The recurrence algorithm for calculating electromagnetic scattering from a multilayer sphere, which was described recently by Wu and Wang [Radio Sci. 26, 1393, (1991)], is derived in a slightly modified form and extended to include a calculation of the internal field and the absorption cross sections of the individual layers. The original algorithm calculates the scattering by a recurrence procedure that propagates the log derivatives of the Debye potentials outward from the core to the outer layer. The extended algorithm then continues the calculation by an inward recurrence procedure that propagates the Debye potentials from the outer layer to the core. Concurrent with the inward propagation, a separate algorithm calculates the absorption cross sections of the imbedded concentric spheres. The results of several example calculations are presented, including the differential cross section and internal electric field of a Luneburg lens.