function [logmax,dL]=nolin1d_grad_optag_3hyp_schur(x,M,y)

% NOLIN1D_GRAD_OPTAG_3HYP_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, and a the data correlation.
%
% This is a fast algorithm using Generalised Schur algorithm to perform
% computation task, impossible by standard MATLAB command.
%
% [logmax,dL] = NOLIN1D_GRAD_OPTAG_3HYP_SCHUR(x,M,y)
%
% Inputs:   M   : one-dimensional input column vector
%           y   : one-dimensional output column vector
%           x   : hyperparameter for GP, x = [a g V] for training
%
% 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);
a=exp(x(1));
g=exp(x(2));
V=exp(x(3));
G=get_eigen_block_trace(M,[a g a*V]);
[trq2,bdQb,dQb,invB,logdet,tr,yQy,yQQy]=schur_invbtrace3(G,y);
logmax=logdet+yQy;
dL=zeros(3,1);
dL(1,1)=N0-yQy;
dL(2,1)=trq2-bdQb;
dL(3,1)=a*V*(tr-yQQy);