Efficient Algorithms for the Matrix Cosine and Sine∗

Several improvements are made to an algorithm of Higham and Smith for com-puting the matrix cosine. The original algorithm scales the matrix by a power of 2 to bring the ∞-norm to 1 or less, evaluates the [8/8] Pade ́ approximant, then uses the double-angle formula cos(2A) = 2 cos2A − I to recover the cosine of the origi-nal matrix. The first improvement is to phrase truncation error bounds in terms of ‖A2‖1/2 instead of the (no smaller and potentially much larger quantity) ‖A‖. The second is to choose the degree of the Pade ́ approximant to minimize the computa-tional cost subject to achieving a desired truncation error. A third improvement is to use an absolute, rather than relative, error criterion in the choice of Padé approximant; this allows the use of higher degree approximants without worsen-ing an a priori error bound. Our theory and experiments show that each of these modifications brings a reduction in computational cost. Moreover, because the modifications tend to reduce the number of double-angle steps they usually result in a more accurate computed cosine in floating point arithmetic. We also derive an algorithm for computing both cos(A) and sin(A), by adapting the ideas developed for the cosine and intertwining the cosine and sine double angle recurrences