The Pisarenko form of the method is consistent but inefficient for estimating sinusoid signals and inconsistent for estimating damped sinusoids or exponential signals. The modified Prony algorithm has the great practical advantage that it will estimate any of these functions according to which best fits the available observations. The algorithm directly estimates the coefficients, g k say, of this difference equation. The residual sum of squares after estimating the a j can be written in terms of the g k . The eigenvalue l is the Lagrange multiplier for the scale of g in the homogeneous difference equation.
Inverse iteration proves very suitable for the actual computations.
Although the algorithm estimates all functions in the same way, the practical considerations and asymptotic arguments differ depending on whether the signals are periodic or transient, real or complex. If rate coefficients are negative, then this represents a function which is transient, i.e., decays to zero as time goes on. This represents the outcome of a chemical experiment or other process in which the reaction completes in a finite amount of time. Real exponential fitting is one of the most important, difficult and frequently occurring problems of applied data analysis. Applications include radioactive decay, compartment models and atmospheric transfer functions.