function [Phi,dPhi,Ind] = adaex(alpha,t)

% function [Phi,dPhi,Ind] = adaex(alpha,t)
% This is a sample user-defined function to be used by varpro.m.
%
% The model for this sample problem is
%
%   eta(t) = c1 exp(-alpha2 t)*cos(alpha3 t) 
%          + c2 exp(-alpha1 t)*cos(alpha2 t)
%          = c1 Phi1 + c2 Phi2.
%
% Given t and alpha, we evaluate Phi, dPhi, and Ind.
%
% Dianne P. O'Leary and Bert W. Rust, September 2010.


% Evaluate Phi1 = exp(-alpha2 t)*cos(alpha3 t),
%          Phi2 = exp(-alpha1 t)*cos(alpha2 t),
% at each of the data points in t.

    Phi(:,1) = exp(-alpha(2)*t).*cos(alpha(3)*t);
    Phi(:,2) = exp(-alpha(1)*t).*cos(alpha(2)*t);

% The nonzero partial derivatives of Phi with respect to alpha are
%              d Phi_1 / d alpha_2 ,
%              d Phi_1 / d alpha_3 ,
%              d Phi_2 / d alpha_1 ,
%              d Phi_2 / d alpha_2 ,
% and this determines Ind.
% The ordering of the columns of Ind is arbitrary but must match dPhi.

    Ind = [1 1 2 2
           2 3 1 2];

% Evaluate the four nonzero partial derivatives of Phi at each of 
% the data points and store them in dPhi.

    dPhi(:,1) = -t .* Phi(:,1);
    dPhi(:,2) = -t .* exp(-alpha(2)*t).*sin(alpha(3)*t);
    dPhi(:,3) = -t .* Phi(:,2);
    dPhi(:,4) = -t .* exp(-alpha(1)*t).*sin(alpha(2)*t);