numerically_evaluate_xt_ARMA_01 = function(theta10,a_t,w_t){
#
# Given a time series of differences in w_t and the initial time 
# series of white noise in a_t and the initial MA(1) parameter theta1
# this routine numerically computes an estimate of the (negative) derivative of 
# a_t with resepect to theta1    
#
# Written by:
# -- 
# John L. Weatherwax                2009-04-21
# 
# email: wax@alum.mit.edu
# 
# Please send comments and especially bug reports to the
# above email address.
# 
#-----
    
source('uncond_sum_of_squares_ARMA_01.R')    

param_step = 0.02

if( theta10 != 0 ){ 
  delta_theta1 = param_step * theta10 # take a step in theta1 ... call S() using arguments in lambda
}else{
  delta_theta1 = param_step 
}
new_theta1 = theta10 + delta_theta1
    
res = uncond_sum_of_squares_ARMA_01(new_theta1,w_t)
    
new_a_t = res[[3]] 
    
x_t = ( a_t - new_a_t )/delta_theta1
    
return( x_t ) 
    
}