A Fast Björck–Pereyra-Type Algorithm for Solving Hessenberg-Quasiseparable-Vandermonde Systems

Abstract

In this paper we derive a fast O(n 2) algorithm for solving linear systems where the coefficient matrix is a polynomial-Vandermonde matrix V R (x) = [r j−1 (x i)] with polynomials {r k (x)} related to a Hessenberg quasiseparable matrix. The result generalizes the well-known Björck-Pereyra algorithm for classical Vandermonde systems involving monomials. It also generalizes the algorithms of [RO91] for V R (x) involving Chebyshev polynomials, of [H90] for V R (x) involving real orthogonal polynomials, and of [BEGKO07] for V R (x) involving Szegö polynomials. The new algorithm applies to a fairly general class of H-k-q.s.-polynomials (Hessenberg order k quasiseparable) that includes (along with the above mentioned classes of real orthogonal and Szegö polynomials) several other important classes of polynomials. Preliminary numerical experiments are presented comparing the algorithm to standard structure-ignoring methods.

A Fast Algorithm for the Inversion of Quasiseparable Vandermonde-like Matrices

“…inversion inversion system formula algorithm solver Classical-V monomials P [25], Tr [28], GO [16] P [25], Tr [28] BP [9] Chebychev-V Chebychev poly GO [14] GO [14] RO [26] Three-Term-V Real orthogonal poly Vs [29], GO [14] CR [10] Hi [27] Szegö-V Szegö polynomial O [23] O [24] BEGKO [1] Quasiseparable Quasiseparable BEGOT [4] BEGOT [6], BEGKO [2] Vandermonde polynomial BEGOT [5] BEGOTZ [7] Table 1: Fast O(n 2 ) inversion for polynomial-Vandermonde matrices.…”

Computations with quasiseparable polynomials and matrices

“…Specifically, since quasiseparable matrices are of the ''recurrence type'', it is of interest to study their so-called quasiseparable polynomials. 3 The point is that historically, algorithm development is more advanced for real orthogonal polynomials. Recently, several important algorithms originally derived for real orthogonal polynomials have been carried over to the class of Szegö polynomials (however, such new algorithms tend to be valid only for the Szegö polynomials; they are analogues and not generalizations of the original algorithms).…”

“…A new Björck-Pereyra-type algorithm, obtained recently in [3,1], uses the superclass of quasiseparable polynomials and hence it is a generalization of all of previous work listed in Table 6. The new algorithm is based on the following theorem.…”

“…For L k it follows immediately from its sparsity. For U k the linear cost of matrix-vector multiplication is possible thanks to the quasiseparable structure of its block A (n−k)×(n−k) , see, e.g., [3,1] for details.…”

Numerical conditioning

“…The first such algorithm is an O(n 2 ) operation algorithm for solving V n x = b of Björck and Pereyra [3], whose error analysis was given by Higham [21]. There is now a long list of generalizations of this algorithm in various directions, of which we mention just Demmel and Koev [5] and Bella et al [2]; various other algorithms up to 2002 are described or cited in the chapter "Vandermonde systems" in [22].…”