function [logmax,dL]=nolin1d_grad_vonly_schur(x,M,y,g)

% NOLIN1D_GRAD_VONLY_SCHUR      This optimisation script uses only GRADIENT
% information to perform Gaussian Process prior model optimisation for
% time-series data, i.e. one-dimensional data, [M,y] with M as the input
% and y as the output. Hessian is approximated by finite-differencing using
% MATLAB operation.
% The hyperparamters, g represents the length-scale, and V, the noise
% variance.
%
% This is a fast algorithm using Generalised Schur algorithm to perform
% computation task, impossible by standard MATLAB command. Note that, only
% V is trained with g supplied as one of the variables.
%
% [logmax,dL] = NOLIN1D_GRAD_VONLY_SCHUR(x,M,y,g)
%
% Inputs:   M   : one-dimensional input column vector
%           y   : one-dimensional output column vector
%           x   : hyperparameter for GP, x = [V] for training
%           g   : hyperparameter for GP, g to be supplied
%
% Outputs:  logmax  = log likelihood function value
%           dL      = gradient vector value
%
%
% (C) Gaussian Process Schur Toolbox 1.0
% (C) Copyright 2005-2007, Keith Neo <keith.neo@nuim.ie>
%                          http://www.hamilton.ie

N0=size(M,1);
V=exp(x);
G=get_eigen_block_a(M,[g V]);
[logdet,yQy,trQ,yQQy]=schur_invbfastv3(G,y);
logmax=logdet+N0*log(yQy);
dL=V*(trQ-N0*yQQy/yQy);