TY - JOUR

T1 - THE BISECTION EIGENVALUE METHOD FOR UNITARY HESSENBERG MATRICES VIA THEIR QUASISEPARABLE STRUCTURE_

AU - Eidelman, Yuli

AU - Haimovici, Iulian

N1 - Publisher Copyright:
© 2023, Kent State University.

PY - 2023

Y1 - 2023

N2 - If N0 is a normal matrix, then the Hermitian matrices 1 2 (N0 +N∗ 0 ) and i 2 (N∗ 0 -N0) have the same eigenvectors as N0. Their eigenvalues are the real part and the imaginary part of the eigenvalues of N0, respectively. If N0 is unitary, then only the real part of each of its eigenvalues and the sign of the imaginary part is needed to completely determine the eigenvalue, since the sum of the squares of these two parts is known to be equal to 1. Since a unitary upper Hessenberg matrix U has a quasiseparable structure of order one and we express the matrix A = 1 2 (U + U∗) as quasiseparable matrix of order two , we can find the real part of the eigenvalues and, when needed, a corresponding eigenvector x, by using techniques that have been established in the paper by Eidelman and Haimovici [Oper. Theory Adv. Appl., 271 (2018), pp. 181-200]. We describe here a fast procedure, which takes only 1:7% of the bisection method time, to find the sign of the imaginary part. For instance, in the worst case only, we build one row of the quasiseparable matrix U and multiply it by a known eigenvector of A, as the main part of the procedure. This case occurs for our algorithm when among the 4 numbers ±cos t ± i sin t there are exactly 2 eigenvalues and they are opposite, so that we have to distinguish between the case λ;-λ and the case λ;-λ. The performance of the developed algorithm is illustrated by a series of numerical tests. The algorithm is more accurate and many times faster (when executed in Matlab) than for general Hermitian matrices of quasiseparable order two, because the action of the quasiseparable generators, which are small matrices in the previous cited paper, can be replaced by scalars, most of them real numbers.

AB - If N0 is a normal matrix, then the Hermitian matrices 1 2 (N0 +N∗ 0 ) and i 2 (N∗ 0 -N0) have the same eigenvectors as N0. Their eigenvalues are the real part and the imaginary part of the eigenvalues of N0, respectively. If N0 is unitary, then only the real part of each of its eigenvalues and the sign of the imaginary part is needed to completely determine the eigenvalue, since the sum of the squares of these two parts is known to be equal to 1. Since a unitary upper Hessenberg matrix U has a quasiseparable structure of order one and we express the matrix A = 1 2 (U + U∗) as quasiseparable matrix of order two , we can find the real part of the eigenvalues and, when needed, a corresponding eigenvector x, by using techniques that have been established in the paper by Eidelman and Haimovici [Oper. Theory Adv. Appl., 271 (2018), pp. 181-200]. We describe here a fast procedure, which takes only 1:7% of the bisection method time, to find the sign of the imaginary part. For instance, in the worst case only, we build one row of the quasiseparable matrix U and multiply it by a known eigenvector of A, as the main part of the procedure. This case occurs for our algorithm when among the 4 numbers ±cos t ± i sin t there are exactly 2 eigenvalues and they are opposite, so that we have to distinguish between the case λ;-λ and the case λ;-λ. The performance of the developed algorithm is illustrated by a series of numerical tests. The algorithm is more accurate and many times faster (when executed in Matlab) than for general Hermitian matrices of quasiseparable order two, because the action of the quasiseparable generators, which are small matrices in the previous cited paper, can be replaced by scalars, most of them real numbers.

KW - Sturm property

KW - bisection

KW - eigenstructure

KW - quasiseparable

KW - unitary Hessenberg

UR - http://www.scopus.com/inward/record.url?scp=85162006744&partnerID=8YFLogxK

U2 - 10.1553/etna_vol59s60

DO - 10.1553/etna_vol59s60

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AN - SCOPUS:85162006744

SN - 1068-9613

VL - 2023

SP - 60

EP - 88

JO - Electronic Transactions on Numerical Analysis

JF - Electronic Transactions on Numerical Analysis

IS - 59

ER -