TY - JOUR

T1 - Eigenstructure of order-one-quasiseparable matrices. Three-term and two-term recurrence relations

AU - Eidelman, Y.

AU - Gohberg, I.

AU - Olshevsky, Vadim

N1 - Funding Information:
∗ Corresponding author. E-mail addresses: eideyu@post.tau.ac.il (Y. Eidelman), gohberg@post.tau.ac.il (I. Gohberg), olshevsky@math.uconn.edu (V. Olshevsky). 1 Supported by the NSF grant 0242518.

PY - 2005/8/1

Y1 - 2005/8/1

N2 - This paper presents explicit formulas and algorithms to compute the eigenvalues and eigenvectors of order-one-quasiseparable matrices. Various recursive relations for characteristic polynomials of their principal submatrices are derived. The cost of evaluating the characteristic polynomial of an N × N matrix and its derivative is only O(N). This leads immediately to several versions of a fast quasiseparable Newton iteration algorithm. In the Hermitian case we extend the Sturm property to the characteristic polynomials of order-one-quasiseparable matrices which yields to several versions of a fast quasiseparable bisection algorithm. Conditions guaranteeing that an eigenvalue of a order-one-quasiseparable matrix is simple are obtained, and an explicit formula for the corresponding eigenvector is derived. The method is further extended to the case when these conditions are not fulfilled. Several particular examples with tridiagonal, (almost) unitary Hessenberg, and Toeplitz matrices are considered. The algorithms are based on new three-term and two-term recurrence relations for the characteristic polynomials of principal submatrices of order-one-quasiseparable matrices R. It turns out that the latter new class of polynomials generalizes and includes two classical families: (i) polynomials orthogonal on the real line (that play a crucial role in a number of classical algorithms in numerical linear algebra), and (ii) the Szegö polynomials (that play a significant role in signal processing). Moreover, new formulas can be seen as generalizations of the classical three-term recurrence relations for the real orthogonal polynomials and of the two-term recurrence relations for the Szegö polynomials.

AB - This paper presents explicit formulas and algorithms to compute the eigenvalues and eigenvectors of order-one-quasiseparable matrices. Various recursive relations for characteristic polynomials of their principal submatrices are derived. The cost of evaluating the characteristic polynomial of an N × N matrix and its derivative is only O(N). This leads immediately to several versions of a fast quasiseparable Newton iteration algorithm. In the Hermitian case we extend the Sturm property to the characteristic polynomials of order-one-quasiseparable matrices which yields to several versions of a fast quasiseparable bisection algorithm. Conditions guaranteeing that an eigenvalue of a order-one-quasiseparable matrix is simple are obtained, and an explicit formula for the corresponding eigenvector is derived. The method is further extended to the case when these conditions are not fulfilled. Several particular examples with tridiagonal, (almost) unitary Hessenberg, and Toeplitz matrices are considered. The algorithms are based on new three-term and two-term recurrence relations for the characteristic polynomials of principal submatrices of order-one-quasiseparable matrices R. It turns out that the latter new class of polynomials generalizes and includes two classical families: (i) polynomials orthogonal on the real line (that play a crucial role in a number of classical algorithms in numerical linear algebra), and (ii) the Szegö polynomials (that play a significant role in signal processing). Moreover, new formulas can be seen as generalizations of the classical three-term recurrence relations for the real orthogonal polynomials and of the two-term recurrence relations for the Szegö polynomials.

KW - Eigenvalue problem

KW - Orthogonal polynomials

KW - Quasiseparable matrices

KW - Recurrence relations

KW - Schur algorithm

KW - Schur-Cohn recursions

KW - Semiseparable matrices

KW - Szego polynomials

KW - Tridiagonal matrices

KW - Unitary hessenberg matrices

UR - http://www.scopus.com/inward/record.url?scp=21244440740&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2005.02.039

DO - 10.1016/j.laa.2005.02.039

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AN - SCOPUS:21244440740

SN - 0024-3795

VL - 405

SP - 1

EP - 40

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 1-3

ER -