Real values of the W -function

Abstract

Approximations for real values of W(x) , where W is defined by solutions of W exp( W ) = x , are presented. All of the approximations have maximum absolute (| W |>1) or relative (| W |<1) errors of O (10 −4 ). With these approximations an efficient algorithm, consisting of a single iteration of a rapidly converging iteration scheme, gives estimates of W(x) accurate to at least 16 significant digits (15 digits if double precision is used). The Fortran code resulting from the algorithm is written to account for the different floating-point-number mantissa lengths on different computers, so that W(x) is computed to the floating-point precision available on the host machine.