est_phi1_theta1_AR1MA1 = function(r1,r2){ 
#
# Estimates \phi_1 and \theta_1 in the ARIMA(1,d,1) model:
#
# w_t = \phi_1 w_{t-1} + a_t - \theta_1 a_{t-1} 
#
# using a very simple version of Newton's method.
#
# Inputs: 
#   r1,r2 = autocorrelation of lag 1 and 2 
#
# 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.
# 
#-----

# The estimate for phi_1 is simple: 
#
phi1 = r2/r1 

# For theta_2 we must solve for x in a nonlinear function (f(x)=0) 
# 
f      = function(x) ( (1-x*phi1) * ( phi1 - x ) - r1 * ( 1 + x^2 - 2 * phi1 * x ) )
fprime = function(x) ( - phi1 * ( phi1 - x ) - ( 1 - x * phi1 ) - r1 * ( 2 * x - 2 * phi1 ) )
x0 = 0.0 
for( ii in 1:20 ){
  x1 = x0 - f(x0)/fprime(x0)
  x0 = x1 
}
theta1 = x0

c(phi1,theta1) # return the results

}