Parallelizing a Code for Counting and Computing Eigenvalues of Complex Tridiagonal Matrices and Roots of Complex Polynomials

Affiliation(s)

Department of Physics, University of Patras, Patras, Greece.

Department of Mathematics, University of Patras, Patras, Greece.

Department of Physics, University of Patras, Patras, Greece.

Department of Mathematics, University of Patras, Patras, Greece.

ABSTRACT

A code developed recently by the authors, for counting and computing the eigenvalues of a complex tridiagonal matrix, as well as the roots of a complex polynomial, which lie in a given region of the complex plane, is modified to run in parallel on multi-core machines. A basic characteristic of this code (eventually pointing to its parallelization) is that it can proceed with: 1) partitioning the given region into an appropriate number of subregions; 2) counting eigenvalues in each subregion; and 3) computing (already counted) eigenvalues in each subregion. Consequently, theoretically speaking, the whole code in itself parallelizes ideally. We carry out several numerical experiments with random complex tridiagonal matrices, and random complex polynomials as well, in order to study the behaviour of the parallel code, especially the degree of declination from theoretical expectations.

Cite this paper

V. Geroyannis and F. Valvi, "Parallelizing a Code for Counting and Computing Eigenvalues of Complex Tridiagonal Matrices and Roots of Complex Polynomials,"*Applied Mathematics*, Vol. 4 No. 5, 2013, pp. 797-802. doi: 10.4236/am.2013.45109.

V. Geroyannis and F. Valvi, "Parallelizing a Code for Counting and Computing Eigenvalues of Complex Tridiagonal Matrices and Roots of Complex Polynomials,"

References

[1] F. N. Valvi and V. S. Geroyannis, “Counting and Computing the Eigenvalues of a Complex Tridiagonal Matrix, Lying in a Given Region of the Complex Plane,” International Journal of Modern Physics C, Vol. 24, No. 2, 2013, Article ID: 1350008(1-10). doi:10.1142/S0129183113500083

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[6] S. Rombouts and K. Heyde, “An Accurate and Efficient Algorithm for the Computation of the Characteristic Polynomial of a General Square Matrix,” Journal of Computational Physics, Vol. 140, No. 2, 1998, pp. 453-458. doi:10.1006/jcph.1998.5909

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[1] F. N. Valvi and V. S. Geroyannis, “Counting and Computing the Eigenvalues of a Complex Tridiagonal Matrix, Lying in a Given Region of the Complex Plane,” International Journal of Modern Physics C, Vol. 24, No. 2, 2013, Article ID: 1350008(1-10). doi:10.1142/S0129183113500083

[2] W. W. Chen, “Introduction to Complex Analysis,” 2008. http://rutherglen.science.mq.edu.au/wchen/lnicafolder/lnica.html

[3] V. S. Geroyannis and F. N. Valvi, “A Runge-Kutta Fehlberg Code for the Complex Plane: Comparing with Similar Codes by Applying to Polytropic Models,” International Journal of Modern Physics C, Vol. 23, No. 5, 2012, Article ID: 1250038(1-15). doi:10.1142/S0129183112500386

[4] D. H. Bailey, “A Fortran 90-Based Multiprecision System,” ACM Transactions on Mathematical Software, Vol. 21, No. 4, 1995, pp. 379-387. doi:10.1145/212066.212075

[5] D. H. Bailey, “A Fortran-90 Based Multiprecision System,” RNR Technical Report, RNR-94-013, 1995, pp. 1-10.

[6] S. Rombouts and K. Heyde, “An Accurate and Efficient Algorithm for the Computation of the Characteristic Polynomial of a General Square Matrix,” Journal of Computational Physics, Vol. 140, No. 2, 1998, pp. 453-458. doi:10.1006/jcph.1998.5909

[7] J. Skowron and A. Gould, “General Complex Polynomial Root Solver and Its Further Optimization for Binary Microlenses,” arXiv:1203.1034v1 [astro-ph.EP], 2012, pp. 1-29.