the inner loop pbest is
## actually identical to p, since once they deviate, the outer
## loop will not be repeated. Though the inner loop can still be
## repeated in this case, pbest is not used in it. Since pprev
## is set from pbest in the outer loop before the inner loop, it
## is also identical to p up to here.
p=chg+pprev;
## since the projection method may have slightly violated
## constraints due to inaccuracy, correct parameters to bounds
## --- but only if no further constraints are given, otherwise
## the inaccuracy in honoring them might increase by this
skipped = false;
if (~have_constraints_except_bounds)
lidx = p < lbound;
uidx = p > ubound;
p(lidx, 1) = lbound(lidx, 1);
p(uidx, 1) = ubound(uidx, 1);
chg(lidx, 1) = p(lidx, 1) - pprev(lidx, 1);
chg(uidx, 1) = p(uidx, 1) - pprev(uidx, 1);
endif
##
f = F (p);
r = wt .* f;
r = r(:);
if (~isreal (r))
error ("weighted residuals are not real");
endif
ss = r.' * r;
deb_printf (testing, "tried parameters:\n");
deb_printf (testing, "%.16e\n", p);
deb_printf (testing, "sbest: %.16e\n", sbest);
deb_printf (testing, "sgoal: %.16e\n", sgoal);
deb_printf (testing, " ss: %.16e\n", ss);
if (ss<sbest)
pbest=p;
fbest=f;
sbest=ss;
endif
if (ss < sgoal) # <, not <=, since sgoal can be equal to sprev
# if TolFun <= eps
break;
endif
else
skipped = true;
break;
endif #---
endfor
## printf ("epsL no.: %i\n", jjj); # for testing
epsLlast = epsL;
hook.plot_cmd (f);
if (([stop, outp.user_interaction] = ...
__do_user_interaction__ (user_interaction, p,
struct ("iteration", iter,
"residual", f),
"iter")))
outp.niter = iter;
resid = f;
cvg = -1;
return;
endif
if (skipped)
cvg = 2;
break;
endif
if (ss < eps) # in this case ss == sbest
cvg = 3; # there is no more suitable flag for this
break;
endif
if (ss >= sgoal) # >=, not >, since sgoal can be equal to sprev if
# TolFun <= eps
cvg = 3;
break;
endif
aprec = pprec .* (abs (pbest) + add_pprec);
## [aprec, chg, chgprev]
if (all(abs(chg) <= aprec) && all(abs(chgprev) <= aprec))
cvg = 2;
if (verbose)
fprintf("Parameter changes converged to specified precision\n");
endif
break;
else
chgprev=chg;
endif
endfor
## set further return values
##
p = pbest;
resid = fbest;
outp.niter = iter;
if (([stop, outp.user_interaction] = ...
__do_user_interaction__ (user_interaction, pbest,
struct ("iteration", iter,
"residual", fbest),
"done")))
cvg = -1;
endif
endfunction
function deb_printf (do_printf, varargin)
## for testing
if (do_printf)
printf (varargin{:})
endif
endfunction
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