You are an expert in writing code in many different languages. Your goal is to perform code completion for the following code, keeping in mind the rest of the code and the file meta data.
    Metadata details:
    filename: spm_dx_eig.m
    path of file: ./repos/bspm/thirdparty/spm8
    
    the code of the file until where you have to start completion: function [dx] = spm_dx_eig(dfdx,f,t)
% returns dx(t) = (expm(dfdx*t) - I)*inv(dfdx)*f; using an eigensystem
% FORMAT [dx] = spm_dx_eig(dfdx,f,[t])
% dfdx   = df/dx
% f      = dx/dt
% t      = integration time: (default t = Inf);
%          if t is a cell (i.e., {t}) then t is set ti:
%          exp(t - log(diag(-dfdx))
%
% dx     = x(t) - x(0)
%--------------------------------------------------------------------------
% Integration of a dynamic system using local linearization.  This scheme

    
function [dx] = spm_dx_eig(dfdx,f,t)
% returns dx(t) = (expm(dfdx*t) - I)*inv(dfdx)*f; using an eigensystem
% FORMAT [dx] = spm_dx_eig(dfdx,f,[t])
% dfdx   = df/dx
% f      = dx/dt
% t      = integration time: (default t = Inf);
%          if t is a cell (i.e., {t}) then t is set ti:
%          exp(t - log(diag(-dfdx))
%
% dx     = x(t) - x(0)
%--------------------------------------------------------------------------
% Integration of a dynamic system using local linearization.  This scheme
% accommodates nonlinearities in the state equation by using a functional of
% f(x) = dx/dt.  This uses the equality
%
%             expm([0   0     ]) = expm(t*dfdx) - I)*inv(dfdx)*f
%                  [t*f t*dfdx]
%
% When t -> Inf this reduces to
%
%              dx(t) = -inv(dfdx)*f
%
% These are the solutions to the gradient ascent ODE
%
%            dx/dt   = k*f = k*dfdx*x =>
%
%            dx(t)   = expm(t*k*dfdx)*x(0)
%                    = expm(t*k*dfdx)*inv(dfdx)*f(0) -
%                      expm(0*k*dfdx)*inv(dfdx)*f(0)
%
% When f = dF/dx (and dfdx = dF/dxdx), dx represents the update from a
% Gauss-Newton ascent on F.  This can be regularised by specifying {t}
% A heavy regularization corresponds to t = -4 and a light
% regularization would be t = 4. This version of spm_dx uses the
% eigensystem of dfdx (i.e., natural gradients).
%
% references:
% 
% Friston K, Mattout J, Trujillo-Barreto N, Ashburner J, Penny W. (2007).
% Variational free energy and the Laplace approximation. NeuroImage.
% 34(1):220-34
% 
% Ozaki T (1992) A bridge between nonlinear time-series models and
% nonlinear stochastic dynamical systems: A local linearization approach.
% Statistica Sin. 2:113-135.
% 
%__________________________________________________________________________
% Copyright (C) 2008 Wellcome Trust Centre for Neuroimaging

% Karl Friston
% $Id: spm_dx_eig.m 1277 2008-03-28 18:36:49Z karl $

% defaults
%--------------------------------------------------------------------------
if nargin < 3, t = Inf; end