Recently, an efficient, numerically robust algorithm for calculating acoustic normal modes called MODELAB was implemented [J. Acoust. Soc. Am.97, 1576–1585 (1995)]. In this paper, a new approach is introduced and shown to significantly reduce the most computationally intensive aspect of MODELAB: the requirement of fine wave number presampling during the search for eigenvalues. Analytic derivatives (up to sixth order) are efficiently calculated using a new recursion relation for the characteristic equation. The derivatives are used to form Padé approximants, which in turn, are used to locally extrapolate the characteristic equation to find successive eigenvalues. Results indicate that an algorithm using this technique can be more than a factor of five times faster than the MODELAB algorithm, and the new technique is shown to be particularly efficient at near‐degenerate eigenvalues, which previously had been the most difficult to identify.