Efficient evaluation of incomplete elliptic integrals and functions (Q1117368)

These integrals with real parameters (with emphasis on Legendre's F(\(\theta\),k)) may be evaluated very efficiently (better than by the expansions given in the Abramowitz-Stegun book) by a Newton-Raphson scheme if one uses relations between Jacobi's elliptic functions and theta functions (motivated by experiences with complete elliptic integrals). Algorithms and a few numerical values are given. Related methods (AGM etc.) are discussed.