% Chapter 5: Numerical Techniques
%            The One Dimensional Problem
%--------------------------------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Applied Optimization with Matlab Programming
% Dr. P.Venkataraman
% Second Edition,  John Wiley
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--------------------------------------------------
% Generic Scanning Procedure - Single Variable
%  Section 5.2.3
% An m-file to bracket the minimum
%	of a function of a single
%   Lower bound is known
%   only upper bound is found by this program
%------------------------------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% Usage
%  UpperBound_1Var(functname,a0,da,ns)
%
%%% the following information are passed to the function
% the name of the function 			'functname'
% the function should be available as a function m-file
% and shoukd return the value of the function
%
%%% the inputs
% the initial value							a0
% the incremental value 					da
% the number of scanning steps	    	    ns
%
%
%%% Example
%  UpperBound_1Var('Example5_1',0,1,10)
%
%
%%%   Data %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function ReturnValue = UpperBound(functname,a0,da,ns)

%%% management functions
format compact  % avoid skipping a line when writing to the command window
warning off  % don't report any warnings like divide by zero etc.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	ntrials are used to bisect/double values of da
if (ns ~= 0) ntrials = ns;
else ntrials = 20;   % default
end

if (da ~= 0) das = da;
else das = 1;  %default
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%  for a minimum  and to start the scanning
%%%  we need the initial point to have a negative slope
%%%   this additional code should assist
% modification for a positive slope
% check if initial slope is positive
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
fstart = feval(functname,a0);
fnext = feval(functname,(a0 + 0.1*das));
slope =(fnext-fstart)/(0.1*das);
if (slope > 0)
   for i = 2:ntrials
      astart = a0 + i*das;
      anext = astart + 0.1*das;
      fstart = feval(functname,astart);
      fnext = feval(functname,anext);
      newslope = (fnext - fstart)/(anext - astart);
      if (newslope < 0)
         a0= astart;
         break;
      end
   end 
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for i = 1:ntrials;
    j = 0;	dela = j*das;	a00 = a0 + dela;
    f0 = feval(functname,a00);

    j = 1;	dela = j*das;	a01 = a0 + dela;
    f1 = feval(functname,a01);
    f1s =f1;
    if f1 < f0
        for j = 2:ntrials
            aa01 = a0 + j*das;
            af1 = feval(functname,aa01);
            f1s=min(f1s,af1);
            if af1 > f1s
                ReturnValue = [aa01 af1];
                return;
            end
        end
        % after ntrials the value is still less than start
        %return last value
        fprintf('\n cannot increase function value\n')
        ReturnValue = [aa01 af1];
        return;

    else

        das = 0.5*das;


    end
end
fprintf('\ncannot decrease function value \n')

fprintf('\ninitial slope may be greater than zero')
fprintf('\nlower bound needs adjusting \n')

% return start value
ReturnValue =[a0 f0];