function [x, options, flog, pointlog] = quasinew(f, x, options, gradf, ...                                    varargin)%QUASINEW Quasi-Newton optimization.%%	Description%	[X, OPTIONS, FLOG, POINTLOG] = QUASINEW(F, X, OPTIONS, GRADF)  uses a%	quasi-Newton algorithm to find a local minimum of the function F(X)%	whose gradient is given by GRADF(X).  Here X is a row vector and F%	returns a scalar value.   The point at which F has a local minimum is%	returned as X.  The function value at that point is returned in%	OPTIONS(8). A log of the function values after each cycle is%	(optionally) returned in FLOG, and a log of the points visited is%	(optionally) returned in POINTLOG.%%	QUASINEW(F, X, OPTIONS, GRADF, P1, P2, ...) allows  additional%	arguments to be passed to F() and GRADF().%%	The optional parameters have the following interpretations.%%	OPTIONS(1) is set to 1 to display error values; also logs error%	values in the return argument ERRLOG, and the points visited in the%	return argument POINTSLOG.  If OPTIONS(1) is set to 0, then only%	warning messages are displayed.  If OPTIONS(1) is -1, then nothing is%	displayed.%%	OPTIONS(2) is a measure of the absolute precision required for the%	value of X at the solution.  If the absolute difference between the%	values of X between two successive steps is less than OPTIONS(2),%	then this condition is satisfied.%%	OPTIONS(3) is a measure of the precision required of the objective%	function at the solution.  If the absolute difference between the%	objective function values between two successive steps is less than%	OPTIONS(3), then this condition is satisfied. Both this and the%	previous condition must be satisfied for termination.%%	OPTIONS(9) should be set to 1 to check the user defined gradient%	function.%%	OPTIONS(10) returns the total number of function evaluations%	(including those in any line searches).%%	OPTIONS(11) returns the total number of gradient evaluations.%%	OPTIONS(14) is the maximum number of iterations; default 100.%%	OPTIONS(15) is the precision in parameter space of the line search;%	default 1E-2.%%	See also%	CONJGRAD, GRADDESC, LINEMIN, MINBRACK, SCG%%	Copyright (c) Ian T Nabney (1996-2001)%  Set up the options.if length(options) < 18  error('Options vector too short')endif(options(14))  niters = options(14);else  niters = 100;end% Set up options for line searchline_options = foptions;% Don't need a very precise line searchif options(15) > 0  line_options(2) = options(15);else  line_options(2) = 1e-2;  % Defaultend% Minimal fractional change in f from Newton step: otherwise do a line searchmin_frac_change = 1e-4;	display = options(1);% Next two lines allow quasinew to work with expression stringsf = fcnchk(f, length(varargin));gradf = fcnchk(gradf, length(varargin));% Check gradientsif (options(9))  feval('gradchek', x, f, gradf, varargin{:});endnparams = length(x);fnew = feval(f, x, varargin{:});options(10) = options(10) + 1;gradnew = feval(gradf, x, varargin{:});options(11) = options(11) + 1;p = -gradnew;		% Search directionhessinv = eye(nparams); % Initialise inverse Hessian to be identity matrixj = 1;if nargout >= 3  flog(j, :) = fnew;  if nargout == 4    pointlog(j, :) = x;  endendwhile (j <= niters)  xold = x;  fold = fnew;  gradold = gradnew;  x = xold + p;  fnew = feval(f, x, varargin{:});  options(10) = options(10) + 1;  % This shouldn't occur, but rest of code depends on sd being downhill  if (gradnew*p' >= 0)    p = -p;    if options(1) >= 0      warning('search direction uphill in quasinew');    end  end  % Does the Newton step reduce the function value sufficiently?  if (fnew >= fold + min_frac_change * (gradnew*p'))    % No it doesn't    % Minimize along current search direction: must be less than Newton step    [lmin, line_options] = feval('linemin', f, xold, p, fold, ...      line_options, varargin{:});    options(10) = options(10) + line_options(10);    options(11) = options(11) + line_options(11);    % Correct x and fnew to be the actual search point we have found    x = xold + lmin * p;    p = x - xold;    fnew = line_options(8);  end  % Check for termination  if (max(abs(x - xold)) < options(2) & max(abs(fnew - fold)) < options(3))    options(8) = fnew;    return;  end  gradnew = feval(gradf, x, varargin{:});  options(11) = options(11) + 1;  v = gradnew - gradold;  vdotp = v*p';  % Skip update to inverse Hessian if fac not sufficiently positive  if (vdotp*vdotp > eps*sum(v.^2)*sum(p.^2))     Gv = (hessinv*v')';    vGv = sum(v.*Gv);    u = p./vdotp - Gv./vGv;    % Use BFGS update rule    hessinv = hessinv + (p'*p)/vdotp - (Gv'*Gv)/vGv + vGv*(u'*u);  end  p = -(hessinv * gradnew')';  if (display > 0)    fprintf(1, 'Cycle %4d  Function %11.6f\n', j, fnew);  end  j = j + 1;  if nargout >= 3    flog(j, :) = fnew;    if nargout == 4      pointlog(j, :) = x;    end  endend% If we get here, then we haven't terminated in the given number of % iterations.options(8) = fold;if (options(1) >= 0)  disp('Warning: Maximum number of iterations has been exceeded in quasinew');end