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

Abstract 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 Szego 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 Szego polynomials.

[1]  F. Wubs.  Notes on numerical fluid mechanics , 1985 .

[2]  P. Dewilde,et al.  Time-Varying Systems and Computations , 1998 .

[3]  Israel Koltracht,et al.  Linear complexity algorithm for semiseparable matrices , 1985 .

[4]  I. Gohberg,et al.  A modification of the Dewilde-van der Veen method for inversion of finite structured matrices , 2002 .

[5]  Marc Van Barel,et al.  Divide-and-conquer algorithms for computing the eigendecomposition of diagonal-plus-semiseparable matrices , 2002 .

[6]  S. Chandrasekaran,et al.  Fast and stable eigendecomposition of symmetric banded plus semi-separable matrices , 2000 .

[7]  Arnold Neumaier,et al.  Introduction to Numerical Analysis , 2001 .

[8]  Marc Van Barel,et al.  Two fast algorithms for solving diagonal-plus-semiseparable linear systems , 2004 .

[9]  I. Gohberg,et al.  Linear complexity inversion algorithms for a class of structured matrices , 1999 .

[10]  William B. Gragg,et al.  The QR algorithm for unitary Hessenberg matrices , 1986 .

[11]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[12]  Shivkumar Chandrasekaran,et al.  A divide-and-conquer algorithm for the eigendecomposition of symmetric block-diagonal plus semiseparable matrices , 2004, Numerische Mathematik.

[13]  Raf Vandebril,et al.  A note on the representation and definition of semiseparable matrices , 2005, Numer. Linear Algebra Appl..

[14]  F. V. Atkinson,et al.  Discrete and Continuous Boundary Problems , 1964 .

[15]  F. Gantmacher,et al.  Oscillation matrices and kernels and small vibrations of mechanical systems , 1961 .

[16]  Eugene E. Tyrtyshnikov.  Mosaic Ranks for Weakly Semiseparable Matrices , 1999, LSSC.

[17]  D. Fasino,et al.  Fast and stable algorithms for reducing diagonal plus semiseparable matrices to tridiagonal and bidiagonal form , 2001 .

[18]  N. Mastronardi,et al.  Fast and Stable Reduction of Diagonal Plus Semi-Separable Matrices to Tridiagonal and Bidiagonal Form , 2001 .

[19]  I. Gohberg,et al.  On a new class of structured matrices , 1999 .

[20]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[21]  Dario Fasino,et al.  Direct and Inverse Eigenvalue Problems for Diagonal-Plus-Semiseparable Matrices , 2003, Numerical Algorithms.

[22]  Victor Y. Pan,et al.  TR-2003013: QR-Like Algorithms for Generalized Semiseparable Matrices , 2003 .

[23]  Thomas Kailath,et al.  Linear complexity algorithms for semiseparable matrices , 1985 .

[24]  Israel Gohberg,et al.  Fast inversion algorithms for diagonal plus semiseparable matrices , 1997 .