TY - JOUR

T1 - The fast bisection Eigenvalue method for Hermitian order one Quasiseparable matrices and computations of norms

AU - Eidelman, Yuli

AU - Haimovici, Iulian

N1 - Publisher Copyright:
Copyright © 2015, Kent State University.

PY - 2015

Y1 - 2015

N2 - Since we can evaluate the characteristic polynomial of an N × N order one quasiseparable Hermitian matrix A in less than 10N arithmetical operations by sharpening results and techniques from Eidelman, Gohberg, and Olshevsky [Linear Algebra Appl., 405 (2005), pp. 1-40], we use the Sturm property with bisection to compute all or selected eigenvalues of A. Moreover, linear complexity algorithms are established for computing norms, in particular, the Frobenius norm ∥A∥F and ∥A∥∞, ∥A∥1, and other bounds for the initial interval to be bisected. Upper and lower bounds for eigenvalues are given by the Gershgorin Circle Theorem, and we describe an algorithm with linear complexity to compute them for quasiseparable matrices.

AB - Since we can evaluate the characteristic polynomial of an N × N order one quasiseparable Hermitian matrix A in less than 10N arithmetical operations by sharpening results and techniques from Eidelman, Gohberg, and Olshevsky [Linear Algebra Appl., 405 (2005), pp. 1-40], we use the Sturm property with bisection to compute all or selected eigenvalues of A. Moreover, linear complexity algorithms are established for computing norms, in particular, the Frobenius norm ∥A∥F and ∥A∥∞, ∥A∥1, and other bounds for the initial interval to be bisected. Upper and lower bounds for eigenvalues are given by the Gershgorin Circle Theorem, and we describe an algorithm with linear complexity to compute them for quasiseparable matrices.

KW - Bisection

KW - Eigenvalues

KW - Hermitian

KW - Matrix norm

KW - Quasiseparable

KW - Sturm property

UR - http://www.scopus.com/inward/record.url?scp=84937416508&partnerID=8YFLogxK

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AN - SCOPUS:84937416508

SN - 1068-9613

VL - 44

SP - 342

EP - 366

JO - Electronic Transactions on Numerical Analysis

JF - Electronic Transactions on Numerical Analysis

ER -