Abstract
Abstract Let L be the infinite lower triangular Toeplitz matrix with first column (µ, a 1, a 2, ..., ap , a 1, ..., ap , ...) T and let D be the infinite diagonal matrix whose entries are 1, 2, 3, . . . Let A := L + D be the sum of these two matrices. Bünger and Rump have shown that if p = 2 and certain linear inequalities between the parameters µ, a 1, a 2, are satisfied, then the singular values of any finite left upper square submatrix of A can be bounded from below by an expression depending only on those parameters, but not on the matrix size. By extending parts of their reasoning, we show that a similar behaviour should be expected for arbitrary p and a much larger range of values for µ, a 1, ..., ap . It depends on the asymptotics in µ of the l 2-norm of certain sequences defined by linear recurrences, in which these parameters enter. We also consider the relevance of the results in a numerical analysis setting and moreover a few selected numerical experiments are presented in order to show that our bounds are accurate in practical computations.
Highlights
Introduction and preliminaries) is an expression in whose de niton n does not enter and which, is independent of n
Introduction and preliminariesGiven p real numbers a, a, ..., ap, μ, denote for an integer k by k = k mod p the residue modulo p of k.De ne the in nite array A =i,j=,... by aij =μ + i, if i = j, if i < j, a +i−j−, if i > j.The left upper n × n subarrays of A de ne matrices which we denote by A(μ, a, ..., ap, n)
Since the structures studied in this note are encountered in queuing theory, Markov chains, spectral factorizations and the solution of Toeplitz related linear systems, our results can be useful in those areas
Summary
) is an expression in whose de niton n does not enter and which, is independent of n With this they solved a problem posed by Yoshitaka Watanabe from Kyushu University at the Open Problems session of the workshop Numerical Veri cation (NIVEA) 2019 in Hokkaido. Since the structures studied in this note are encountered in queuing theory, Markov chains, spectral factorizations and the solution of Toeplitz related linear systems, our results can be useful in those areas.
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