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The scaling and squaring method for the matrix exponential is based on the approximation e A ≈ (rm(2 −s A)) 2 s , where rm(x) is the [m/m] Padé approximant to e x 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… (More)

A new algorithm is developed for computing e tA B, where A is an n × n matrix and B is n × n 0 with n 0 ≪ n. The algorithm works for any A, its computational cost is dominated by the formation of products of A with n × n 0 matrices, and the only input parameter is a backward error tolerance. The algorithm can return a single matrix e tA B or a sequence e t… (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 e A to perturbations in A and its norm determines a condition number for e A. Among the numerous methods for computing e A the scaling and squaring method is the most widely used. We… (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 of the scaling and squaring… (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)

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. 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 a backward stable algorithm… (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)

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