A Parllelization of Parlett's Algorithm for Functions of Triangular Matrices

We present a parallelization of Parlett s algorithm for computing arbitrary functions of upper triangular matrices The parallel algorithm preserves the numerical stability properties of the serial algorithm and is suitable for implementation on coarse grain parallel computers The algorithm ob tains a speedup of for matrices of size greater than as our analysis and actual implementation on a processor Meiko CS indicate Computing Matrix Functions Computing functions of square matrices is an important topic in linear algebra engineering and applied mathematics There are several methods for this task Jordan decomposition Schur decom position and approximation methods e g Taylor expansion and rational Pad e approximations The approximation methods may not be suitable for arbitrary functions since speci c properties of the function are exploited The Jordan and Schur decomposition methods are more general in the sense that an arbitrary function of a given square matrix can be computed using these algorithms Let A be an n n matrix with entries from the real or complex eld and f be the function The Jordan decomposition algorithm is used to obtain A MJM and then f A is computed using the formula f A Mf J M However there are some computational di culties with the Jor dan decomposition approach unless A can be diagonalized and has well conditioned eigenvectors The Schur decomposition on the other hand is more stable and can be used for computing arbitrary functions of matrices Let A QTQ be the Schur decomposition of the full matrix A then f A Qf T Q where T is an upper triangular matrix This way the computation of f A for an arbitrary matrix A is reduced to the computation of f T for an upper triangular matrix T Let F f T and fij and tij be the elements in the ith row and jth columns of the upper triangular matrices F and T respectively One approach in computing the entries fij is to obtain explicit expressions for each fij in terms of tij for all possible values of i and j However these expressions become very complicated as we move away from the main diagonal and do not allow cost e ective computation of the entries of F Let i tii It is shown in that fii f i for i n and fij for j i n Furthermore for all i j n we have This work is supported in part by the National Science Foundation under grant ECS and CDA