Fast Inversion of Polynomial-Vandermonde Matrices for Polynomial Systems Related to Order One Quasiseparable Matrices

Bella, T.; Eidelman, Yuli; Olshevsky, Vadim; Тыртышников, Е. Е.

doi:10.1007/978-3-0348-0639-8_8

Cited by 3 publications

(3 citation statements)

References 20 publications

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“…Comparing this with the recurrence relations (5), one can see that matrices M Q and N Q are defined by (28).…”

Section: Quasiseparable Polynomialsmentioning

confidence: 81%

“…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.…”

Section: Vandermondementioning

confidence: 99%

“…Note that the matrix on the right-hand side of equation ( 7) is of rank one. Thus when the system of polynomials {Q} satisfies the recurrence relations (5), the class of polynomial Vandermonde matrices can be generalized by allowing the displacement rank to be one. We say that matrix R is Vandermonde-like if it satisfies the displacement equation:…”

Section: Displacement Operator Based On Recurrence Relationsmentioning

confidence: 99%

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

Perera¹,

Bonik²,

Olshevsky³

2014

Preprint

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The results on Vandermonde-like matrices were introduced as a generalization of polynomial Vandermonde matrices, and the displacement structure of these matrices was used to derive an inversion formula. In this paper we first present a fast Gaussian elimination algorithm for the polynomial Vandermonde-like matrices. Later we use the said algorithm to derive fast inversion algorithms for quasiseparable, semiseparable and well-free Vandermonde-like matrices having O(n 2 ) complexity. To do so we identify structures of displacement operators in terms of generators and the recurrence relations(2-term and 3-term) between the columns of the basis transformation matrices for quasiseparable, semiseparable and well-free polynomials. Finally we present an O(n 2 ) algorithm to compute the inversion of quasiseparable Vandermonde-like matrices.

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“…Comparing this with the recurrence relations (5), one can see that matrices M Q and N Q are defined by (28).…”

Section: Quasiseparable Polynomialsmentioning

confidence: 81%