Exponential approximation via closed-form Gauss-Newton method

A convergent gradient descent technique is provided for the approximation of functions of time from their Laplace transforms. The approximation is composed of a finite series of exponential functions. Both the complex coefficients and exponents of the exponential terms are adjusted to reduce the integral squared error at each iteration. The novel feature of the present method is that all parameters are adjusted through the use of explicit formulas. This overcomes the usual numerical difficulties associated with the inversion of ill-conditioned matrices. Finally, the utility of the proposed method is demonstrated by its performance on certain examples chosen to allow comparison with previously developed methods.