The star-matrix models are difficult to solve due to the multiple powers of the Vandermonde determinants in the partition function. We apply to these models a modified Q-matrix approach and we get results consistent with those obtained by other methods.As examples we study the inhomogenous gaussian model on Bethe tree and matrix q-Potts-like model. For the last model in the special cases q = 2 ...

This paper presents Levinson-type algorithms for (i) polynomial fitting, (ii) obtaining a Q decomposition of Vandermonde matrices and a Cholesky factorization of Hankel matrices and (iii) obtaining the inverse of Hankel matrices. The algorithm for the least-squares solution of Hankel systems of equations, requires 3n + 9n+ 3 multiply and divide operation (MD0). The algorithm for obtaining an or...

Abstract We review several determinant and Pfaffian identities, which generalize the evaluation formulae of Cauchy’s determinant det (1/(xi + yj)) and Schur’s Pfaffian Pf ((xj − xi)/(xj + xi)). As a multi-variable generalization, we consider Cauchytype determinants and Schur-type Pfaffians of matrices with entries involving some generalized Vandermonde determinants. Also we give an elliptic gen...

Let K,M ∈ N with K < M , and define a square K × K Vandermonde matrix A = A τ,−→n with nodes on the unit circle: Ap,q = exp (−j2πpnqτ/K) ; p, q = 0, 1, ...,K − 1, where nq ∈ {0, 1, ...,M − 1} and n0 < n1 < .... < nK−1. Such matrices arise in some types of interpolation problems. In this paper, necessary and sufficient conditions are presented on the vector −→n so that a value of τ ∈ R can be fo...

Let K,M ∈ N with K < M , and define a square K × K Vandermonde matrix A = A τ,−→n with nodes on the unit circle: Ap,q = exp (−j2πpnqτ/K) ; p, q = 0, 1, ...,K − 1, where nq ∈ {0, 1, ...,M − 1} and n0 < n1 < .... < nK−1. Such matrices arise in some types of interpolation problems. In this paper, necessary and sufficient conditions are presented on the vector −→n so that a value of τ ∈ R can be fo...

We construct the conditional version of k independent and identically distributed random walks on R given that they stay in strict order at all times. This is a generalisation of so-called non-colliding or non-intersecting random walks, the discrete variant of Dyson’s Brownian motions, which have been considered yet only for nearest-neighbor walks on the lattice. Our only assumptions are moment...

We construct the conditional version of k independent and identically distributed random walks on R given that they stay in strict order at all times. This is a generalisation of so-called non-colliding or non-intersecting random walks, the discrete variant of Dyson’s Brownian motions, which have been considered yet only for nearest-neighbor walks on the lattice. Our only assumptions are moment...

||||||||||||||||||||||||||||||||||||||| ABSTRACT In this paper we introduce a new class of what we shall call polynomial Vandermonde-like matrices. This class generalizes the polynomial Vandermonde matrices studied earlier by various authors, who derived explicit inversion formulas and fast algorithms for inversion and for solving the associated linear systems. A displacement structure approach...

A general scheme is proposed for computing the QR-factorization of certain displacement structured matrices, including Cauchy-like, Vandermonde-like, Toeplitz-like and Hankel-like matrices, hereby extending some earlier work for the QR-factorization of the Cauchy matrix. The algorithm employs a chasing scheme for the recursive construction of a diagonal plus semiseparable matrix of semiseparabi...