Inverting block Toeplitz matrices in block Hessenberg form by means of displacement operators: Application to queueing problems

Abstract The concept of displacement rank is used to devise an algorithm for the inversion of an n × n block Toeplitz matrix in block Hessenberg form H n having m × m block entries. This kind of matrices arises in many important problems in queueing theory. We explicitly relate the first and last block rows and block columns of H −1 n with the corresponding ones of H −1 n /2 . These block vectors fully define all the entries of H −1 n by means of a Gohberg-Semencul-like formula. In this way we obtain a doubling algorithm for the computation of H −1 2 i , i = 0, 1,…, q , n = 2 q , where at each stage of the doubling procedure only a few convolutions of block vectors must be computed. The overall cost of this computation is O ( m 2 n log n + m 3 n ) arithmetic operations with a moderate overhead constant. The same technique can be used for solving the linear system H n x = b within the same computational cost. The case where H n is in addition to a scalar Toeplitz matrix is analyzed as well. An application to queueing problems is presented, and comparisons with existing algorithms are performed showing the higher efficiency and reliability of this approach

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