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.
Highlights
A code for “counting and computing the eigenvalues of a complex tridiagonal matrix, as well as the roots of a complex polynomial, lying in a given region of the complex plane” (CCE) has been developed by the authors in [1]
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
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
Summary
A code for “counting and computing the eigenvalues of a complex tridiagonal matrix, as well as the roots of a complex polynomial, lying in a given region of the complex plane” (CCE) has been developed by the authors in [1]. We use the Fortran package dcrkf54.f95: a Runge-KuttaFehlberg code of fourth and fifth order modified for the purpose of solving complex IVPs, which are allowed to have high complexity in the definition of their ODEs, along contours prescribed as continuous chains of straightline segments; interested readers can find full details on dcrkf54.f95 in [3]. This package contains the subroutine DCRKF54, used in this study with KIND = 10, i.e. with high precision, and with input parameters as given in [1] (Section 3.1). As discussed in [1] (Sections 3.4, 4), instead of the elements of a tridiagonal matrix, CCE can readily accept the coefficients of a complex polynomial, eventually of very high degree, with several roots of high multiplicity, as well as with closely spaced roots
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