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Abstract: In this paper a new method for computing the action of the matrix exponential on a vector , where is a complex matrix and is a positive real number, is proposed. Our approach is based on vector valued rational approximation where the approximants are determined by the denominator polynomials whose coefficients are obtained by solving an inexpensive linear least-squares problem. No matrix multiplications or divisions but matrix-vector products are required in the whole process. A technique of scaling and recurrence enables our method to be more effective when the problem is for fixed and many values of . We also give a backward error analysis in exact arithmetic for the truncation errors to derive our new algorithm. Preliminary numerical results illustrate that the new algorithm performs well. [Copyright &y& Elsevier]

Let x 0 , x 1 , ... , x n , be a set of n+1 distinct real numbers (i.e., x i ≠ x j , for i ≠ j) and y m , k , for m = 0 , 1 , ... , n , and k = 0 , 1 , ... , n m , with n m ∈ N , be given of real numbers, we know that there exists a unique polynomial p N − 1 of degree N − 1 where N = ∑ i = 0 n (n i + 1) , such that p N − 1 (k) (x m) = y m , k , for m = 0 , 1 , ... , n and k = 0 , 1 , ... , n m . p N − 1 is the Hermite interpolation polynomial for the set { (x m , y m , k) , m = 0 , 1 , ... , n , k = 0 , 1 , ... , n m }. The polynomial p N − 1 can be computed by using the Lagrange generalized polynomials. Recently Messaoudi et al. (2018) presented a new algorithm for computing the Hermite interpolation polynomials, for a general case, called Generalized Recursive Polynomial Interpolation Algorithm (GRPIA), this algorithm has been developed without using the Matrix Recursive Interpolation Algorithm (Jbilou and Messaoudi, 1999). Messaoudi et al. (2017) presented also a new algorithm called Matrix Recursive Polynomial Interpolation Algorithm (MRPIA), for a particular case where n m = μ = 1 , for m = 0 , 1 , ... , n. In this paper we will give the version of the MRPIA for a particular case n m = μ ≥ 0 , for m = 0 , 1 , ... , n. We will recall the result of the existence of the polynomial p N − 1 for this case, some of its properties will also be given. Using the MRPIA, a method will be proposed for the general case, where n m , for some m , are different and some examples will also be given. [ABSTRACT FROM AUTHOR]