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On Cyclic Reduction Applied to a Class of Toeplitz-Like Matrices Arising in Queueing Problems

  • Dario Bini
  • Beatrice Meini

Abstract

We observe that the cyclic reduction algorithm leaves unchanged the structure of a block Toeplitz matrix in block Hessenberg form. Based on this fact, we devise a fast algorithm for computing the probability invariant vector of stochastic matrices of a wide class of Toeplitz-like matrices arising in queueing problems. We prove that for any block Toeplitz matrix H in block Hessenberg form it is possible to carry out the cyclic reduction algorithm with O(k 3 n + k 2 n log n) arithmetic operations, where k is the size of the blocks and n is the number of blocks in each row and column of H. The probability invariant vector is computed within the same cost. This substantially improves the O(k 3 n 2) arithmetic cost of the known methods based on Gaussian elimination. The algorithm, based on the FFT, is numerically weakly stable. In the case of semi-infinite matrices the cyclic reduction algorithm is rephrased in functional form by means of the concept of generating function and a convergence result is proved.

Keywords

Stochastic Matrice Block Column Cyclic Reduction Triangular Block Block Toeplitz Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • Dario Bini
    • 1
  • Beatrice Meini
    • 1
  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly

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