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Comparison theorems for splittings of M-matrices in (block) Hessenberg form

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Abstract

Some variants of the (block) Gauss–Seidel iteration for the solution of linear systems with M-matrices in (block) Hessenberg form are discussed. Comparison results for the asymptotic convergence rate of some regular splittings are derived: in particular, we prove that for a lower-Hessenberg M-matrix \(\rho (P_{GS})\ge \rho (P_S)\ge \rho (P_{AGS})\), where \(P_{GS}, P_S, P_{AGS}\) are the iteration matrices of the Gauss–Seidel, staircase, and anti-Gauss–Seidel method. This is a result that does not seem to follow from classical comparison results, as these splittings are not directly comparable. It is shown that the concept of stair partitioning provides a powerful tool for the design of new variants that are suited for parallel computation.

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References

  1. Amodio, P., Mazzia, F.: A parallel Gauss–Seidel method for block tridiagonal linear systems. SIAM J. Sci. Comput. 16(6), 1451–1461 (1995). https://doi.org/10.1137/0916084

    Article  MathSciNet  MATH  Google Scholar 

  2. Berman, A., Plemmons, R.J.: Nonnegative matrices in the mathematical sciences. In: Classics in Applied Mathematics, vol. 9. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1994). https://doi.org/10.1137/1.9781611971262. Revised reprint of the 1979 original

  3. Bini, D., Meini, B.: On cyclic reduction applied to a class of Toeplitz-like matrices arising in queueing problems. In: Stewart, W.J. (ed.) Computations with Markov Chains. Springer, Boston (1995)

    Google Scholar 

  4. Csordas, G., Varga, R.S.: Comparisons of regular splittings of matrices. Numer. Math. 44(1), 23–35 (1984). https://doi.org/10.1007/BF01389752

    Article  MathSciNet  MATH  Google Scholar 

  5. Dongarra, J., Cleary, A.: Implementation in scalapack of divide-and-conquer algorithms for banded and tridiagonal linear systems. Center for Research on Parallel Computation (CRPC), Rice University, Tech. rep. (1997)

  6. Dudin, S., Dudin, A., Kostyukova, O., Dudina, O.: Effective algorithm for computation of the stationary distribution of multi-dimensional level-dependent Markov chains with upper block-Hessenberg structure of the generator. J. Comput. Appl. Math. 366, 112 (2020). https://doi.org/10.1016/j.cam.2019.112425

    Article  MathSciNet  MATH  Google Scholar 

  7. Forsythe, G.E.: Solving linear algebraic equations can be interesting. Bull. Am. Math. Soc. 59, 299–329 (1953). https://doi.org/10.1090/S0002-9904-1953-09718-X

    Article  MathSciNet  MATH  Google Scholar 

  8. Gemignani, L., Lotti, G.: Efficient and stable solution of M-matrix linear systems of (block) Hessenberg form. SIAM J. Matrix Anal. Appl. 24(3), 852–876 (2003). https://doi.org/10.1137/S0895479801387085

    Article  MathSciNet  MATH  Google Scholar 

  9. Grinstead, C.M., Snell, J.L.: Introduction to Probability. AMS (2003). http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html

  10. Kahan, W.M.: Gauss–Seidel methods of solving large systems of linear equations. Ph.D. thesis (1958)

  11. Kalauch, A., Lavanya, S., Sivakumar, K.C.: Singular irreducible \(M\)-matrices revisited. Linear Algebra Appl. 565, 47–64 (2019). https://doi.org/10.1016/j.laa.2018.11.030

    Article  MathSciNet  MATH  Google Scholar 

  12. Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM Series on Statistics and Applied Probability. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; American Statistical Association, Alexandria, VA (1999). https://doi.org/10.1137/1.9780898719734

  13. Li, H.B., Huang, T.Z., Zhang, Y., Liu, X.P., Li, H.: On some new approximate factorization methods for block tridiagonal matrices suitable for vector and parallel processors. Math. Comput. Simul. 79(7), 2135–2147 (2009). https://doi.org/10.1016/j.matcom.2008.09.009

    Article  MathSciNet  MATH  Google Scholar 

  14. Lu, H.: Stair matrices and their generalizations with applications to iterative methods. I. A generalization of the successive overrelaxation method. SIAM J. Numer. Anal 37(1), 1–17 (1999). https://doi.org/10.1137/S0036142998343294

    Article  MathSciNet  MATH  Google Scholar 

  15. Meier-Hellstern, K.: The analysis of a queue arising in overflow models. IEEE Trans. Commun. 37(4), 367–372 (1989). https://doi.org/10.1109/26.20117

    Article  MathSciNet  Google Scholar 

  16. Meurant, G.: Domain decomposition preconditioners for the conjugate gradient method. Calcolo 25(1–2), 103–119 (1989). https://doi.org/10.1007/BF02575749

    Article  MathSciNet  MATH  Google Scholar 

  17. Nelson, R.: Probability, Stochastic Processes, and Queueing Theory. The Mathematics of Computer Performance Modeling, Springer, New York (1995). https://doi.org/10.1007/978-1-4757-2426-4

    Book  MATH  Google Scholar 

  18. O’Leary, D.P.: Iterative methods for finding the stationary vector for Markov chains. In: Linear Algebra, Markov Chains, and Queueing Models (Minneapolis, MN, 1992), IMA Vol. Math. Appl., vol. 48, pp. 125–136. Springer, New York (1993). https://doi.org/10.1007/978-1-4613-8351-2_9

  19. Ortega, J.M., Voigt, R.G.: Solution of partial differential equations on vector and parallel computers. SIAM Rev. 27(2), 149–240 (1985). https://doi.org/10.1137/1027055

    Article  MathSciNet  MATH  Google Scholar 

  20. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2003). https://doi.org/10.1137/1.9780898718003

    Book  MATH  Google Scholar 

  21. Shang, Y.: A distributed memory parallel Gauss–Seidel algorithm for linear algebraic systems. Comput. Math. Appl. 57(8), 1369–1376 (2009). https://doi.org/10.1016/j.camwa.2009.01.034

    Article  MathSciNet  MATH  Google Scholar 

  22. Stewart, G.W.: On the solution of block Hessenberg systems. Numer. Linear Algebra Appl. 2(3), 287–296 (1995). https://doi.org/10.1002/nla.1680020309

    Article  MathSciNet  MATH  Google Scholar 

  23. Varga, R.S.: Matrix Iterative Analysis. Springer Series in Computational Mathematics, vol. 27, expanded Springer, Berlin (2000). https://doi.org/10.1007/978-3-642-05156-2

    Book  MATH  Google Scholar 

  24. Wallin, D., Löf, H., Hagersten, E., Holmgren, S.: Multigrid and Gauss–Seidel smoothers revisited: parallelization on chip multiprocessors. In: Egan, G.K., Muraoka, Y. (eds.) Proceedings of the 20th Annual International Conference on Supercomputing, ICS 2006, Cairns, Queensland, Australia, pp. 145–155. ACM (2006). https://doi.org/10.1145/1183401.1183423

  25. Woźnicki, Z.I.: Matrix splitting principles. Int. J. Math. Math. Sci. 28(5), 251–284 (2001). https://doi.org/10.1155/S0161171201007062

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Informatica, Università di Pisa, Pisa, Italy

    Luca Gemignani & Federico Poloni

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  1. Luca Gemignani
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  2. Federico Poloni
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Correspondence to Federico Poloni.

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Communicated by Daniel Kressner.

The authors are partially supported by INDAM/GNCS and by the project PRA_2020_61 of the University of Pisa.

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Gemignani, L., Poloni, F. Comparison theorems for splittings of M-matrices in (block) Hessenberg form. Bit Numer Math 62, 849–867 (2022). https://doi.org/10.1007/s10543-021-00899-4

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