function [logmax,dL]=nolin1d_grad_optag_durlev(x,M,y)

% NOLIN1D_GRAD_OPTAG_DURLEV     is an optimisation script for the training
% of Gaussian process prior models, in cases where covariance matrices are
% Toeplitz.
%
% [LOGMAX,DL] = NOLIN1D_GRAD_OPTAG_DURLEV(x,M,y)
%
% This optimisation script uses only Gradient information to perform
% optimisation for Gaussian process prior models, by adapting
% hyperparameters to minimise negative the log-likelihood function,
%
%                   logmax = log(det(Q)) + N0*log(y'*inv(Q)*y)
%
% The prior covariance function used is of the form,
%
%                   C(zi,zj) = exp([zi-zj]'*g*[zi-zj]) + v
%
% Inputs:
%       x   = Hyperparameters, [g v]
%       M   = Input explanatory variable, of dimension {N0 x 1}
%       y   = Outcome (vector) from every input of the explantory variable
%
% Outputs:
%       LOGMAX  = Negative log-likelihood function value
%       DL      = Gradient vector values
%
% Note that, this GP optimisation script is meant for strictly
% one-dimensional time-series input explanatory variable only. This script
% does not utilise user-supplied Hessian information.
%
% This function script utilises modified Durbin-Levinson's algorithm to
% construct fast algorithm for computation of task impossible by standard
% MATLAB command.
%
%
% (C) Gaussian Process Toeplitz Toolbox 1.0
% (C) Copyright 2005-2007, Keith Neo <keith.neo@nuim.ie>
%                          http://www.hamilton.ie

g=exp(x(1));
V=exp(x(2));
N0=size(M,1);
p=[1 g V];
ct=toep2_genct(M,p);
invb=toep2_invcy(ct,y);
yQy=y'*invb;
[logdet,ds,vn1]=toep2_logdet(ct);
logmax=logdet+N0*log(yQy);

tmp=-0.5*g*(M'-M(1,1)).^2;
dt=tmp.*exp(tmp);
tmp=toe_times_y(dt,invb);
bdQb=invb'*tmp;
trdQ=toep2_trace(ds,vn1,dt);
dt=[1 zeros(1,N0-1)];
trQ=toep2_trace(ds,vn1,dt);
Ny=N0/yQy;
yQQy=invb'*invb;
dL=zeros(2,1);
dL(1,1)=trdQ-Ny*bdQb;
dL(2,1)=V*(trQ-Ny*yQQy);