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Highlights from
Hybrid Symbolic and Numerical Simulation Studies of Time-fractional Order

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Hybrid Symbolic and Numerical Simulation Studies of Time-fractional Order

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27 Mar 2006 (Updated )

Matlab code to duplicate all figures in the paper, self-containing.

[x,f,options]=solvopt(x,fun,grad,options,func,gradc)
function [x,f,options]=solvopt(x,fun,grad,options,func,gradc)
% Usage:
% [x,f,options]=solvopt(x,fun,grad,options,func,gradc)
% The function SOLVOPT performs a modified version of Shor's r-algorithm in
% order to find a local minimum resp. maximum of a nonlinear function
% defined on the n-dimensional Euclidean space 
% or 
% a solution of a nonlinear constrained problem: 
% min { f(x): g(x) (<)= 0, g(x) in R(m), x in R(n) }
% Arguments:
% x       is the n-vector (row or column) of the coordinates of the starting
%         point,
% fun     is the name of an M-file (M-function) which computes the value 
%         of the objective function <fun> at a point x,
%         synopsis: f=fun(x)
% grad    is the name of an M-file (M-function) which computes the gradient 
%         vector of the function <fun> at a point x,
%         synopsis: g=grad(x)
% func    is the name of an M-file (M-function) which computes the MAXIMAL
%         RESIDUAL(!) for a set of constraints at a point x,
%         synopsis: fc=func(x)
% gradc   is the name of an M-file (M-function) which computes the gradient 
%         vector for the maximal residual consyraint at a point x,
%         synopsis: gc=gradc(x)
% options is a row vector of optional parameters:
%    options(1)= H, where sign(H)=-1 resp. sign(H)=+1 means minimize
%        resp. maximize <fun> (valid only for unconstrained problem)
%        and H itself is a factor for the initial trial step size 
%        (options(1)=-1 by default),
%    options(2)= relative error for the argument in terms of the 
%        infinity-norm (1.e-4 by default),
%    options(3)= relative error for the function value (1.e-6 by default),
%    options(4)= limit for the number of iterations (15000 by default),
%    options(5)= control of the display of intermediate results and error 
%        resp. warning messages (default value is 0, i.e., no intermediate 
%        output but error and warning messages, see more in the manual),
%    options(6)= admissible maximal residual for a set of constraints
%        (options(6)=1e-8 by default, see more in the manual),
%   *options(7)= the coefficient of space dilation (2.5 by default),
%   *options(8)= lower bound for the stepsize used for the difference
%        approximation of gradients (1e-12 by default, see more in the manual).
%  (* ... changes should be done with care)
% Returned values:
% x       is the optimizer (row resp. column),
% f       is the optimum function value,
% options returns the values of the counters
%    options(9),  the number of iterations, if positive,
%        or an abnormal stop code, if negative (see more in the manual), 
%    options(10), the number of objective 
%    options(11), the number of gradient evaluations,
%    options(12), the number of constraint function evaluations,
%    options(13), the number of constraint gradient evaluations.
% ____________________________________________________________________________

% strings: ----{
errmes='SolvOpt error:';
wrnmes='SolvOpt warning:';
error1='No function name and/or starting point passed to the function.';
error2='Argument X has to be a row or column vector of dimension > 1.';
error30='<fun> returns an empty string.';
error31='Function value does not exist (NaN is returned).';
error32='Function equals infinity at the point.';
error40='<grad> returns an improper matrix. Check the dimension.';
error41='Gradient does not exist (NaN is returned by <grad>).';
error42='Gradient equals infinity at the starting point.';
error43='Gradient equals zero at the starting point.';
error50='<func> returns an empty string.';
error51='<func> returns NaN at the point.';
error52='<func> returns infinite value at the point.';
error60='<gradc> returns an improper vector. Check the dimension';
error61='<gradc> returns NaN at the point.';
error62='<gradc> returns infinite vector at the point.';
error63='<gradc> returns zero vector at an infeasible point.';
error5='Function is unbounded.';
error6='Choose another starting point.';
warn1= 'Gradient is zero at the point, but stopping criteria are not fulfilled.';
warn20='Normal re-setting of a transformation matrix.' ;
warn21='Re-setting due to the use of a new penalty coefficient.' ;
warn4= 'Iterations limit exceeded.';
warn31='The function is flat in certain directions.';
warn32='Trying to recover by shifting insensitive variables.';
warn09='Re-run from recorded point.';
warn08='Ravine with a flat bottom is detected.';
termwarn0='SolvOpt: Normal termination.';
termwarn1='SolvOpt: Termination warning:';
appwarn='The above warning may be reasoned by inaccurate gradient approximation';
endwarn=[...
'Premature stop is possible. Try to re-run the routine from the obtained point.               ';...
'Result may not provide the optimum. The function apparently has many extremum points.        ';...
'Result may be inaccurate in the coordinates. The function is flat at the optimum.            ';...
'Result may be inaccurate in a function value. The function is extremely steep at the optimum.'];
% ----}

% ARGUMENTS PASSED ----{
if nargin<2           % Function and/or starting point are not specified
  options(9)=-1; disp(errmes);  disp(error1);   return
end
if nargin<3,   app=1;             % No user-supplied gradients
elseif isempty(grad),  app=1; 
else,  app=0;                     % Exact gradients are supplied
end

% OPTIONS ----{
  doptions=[-1,1.e-4,1.e-6,15000,0,1.e-8,2.5,1e-11];
  if nargin<4,  options=doptions; 
  elseif isempty(options), options=doptions;
  else,
% Replace default options by user specified options:
    ii=find(options~=0);doptions(ii)=options(ii);
    options=doptions;
  end
% Check the values:
  options([2:4,6:8])=abs(options([2:4,6:8]));
  options(2:3)=max(options(2:3),[1.e-12,1.e-12]);
  options(2)=max(options(8)*1.e2,options(2));
  options(2:3)=min(options(2:3),[1,1]);
  options(6)=max(options(6),1e-12);
  options(7)=max([options(7),1.5]);
  options(8)=max(options(8),1e-11);
% ----}

if nargin<5,   constr=0;          % Unconstrained problem
elseif isempty(func),  constr=0;
else,  constr=1;                  % Constrained problem
   if nargin<6, appconstr=1; t=3; % No user-supplied gradients for constraints
   elseif isempty(gradc),  
          appconstr=1;      
   else,  appconstr=0;            % Exact gradients of constraints are supplied
   end
end
% ----}

% STARTING POINT ----{
 if max(size(x))<=1,      disp(errmes);  disp(error2); 
                          options(9)=-2; return
 elseif size(x,2)==1,     n=size(x,1);  x=x'; trx=1;
 elseif size(x,1)==1,     n=size(x,2);        trx=0;
 else,                    disp(errmes);  disp(error2);
                          options(9)=-2; return
 end
% ----}

% WORKING CONSTANTS AND COUNTERS ----{

options(10)=0; options(11)=0;      % function and gradient calculations
if constr
options(12)=0; options(13)=0;      % same for constraints
end
epsnorm=1.e-15;epsnorm2=1.e-30;    % epsilon & epsilon^2

if constr, h1=-1;                  % NLP: restricted to minimization
 cnteps=options(6);                % Max. admissible residual
else, h1=sign(options(1));         % Minimize resp. maximize a function
end

k=0;                               % Iteration counter

wdef=1/options(7)-1;               % Default space transf. coeff.

%Gamma control ---{
  ajb=1+.1/n^2;                    % Base I
  ajp=20;
  ajpp=ajp;                        % Start value for the power 
  ajs=1.15;                        % Base II
  knorms=0; gnorms=zeros(1,10);    % Gradient norms stored
%---}

%Display control ---{
  if options(5)<=0, dispdata=0;  
     if options(5)==-1, dispwarn=0; else, dispwarn=1; end
  else, dispdata=round(options(5)); dispwarn=1;
  end,  ld=dispdata;               
%---}

%Stepsize control ---{
  dq=5.1;                          % Step divider (at f_{i+1}>gamma*f_{i})
  du20=2;du10=1.5;du03=1.05;       % Step multipliers (at certain steps made)
  kstore=3;nsteps=zeros(1,kstore); % Steps made at the last 'kstore' iterations
  if app, des=6.3;                 % Desired number of steps per 1-D search
  else,   des=3.3; end
  mxtc=3;                          % Number of trial cycles (steep wall detect)
%---}
termx=0; limxterm=50;              % Counter and limit for x-criterion

ddx   =max(1e-11,options(8));      % stepsize for gradient approximation

low_bound=-1+1e-4;                 % Lower bound cosine used to detect a ravine

ZeroGrad=n*1.e-16;                 % Lower bound for a gradient norm

nzero=0;                           % Zero-gradient events counter
% Lower bound for values of variables taking into account 
lowxbound=max([options(2),1e-3]);  
% Lower bound for function values to be considered as making difference
lowfbound=options(3)^2;            
krerun=0;                          % Re-run events counter
detfr=options(3)*100;              % relative error for f/f_{record}
detxr=options(2)*10;               % relative error for norm(x)/norm(x_{record})

warnno=0;                          % the number of warn.mess. to end with

kflat=0;                           % counter for points of flatness
stepvanish=0;                      % counter for vanished steps
stopf=0;
% ----}  End of setting constants
% ----}  End of the preamble

% COMPUTE THE FUNCTION  ( FIRST TIME ) ----{
   if trx,  f=feval(fun,x');  
   else,    f=feval(fun,x);  end
   options(10)=options(10)+1; 
   if isempty(f),      if dispwarn,disp(errmes);disp(error30);end
                       options(9)=-3; if trx, x=x';end, return
   elseif isnan(f),    if dispwarn,disp(errmes);disp(error31);disp(error6);end
                       options(9)=-3; if trx, x=x';end, return
   elseif abs(f)==Inf, if dispwarn,disp(errmes);disp(error32);disp(error6);end
                       options(9)=-3; if trx, x=x';end, return
   end
   xrec=x; frec=f;     % record point and function value
% Constrained problem   
   if constr,  fp=f; kless=0;
      if trx,  fc=feval(func,x');  
      else,    fc=feval(func,x);  end
      if isempty(fc),  
             if dispwarn,disp(errmes);disp(error50);end
             options(9)=-5; if trx, x=x';end, return
      elseif isnan(fc),
             if dispwarn,disp(errmes);disp(error51);disp(error6);end
             options(9)=-5; if trx, x=x';end, return
      elseif abs(fc)==Inf, 
             if dispwarn,disp(errmes);disp(error52);disp(error6);end
             options(9)=-5; if trx, x=x';end, return
      end
      options(12)=options(12)+1; 
      PenCoef=1;                              % first rough approximation
      if fc<=cnteps,  FP=1; fc=0;             % feasible point 
      else,           FP=0;                   % infeasible point
      end
      f=f+PenCoef*fc;
   end   
% ----}
% COMPUTE THE GRADIENT ( FIRST TIME ) ----{
   if app,   deltax=h1*ddx*ones(size(x));
     if constr, if trx, g=apprgrdn(x',fp,fun,deltax',1); 
                else,   g=apprgrdn(x ,fp,fun,deltax,1); end
     else,      if trx, g=apprgrdn(x',f,fun,deltax',1); 
                else,   g=apprgrdn(x ,f,fun,deltax,1); end
     end, options(10)=options(10)+n;
   else,     if trx,  g=feval(grad,x');  
             else,    g=feval(grad,x);   end
             options(11)=options(11)+1;
   end
   if size(g,2)==1, g=g'; end, ng=norm(g);  
   if size(g,2)~=n,    if dispwarn,disp(errmes);disp(error40);end
                       options(9)=-4; if trx, x=x';end, return
   elseif isnan(ng),   if dispwarn,disp(errmes);disp(error41);disp(error6);end
                       options(9)=-4; if trx, x=x';end, return
   elseif ng==Inf,     if dispwarn,disp(errmes);disp(error42);disp(error6);end
                       options(9)=-4; if trx, x=x';end, return
   elseif ng<ZeroGrad, if dispwarn,disp(errmes);disp(error43);disp(error6);end
                       options(9)=-4; if trx, x=x';end, return
   end
   if constr, if ~FP
      if appconstr, 
        deltax=sign(x); idx=find(deltax==0); 
        deltax(idx)=ones(size(idx));  deltax=ddx*deltax;
                if trx, gc=apprgrdn(x',fc,func,deltax',0); 
                else,   gc=apprgrdn(x ,fc,func,deltax ,0); end
                options(12)=options(12)+n; 
      else,     if trx,  gc=feval(gradc,x');  
                else,    gc=feval(gradc,x); end
                options(13)=options(13)+1; 
      end
      if size(gc,2)==1, gc=gc'; end, ngc=norm(gc);
      if size(gc,2)~=n,
             if dispwarn,disp(errmes);disp(error60);end
             options(9)=-6; if trx, x=x';end, return
      elseif isnan(ngc),
             if dispwarn,disp(errmes);disp(error61);disp(error6);end
             options(9)=-6; if trx, x=x';end, return
      elseif ngc==Inf, 
             if dispwarn,disp(errmes);disp(error62);disp(error6);end
             options(9)=-6; if trx, x=x';end, return
      elseif ngc<ZeroGrad, 
             if dispwarn,disp(errmes);disp(error63);end
             options(9)=-6; if trx, x=x';end, return
      end
      g=g+PenCoef*gc; ng=norm(g);
   end, end
   grec=g; nng=ng;   
% ----}
% INITIAL STEPSIZE
      h=h1*sqrt(options(2))*max(abs(x));     % smallest possible stepsize
      if abs(options(1))~=1, 
          h=h1*max(abs([options(1),h]));     % user-supplied stepsize
      else,  
          h=h1*max(1/log(ng+1.1),abs(h));    % calculated stepsize
      end

% RESETTING LOOP ----{
while 1,
   kcheck=0;                        % Set checkpoint counter.
   kg=0;                            % stepsizes stored
   kj=0;                            % ravine jump counter
   B=eye(n);                        % re-set transf. matrix to identity
   fst=f; g1=g;  dx=0;
% ----}    
   
% MAIN ITERATIONS ----{

   while 1,
      k=k+1;kcheck=kcheck+1;
       laststep=dx;

% ADJUST GAMMA --{
           gamma=1+max([ajb^((ajp-kcheck)*n),2*options(3)]);
           gamma=min([gamma,ajs^max([1,log10(nng+1)])]);
% --}      
      gt=g*B;   w=wdef;       
% JUMPING OVER A RAVINE ----{      
      if (gt/norm(gt))*(g1'/norm(g1))<low_bound
        if kj==2, xx=x;  end,  if kj==0, kd=4;  end,      
        kj=kj+1;  w=-.9; h=h*2;             % use large coef. of space dilation
        if kj>2*kd,     kd=kd+1;  warnno=1;  
          if any(abs(x-xx)<epsnorm*abs(x)), % flat bottom is detected 
            if dispwarn,disp(wrnmes);disp(warn08); end
          end
        end 
      else, kj=0; 
      end
% ----}
% DILATION ----{      
      z=gt-g1;
      nrmz=norm(z);
      if(nrmz>epsnorm*norm(gt))             
         z=z/nrmz;               
         g1=gt+w*(z*gt')*z;  B=B+w*(B*z')*z;    
      else
         z=zeros(1,n); nrmz=0; g1=gt;
      end
      d1=norm(g1);  g0=(g1/d1)*B';
% ----}
% RESETTING ----{
if kcheck>1
    idx=find(abs(g)>ZeroGrad); numelem=size(idx,2);
    if numelem>0, grbnd=epsnorm*numelem^2;
      if all(abs(g1(idx))<=abs(g(idx))*grbnd) | nrmz==0 
          if dispwarn,  disp(wrnmes);  disp(warn20); end
          if abs(fst-f)<abs(f)*.01, ajp=ajp-10*n; 
          else, ajp=ajpp; end
          h=h1*dx/3; k=k-1; 
          break
      end
    end   
end
% ----}
% STORE THE CURRENT VALUES AND SET THE COUNTERS FOR 1-D SEARCH 
      xopt=x;fopt=f;   k1=0;k2=0;ksm=0;kc=0;knan=0;  hp=h;
      if constr, Reset=0; end
% 1-D SEARCH ----{ 
      while 1,
         x1=x;f1=f;   
         if constr, FP1=FP; fp1=fp; end
         x=x+hp*g0;  
% FUNCTION VALUE         
         if trx, f=feval(fun,x');  
         else,   f=feval(fun,x );  end
         options(10)=options(10)+1;  
         if h1*f==Inf
            if dispwarn, disp(errmes); disp(error5); end
            options(9)=-7; if trx, x=x';end, return
         end
         if constr, fp=f;
           if trx,fc=feval(func,x');
           else,  fc=feval(func,x);end
           options(12)=options(12)+1; 
           if  isnan(fc),
                  if dispwarn,disp(errmes);disp(error51);disp(error6);end
                  options(9)=-5; if trx, x=x';end, return
           elseif abs(fc)==Inf, 
                  if dispwarn,disp(errmes);disp(error52);disp(error6);end
                  options(9)=-5; if trx, x=x';end, return
           end
           if fc<=cnteps,   FP=1; fc=0; 
           else,            FP=0;       
            fp_rate=(fp-fp1); 
            if fp_rate<-epsnorm
             if ~FP1 
              PenCoefNew=-15*fp_rate/norm(x-x1);
              if PenCoefNew>1.2*PenCoef, 
                 PenCoef=PenCoefNew; Reset=1; kless=0; f=f+PenCoef*fc; break
              end
             end 
            end
           end
           f=f+PenCoef*fc;
         end
         if abs(f)==Inf | isnan(f)
             if dispwarn, disp(wrnmes);  
             if isnan(f), disp(error31); else, disp(error32); end 
             end
             if ksm | kc>=mxtc, options(9)=-3; if trx, x=x';end, return
             else, k2=k2+1;k1=0; hp=hp/dq; x=x1;f=f1; knan=1; 
                   if constr, FP=FP1; fp=fp1; end
             end
% STEP SIZE IS ZERO TO THE EXTENT OF EPSNORM
         elseif all(abs(x-x1)<abs(x)*epsnorm), 
                stepvanish=stepvanish+1;
                if stepvanish>=5,
                    options(9)=-14;
                    if dispwarn, disp(termwarn1);        
                                       disp(endwarn(4,:)); end
                    if trx,x=x';end,  return
                else, x=x1; f=f1; hp=hp*10; ksm=1;
                      if constr, FP=FP1; fp=fp1; end
                end
% USE SMALLER STEP
         elseif h1*f<h1*gamma^sign(f1)*f1
             if ksm,break,end,  k2=k2+1;k1=0; hp=hp/dq; x=x1;f=f1; 
                                if constr, FP=FP1; fp=fp1; end
             if kc>=mxtc, break, end
% 1-D OPTIMIZER IS LEFT BEHIND
         else   if h1*f<=h1*f1, break,  end
% USE LARGER STEP
            k1=k1+1; if k2>0, kc=kc+1; end, k2=0;
            if k1>=20,      hp=du20*hp; 
            elseif k1>=10,  hp=du10*hp;
            elseif k1>=3,   hp=du03*hp;
            end
         end
      end
% ----}  End of 1-D search
% ADJUST THE TRIAL STEP SIZE ----{
      dx=norm(xopt-x);
      if kg<kstore,  kg=kg+1;  end
      if kg>=2,  nsteps(2:kg)=nsteps(1:kg-1); end
      nsteps(1)=dx/(abs(h)*norm(g0));
      kk=sum(nsteps(1:kg).*[kg:-1:1])/sum([kg:-1:1]);
        if     kk>des, if kg==1,  h=h*(kk-des+1); 
                       else,   h=h*sqrt(kk-des+1); end
        elseif kk<des,         h=h*sqrt(kk/des);  
        end

stepvanish=stepvanish+ksm;
% ----}
% COMPUTE THE GRADIENT ----{
      if app,    
        deltax=sign(g0); idx=find(deltax==0); 
        deltax(idx)=ones(size(idx));  deltax=h1*ddx*deltax;
        if constr,  if trx,  g=apprgrdn(x',fp,fun,deltax',1);
                    else,    g=apprgrdn(x ,fp,fun,deltax ,1);    end
        else,       if trx,  g=apprgrdn(x',f,fun,deltax',1);
                    else,    g=apprgrdn(x ,f,fun,deltax ,1);    end
        end,  options(10)=options(10)+n;
      else
          if trx,  g=feval(grad,x'); 
          else,    g=feval(grad,x ); end
          options(11)=options(11)+1;
      end
      if size(g,2)==1, g=g'; end,    ng=norm(g);
      if isnan(ng), 
       if dispwarn, disp(errmes); disp(error41); end
       options(9)=-4; if trx, x=x'; end, return
      elseif ng==Inf,     
       if dispwarn,disp(errmes);disp(error42);end
       options(9)=-4; if trx, x=x';end, return
      elseif ng<ZeroGrad, 
           if dispwarn,disp(wrnmes);disp(warn1);end
           ng=ZeroGrad;
      end
% Constraints:      
      if constr, if ~FP
         if ng<.01*PenCoef 
           kless=kless+1; 
           if kless>=20, PenCoef=PenCoef/10; Reset=1; kless=0; end
         else, kless=0;
         end  
         if appconstr, 
           deltax=sign(x); idx=find(deltax==0); 
           deltax(idx)=ones(size(idx));  deltax=ddx*deltax;
               if trx, gc=apprgrdn(x',fc,func,deltax',0); 
               else,   gc=apprgrdn(x ,fc,func,deltax ,0); end
               options(12)=options(12)+n; 
         else, if trx,  gc=feval(gradc,x');  
               else,    gc=feval(gradc,x ); end
               options(13)=options(13)+1; 
         end
         if size(gc,2)==1, gc=gc'; end, ngc=norm(gc);
         if     isnan(ngc),
                if dispwarn,disp(errmes);disp(error61);end
                options(9)=-6; if trx, x=x';end, return
         elseif ngc==Inf, 
                if dispwarn,disp(errmes);disp(error62);end
                options(9)=-6; if trx, x=x';end, return
         elseif ngc<ZeroGrad & ~appconstr, 
                if dispwarn,disp(errmes);disp(error63);end
                options(9)=-6; if trx, x=x';end, return
         end
         g=g+PenCoef*gc; ng=norm(g); 
         if Reset, if dispwarn,  disp(wrnmes);  disp(warn21); end
            h=h1*dx/3; k=k-1; nng=ng; break
         end
      end, end
      if h1*f>h1*frec, frec=f; xrec=x; grec=g; end
% ----}
     if ng>ZeroGrad,
      if knorms<10,  knorms=knorms+1;  end
      if knorms>=2,  gnorms(2:knorms)=gnorms(1:knorms-1); end
      gnorms(1)=ng;
      nng=(prod(gnorms(1:knorms)))^(1/knorms);
     end 

% DISPLAY THE CURRENT VALUES ----{
if k==ld
  disp('Iter.# ..... Function ... Step Value ... Gradient Norm ');
  disp(sprintf('%5i   %13.5e   %13.5e     %13.5e',k,f,dx,ng));
  ld=k+dispdata;
end
%----}
% CHECK THE STOPPING CRITERIA ----{
termflag=1;
if constr, if ~FP, termflag=0; end, end
if kcheck<=5, termflag=0; end
if knan, termflag=0; end
if kc>=mxtc, termflag=0; end
% ARGUMENT
 if termflag
     idx=find(abs(x)>=lowxbound);
     if isempty(idx) | all(abs(xopt(idx)-x(idx))<=options(2)*abs(x(idx)))   
          termx=termx+1;
% FUNCTION
          if abs(f-frec)> detfr * abs(f)    & ...
             abs(f-fopt)<=options(3)*abs(f) & ...
             krerun<=3                      & ...
             ~constr
             if any(abs(xrec(idx)-x(idx))> detxr * abs(x(idx)))
                 if dispwarn,disp(wrnmes);disp(warn09);end
                 x=xrec; f=frec; g=grec; ng=norm(g); krerun=krerun+1;
                 h=h1*max([dx,detxr*norm(x)])/krerun;
                 warnno=2; break
             else, h=h*10;
             end       
          elseif  abs(f-frec)> options(3)*abs(f)    & ...
                  norm(x-xrec)<options(2)*norm(x) & constr 
                  
          elseif  abs(f-fopt)<=options(3)*abs(f)  | ...
                  abs(f)<=lowfbound               | ...
                  (abs(f-fopt)<=options(3) & termx>=limxterm )
                  if stopf
                   if dx<=laststep
                    if warnno==1 & ng<sqrt(options(3)), warnno=0; end
                    if ~app, if any(abs(g)<=epsnorm2), warnno=3; end, end
                    if warnno~=0, options(9)=-warnno-10;
                       if dispwarn, disp(termwarn1); 
                           disp(endwarn(warnno,:)); 
                           if app, disp(appwarn); end
                       end    
                    else, options(9)=k; 
                       if dispwarn, disp(termwarn0); end
                    end
                    if trx,x=x';end,  return
                   end
                  else, stopf=1; 
                  end  
          elseif dx<1.e-12*max(norm(x),1) & termx>=limxterm 
                    options(9)=-14;
                    if dispwarn, disp(termwarn1); disp(endwarn(4,:));
                                    if app, disp(appwarn); end
                    end
                    x=xrec; f=frec;    
                    if trx,x=x';end,  return
          else, stopf=0;          
          end
    end
 end 
% ITERATIONS LIMIT
      if(k==options(4))
          options(9)=-9; if trx, x=x'; end,
          if dispwarn, disp(wrnmes);  disp(warn4); end
          return
      end
% ----}
% ZERO GRADIENT ----{
    if constr 
      if ng<=ZeroGrad,
          if dispwarn,  disp(termwarn1);  disp(warn1); end
          options(9)=-8; if trx,x=x';end,return
      end
    else  
      if ng<=ZeroGrad,        nzero=nzero+1; 
       if dispwarn, disp(wrnmes);  disp(warn1);  end
       if nzero>=3,  options(9)=-8; if trx,x=x';end,return, end
       g0=-h*g0/2;
       for i=1:10,
          x=x+g0;               
          if trx, f=feval(fun,x');  
          else,   f=feval(fun,x ); end
          options(10)=options(10)+1;
          if abs(f)==Inf 
           if dispwarn, disp(errmes);  disp(error32);  end
           options(9)=-3;if trx,x=x';end,return
          elseif isnan(f),
           if dispwarn, disp(errmes);  disp(error32);  end
           options(9)=-3;if trx,x=x';end,return
          end
          if app, 
             deltax=sign(g0); idx=find(deltax==0); 
             deltax(idx)=ones(size(idx));  deltax=h1*ddx*deltax;
             if trx,  g=apprgrdn(x',f,fun,deltax',1);
             else,    g=apprgrdn(x ,f,fun,deltax ,1);    end
             options(10)=options(10)+n;
          else
             if trx,  g=feval(grad,x');
             else,    g=feval(grad,x );   end
             options(11)=options(11)+1;
          end
          if size(g,2)==1, g=g'; end,       ng=norm(g);
          if ng==Inf
              if dispwarn, disp(errmes);  disp(error42); end
              options(9)=-4; if trx, x=x'; end, return
          elseif isnan(ng) 
              if dispwarn, disp(errmes);  disp(error41); end
              options(9)=-4; if trx, x=x'; end, return
          end
          if ng>ZeroGrad, break, end
       end
       if ng<=ZeroGrad,
          if dispwarn,  disp(termwarn1);  disp(warn1); end
          options(9)=-8; if trx,x=x';end,return
       end
       h=h1*dx; break
      end
    end  
% ----}
% FUNCTION IS FLAT AT THE POINT ----{
    if ~constr & abs(f-fopt)<abs(fopt)*options(3) & kcheck>5 & ng<1
     idx=find(abs(g)<=epsnorm2); ni=size(idx,2);
     if ni>=1 & ni<=n/2 & kflat<=3, kflat=kflat+1;
       if dispwarn,  disp(wrnmes); disp(warn31); end, warnno=1;
       x1=x; fm=f;
       for j=idx, y=x(j); f2=fm;
        if y==0, x1(j)=1; elseif abs(y)<1, x1(j)=sign(y); else, x1(j)=y; end
        for i=1:20, x1(j)=x1(j)/1.15;
         if trx, f1=feval(fun,x1');  
         else,   f1=feval(fun,x1 ); end
         options(10)=options(10)+1;
         if abs(f1)~=Inf & ~isnan(f1), 
          if h1*f1>h1*fm,     y=x1(j); fm=f1;
          elseif h1*f2>h1*f1, break
          elseif f2==f1,      x1(j)=x1(j)/1.5;
          end, f2=f1;
         end
        end
        x1(j)=y; 
       end
       if h1*fm>h1*f
        if app,    
          deltax=h1*ddx*ones(size(deltax));
          if trx,  gt=apprgrdn(x1',fm,fun,deltax',1);
          else,    gt=apprgrdn(x1 ,fm,fun,deltax ,1);    end
          options(10)=options(10)+n;
        else
          if trx,  gt=feval(grad,x1'); 
          else,    gt=feval(grad,x1 ); end
          options(11)=options(11)+1;
        end
        if size(gt,2)==1, gt=gt'; end,       ngt=norm(gt);
        if ~isnan(ngt) & ngt>epsnorm2,  
          if dispwarn,  disp(warn32); end
          options(3)=options(3)/5;
          x=x1; g=gt; ng=ngt; f=fm; h=h1*dx/3; break
        end
       end
     end
    end
% ----}
end % iterations
end % restart
% end of the function
%

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