Two alternative methods for computing the matrix cosine

We discuss and analyze two methods which may be used to approximate the cosine of a matrix. The two methods are similar in that they both form a rational approximation R(kA) to the matrix cosine, cos(kA), and generate approximate schemes which are based on the exact representation y(t + k) + y(t - k) = 2 cos(kA) y(t) . The methods also allow for our generated approximation to be corrected. The methods differ in that one forms the rational approximation in a partial-fraction form with only linear factors in the denominator, for computational efficiency, especially in a parallel environment. This method allows for the approximation of the local error at any time-step via simple calculations.

The double angle method is presented so that comparisons may be made between it and the time-stepping procedure.

Numerical experiments are performed in order to confirm that our theoretical predictions of convergence rates are valid.