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

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.