The high performance method for obtaining an ARMA model spectral estimate of a wide-sense stationary time series has been found to provide typically superior performance when compared to such contemporary approaches as the Box-Jenkins and maximum entropy methods. In this dissertation, fast recursive algorithmic implementations of the high performance method are developed. They are recursive in the sense that as a new element of the time series is observed, the parameters characterizing an ARMA spectral estimate are algorithmically updated. The number of multiplications and additions required at each recursive stage are of the order p with p being the number of denominator coefficients of the ARMA model. Methods of modification of the data are applied to achieve a significant computational improvement. The development is predicated on utilization of various projection operators. / Ph. D.