[en] We develop an automatic spectral synthesis technique based on linear programming and apply it to narrow-band spectrophotometry covering lambdalambda3300--10800 for the nuclei of M31 and three giant elliptical galaxies in Virgo. The mean energy distribution of the gE nuclei (r<400 pc) is well defined and, within the uncertainty in E (B-V), does not differ significantly from that for M31 (r<50 pc). The observations are consistent with only a small range of main-sequence turnoffs, which depends on the relative rates of main-sequence and subgiant evolution. Using Iben's (1967) models of stellar interiors, we find the bulk of the stars in gE nuclei formed 8--11 x 10⁹ years ago, with star formation possibly continuing at a much reduced level until approx.4 x 10⁹ years ago. The maximum mean rate of postburst star formation consistent with the data is approx.2.9[L/subv//10¹¹L/subv/(sun)]M/sub sun/yr^-1. There is no evidence of a significant Population II component. Na i and Mg i exhibit distinct enhancements with respect to Ca ii and the Fe-peak, but the evidence for Fe-peak enrichment is not strong.M giants dominate M dwarfs throughout the spectrum, and M/L/subv/>30 (solar units) is excluded if stars with M/subv/>18 are not important. The best models yield 2approximately-less-thanM/L/subv/approximately-less-than15, in agreement with recent dynamical estimates. Infrared spectral features indicate that short-lived, low-gravity phases near the tip of the asymptotic giant branch make significant contributions. Aside from marginal evidence for a weak optical nonthermal component in NGC 4552, there are no marked optical peculiarities associated with the radio sources in the gE's, and their general properties are consistent with a model in which the radio energy is produced by supernovae which form in the nucleus following mass loss during stellar evolution