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We present a practical algorithm to approximate the exponential ofskew-Hermitian matrices up to round-off error based on an efficient computationof Chebyshev polynomials of matrices and the corresponding error analysis. Itis based on Chebyshev polynomials of degrees 2, 4, 8, 12 and 18 which arecomputed with only 1, 2, 3, 4 and 5 matrix-matrix products, respectively. Forproblems of the form $\exp(-iA)$, with $A$ a real and symmetric matrix, animproved version is presented that computes the sine and cosine of $A$ with areduced computational cost. The theoretical analysis, supported by numericalexperiments, indicates that the new methods are more efficient than schemesbased on rational Pad\'e approximants and Taylor polynomials for all tolerancesand time interval lengths. The new procedure is particularly recommended to beused in conjunction with exponential integrators for the numerical timeintegration of the Schr\"odinger equation.

An efficient algorithm to compute the exponential of skew-Hermitian matrices for the time integration of the Schr\"odinger equation