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A real matrix A = ( a i j ) 1 ≤ i , j , ≤ n is said to be almost strictly totally negative if it is almost strictly sign regular with signature ε = ( − 1 , − 1 , … , − 1 ) , which is equivalent to the property that all its nontrivial minors are negative. In this paper an algorithmic characterization of nonsingular almost strictly totally negative matrices is presented. [ABSTRACT FROM AUTHOR]

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]

Abstract: In this paper, we first give the existence of and the general expression for the solution to an inverse eigenproblem defined as follows: given a set of real -vectors and a set of real numbers , and an -by- real generalized reflexive matrix (or generalized anti-reflexive matrix ) such that and are the eigenvectors and eigenvalues of (or ), respectively, we solve the best approximation problem for the inverse eigenproblem. That is, given an arbitrary real -by- matrix , we find a matrix which is the solution to the inverse eigenproblem such that the distance between and is minimized in the Frobenius norm. We give an explicit solution and a numerical algorithm for the best approximation problem over generalized reflexive (or generalized anti-reflexive) matrices. Two numerical examples are also presented to show that our method is effective. [ABSTRACT FROM AUTHOR]

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]