The Newton-Raphson method is widely used in Mossbauer spectroscopy to fit a spectrum to a number of Lorentzian-shaped lines. Although most Mossbauer spectrometers are now built around a personal computer, such computers are generally too slow for the fitting of spectra. The author discusses a concurrent version of the Newton-Raphson algorithm and its implementation on an accelerator based on a network of INMOS transputers, interfaced to a personal computer. A very significant speed-up factor is obtained with a small number of transputers; for example, the fitting time for a six-line 512-channel spectrum is 2.6 s per iteration with a four-transputer accelerator, compared with about 72 s on a Victor AT computer. Although the work has been applied to the fitting of Mossbauer spectra, the results are equally applicable to other least-squares fitting problems.