A Parallel Solution of Hermitian Toeplitz Linear Systems,

  • Pedro Alonso
  • Miguel O. Bernabeu
  • Antonio M. Vidal
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3991)


A parallel algorithm for solving complex hermitian Toeplitz linear systems is presented. The parallel algorithm exploits the special structure of Toeplitz matrices to obtain the solution in a quadratic asymptotical cost. Our parallel algorithm transfors the Toeplitz matrix into a Cauchy–like matrix. Working on a Cauchy–like system lets to work with real arithmetic. The parallel algorithm for the solution of a Cauchy–like matrix has a low amount of communication cost regarding other parallel algorithms that work directly on the Toeplitz system. We use a message–passing programming model. The experimental tests are obtained in a cluster of personal computers.


Parallel Algorithm Diagonal Entry Toeplitz Matrix Toeplitz Matrice Parallel Solution 


  1. 1.
    Alonso, P., Badía, J.M., Vidal, A.M.: Parallel algorithms for the solution of toeplitz systems of linear equations. In: Wyrzykowski, R., Dongarra, J., Paprzycki, M., Waśniewski, J. (eds.) PPAM 2004. LNCS, vol. 3019, pp. 969–976. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  2. 2.
    Alonso, P., Badía, J.M., Vidal, A.M.: An efficient and stable parallel solution for non–symmetric Toeplitz linear systems. In: Daydé, M., Dongarra, J., Hernández, V., Palma, J.M.L.M. (eds.) VECPAR 2004. LNCS, vol. 3402, pp. 685–692. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. 3.
    Alonso, P., Vidal, A.M.: An efficient parallel solution of complex toeplitz linear systems. LNCS (2006) (to appear)Google Scholar
  4. 4.
    Alonso, P., Vidal, A.M.: The symmetric–toeplitz linear system problem in parallel. In: Sunderam, V.S., van Albada, G.D., Sloot, P.M.A., Dongarra, J. (eds.) ICCS 2005. LNCS, vol. 3514, pp. 220–228. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  5. 5.
    Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J., Croz, J.D., Greenbaum, A., Hammarling, S., McKenney, A., Ostrouchov, S., Sorensen, D.: LAPACK Users’ Guide, 2nd edn. SIAM, Philadelphia (1995)Google Scholar
  6. 6.
    Blackford, L., et al.: ScaLAPACK Users’ Guide. SIAM, Philadelphia (1997)MATHCrossRefGoogle Scholar
  7. 7.
    Swarztrauber, P.: Vectorizing the FFT’s. Academic Press, New York (1982)Google Scholar
  8. 8.
    Swarztrauber, P.: FFT algorithms for vector computers. Parallel Computing 1, 45–63 (1984)MATHCrossRefGoogle Scholar
  9. 9.
    Loan, C.V.: Computational Frameworks for the Fast Fourier Transform. SIAM Press, Philadelphia (1992)Google Scholar
  10. 10.
    Kailath, T., Sayed, A.H.: Displacement structure: Theory and applications. SIAM Review 37, 297–386 (1995)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bojanczyk, A.W., Heinig, G.: Transformation techniques for toeplitz and toeplitz-plus-hankel matrices part I. transformations. Technical Report 96-250, Cornell Theory Center (1996)Google Scholar
  12. 12.
    Bojanczyk, A.W., Heinig, G.: Transformation techniques for toeplitz and toeplitz-plus-hankel matrices part II. algorithms. Technical Report 96-251, Cornell Theory Center (1996)Google Scholar
  13. 13.
    Gohberg, I., Kailath, T., Olshevsky, V.: Fast Gaussian elimination with partial pivoting for matrices with displacement structure. Mathematics of Computation 64, 1557–1576 (1995)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Thirumalai, S.: High performance algorithms to solve Toeplitz and block Toeplitz systems. Ph.d. thesis, Graduate College of the University of Illinois at Urbana-Champaign (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Pedro Alonso
    • 1
  • Miguel O. Bernabeu
    • 1
  • Antonio M. Vidal
    • 1
  1. 1.Universidad Politécnica de ValenciaValenciaSpain

Personalised recommendations