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 -