COUNTING AND COMPUTING THE EIGENVALUES OF A COMPLEX TRIDIAGONAL MATRIX, LYING IN A GIVEN REGION OF THE COMPLEX PLANE

F. N. Valvi () and V. S. Geroyannis ()
Additional contact information
F. N. Valvi: Department of Mathematics, University of Patras, Greece
V. S. Geroyannis: Astronomy Laboratory, Department of Physics, University of Patras, Greece

International Journal of Modern Physics C (IJMPC), 2013, vol. 24, issue 02, 1-10

Abstract: We present a numerical technique for counting and computing the eigenvalues of a complex tridiagonal matrix, lying in a given region of the complex plane. First, we evaluate the integral of the logarithmic derivativep′(λ)/p(λ), wherep(λ)is the characteristic polynomial of the tridiagonal matrix, on a simple closed contour, being the closure of that region. The problem of evaluating this integral is transformed into the equivalent problem of numerically solving a complex initial value problem defined on an ordinary first-order differential equation, integrated along this contour, and solved by the Fortran package dcrkf54.f95 (developed recently by the authors). In accordance with the "argument principle," the value of the contour integral, divided by2πi, counts the eigenvalues lying in the region. Second, a Newton–Raphson (NR) method, fed by random guesses lying in the region, computes the roots one by one. If, however, NR signals that|p′(λ)|converges to zero, i.e. a multiple root probably exists at the current value λ, then counting the roots within an elementary square centered at λ reveals the multiplicity of this root.

Keywords: Complex polynomial; complex tridiagonal matrix; eigenvalues; numerical methods; polynomial root-finding; root-polishing (search for similar items in EconPapers)
Date: 2013
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://www.worldscientific.com/doi/abs/10.1142/S0129183113500083
Access to full text is restricted to subscribers

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:wsi:ijmpcx:v:24:y:2013:i:02:n:s0129183113500083

Ordering information: This journal article can be ordered from

DOI: 10.1142/S0129183113500083

Access Statistics for this article

International Journal of Modern Physics C (IJMPC) is currently edited by H. J. Herrmann

More articles in International Journal of Modern Physics C (IJMPC) from World Scientific Publishing Co. Pte. Ltd.
Bibliographic data for series maintained by Tai Tone Lim ().