Bounds for variable degree rational L∞ approximations to the matrix cosine

In this work we derive new alternatives for efficient computation of the matrix cosine which is useful when solving second order Initial Value Problems such as free vibration. We focus especially on the two classes of normal and nonnegative matrices and we present intervals of applications for rational L∞ approximations of various degrees for these types of matrices in the lines of Hargreaves and Higham (2005). Our method relies on Remez algorithm for rational approximation while the innovation here is the choice of the starting set of non-symmetrical Chebyshev points. Only one Remez iteration is then usually enough to quickly approach the actual L∞ approximant.