A new algorithm OMI (Optimization in Multiple Intervals) for the computation of the logarithmic decrement from exponentially damped harmonic oscillations is described. This method is shown to be effective and computationally compact for high damping materials. A comparison between the OMI algorithm and the four classical methods usually used in the computation of the logarithmic decrement is reported. The OMI algorithm yields high precision in the computation of the logarithmic decrement and the smallest dispersion of experimental points on the plots of mechanical loss spectra. The effect of the acquisition parameters and the experimental conditions on the results of computations of the logarithmic decrement and the relative error is discussed.