Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators

Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators

A new algorithm is developed for computing etAB, where A is an n× n matrix and B is n×n0 with n0 ≪ n. The algorithm works for any A, its computational cost is dominated by the formation of products of A with n× n0 matrices, and the only input parameter is a backward error tolerance. The algorithm can return a single matrix etAB or a sequence etkAB on an equally spaced grid of points tk. It uses...

Computing a Matrix Function for Exponential Integrators

An efficient numerical method is developed for evaluating φ(A), where A is a symmetric matrix and φ is the function defined by φ(x) = (ex − 1)/x = 1+ x/2 + x2/6+ .... This matrix function is useful in the so-called exponential integrators for differential equations. In particular, it is related to the exact solution of the ODE system dy/dt = Ay + b, where A and b are t-independent. Our method a...

Computing analytic matrix functions for a class of exponential integrators

Different methods for numerically stable computation of functions which are closely related to the exponential function are discussed. Such functions appear in the format of the most often used exponential integrators. A generalization of the method base on the tridiagonal reduction is proposed. The new approach, allows to compute all the functions included in the format of a general exponentia...

The Matrix Unwinding Function, with an Application to Computing the Matrix Exponential

A new matrix function corresponding to the scalar unwinding number of Corless, Hare, and Jeffrey is introduced. This matrix unwinding function, U , is shown to be a valuable tool for deriving identities involving the matrix logarithm and fractional matrix powers, revealing, for example, the precise relation between logAα and α logA. The unwinding function is also shown to be closely connected w...

Computing a Matrix Fun tion for Exponential Integrators