Awad H. Al-Mohy

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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(More)
The scaling and squaring method for the matrix exponential is based on the approximation eA ≈ (rm(2−sA))2s , where rm(x) is the [m/m] Padé approximant to ex and the integers m and s are to be chosen. Several authors have identified a weakness of existing scaling and squaring algorithms termed overscaling, in which a value of s much larger than necessary is(More)
The matrix exponential is a much-studied matrix function having many applications. The Fréchet derivative of the matrix exponential describes the first-order sensitivity of eA to perturbations in A and its norm determines a condition number for eA. Among the numerous methods for computing eA the scaling and squaring method is the most widely used. We show(More)
The need to evaluate a function f (A) ∈ C n×n of a matrix A ∈ C n×n arises in a wide and growing number of applications, ranging from the numerical solution of differential equations to measures of the complexity of networks. We give a survey of numerical methods for evaluating matrix functions, along with a brief treatment of the underlying theory and a(More)
The most popular method for computing the matrix logarithm is the inverse scaling and squaring method, which is the basis of the recent algorithm of [A. H. Al-Mohy and N. J. Higham, Improved inverse scaling and squaring algorithms for the matrix logarithm, SIAM J. Sci. Comput., 34 (2012), pp. C152–C169]. We show that by differentiating the latter algorithm(More)
A popular method for computing the matrix logarithm is the inverse scaling and squaring method, which essentially carries out the steps of the scaling and squaring method for the matrix exponential in reverse order. Here we make several improvements to the method, putting its development on a par with our recent version [SIAM J. Matrix Anal. Appl., 31(More)
We show that the Fréchet derivative of a matrix function f at A in the direction E, where A and E are real matrices, can be approximated by Im f(A + ihE)/h for some suitably small h. This approximation, requiring a single function evaluation at a complex argument, generalizes the complex step approximation known in the scalar case. The approximation is(More)
A new approach for computing an expression of the form $a^{1/2^k}-1$ is presented that avoids the danger of subtractive cancellation in floating point arithmetic, where a is a complex number not belonging to the closed negative real axis and k is a nonnegative integer. We also derive a condition number for the problem. The algorithm therefore allows highly(More)
Several existing algorithms for computing the matrix cosine employ polynomial or rational approximations combined with scaling and use of a double angle formula. Their derivations are based on forward error bounds. We derive new algorithms for computing the matrix cosine, the matrix sine, and both simultaneously, that are backward stable in exact arithmetic(More)