Related papers: Confluent Vandermonde matrices, divided difference…
In this note, we explore the connections between the confluent Vandermonde matrix over an arbitrary field and several mathematical topics, including interpolation polynomials, Hasse derivatives, LU factorization, companion matrices and…
The inverse of the Vandermonde and confluent Vandermonde matrices are presented. In the case of the Vandermonde matrix, we present a decomposition in three factors, one of them a diagonal matrix. The evaluation of such inverse matrices is a…
We give explicit formulas as well as a quadratic time algorithm to solve (so called) generalized Vandermonde's systems of p linear equations and n variables. It allows in particular to find all (so called Lagrange's) interpolation polynoms…
In this study, we introduce the concept of commutative quaternions and commutative quaternion matrices. Firstly, we give some properties of commutative quaternions and their Hamilton matrices. After that we investigate commutative…
In this paper, we present a novel method to compute an explicit formula for the inverse of the confluent Vandermonde matrices. Our proposed results may have many interesting perspectives in diverse areas of mathematics and natural sciences,…
Different types of convolution operations involving large Vandermonde matrices are considered. The convolutions parallel those of large Gaussian matrices and additive and multiplicative free convolution. First additive and multiplicative…
A new class of structured matrices is presented and a closed form formula for their determinant is established. This formula has strong connections with the one for Vandermonde matrices.
This paper proposes a novel matrix rank-one decomposition for quaternion Hermitian matrices, which admits a stronger property than the previous results in (sturm2003cones,huang2007complex,ai2011new). The enhanced property can be used to…
We present a method to derive new explicit expressions for bidiagonal decompositions of Vandermonde and related matrices such as the (q-, h-) Bernstein-Vandermonde ones, among others. These results generalize the existing expressions for…
This paper considers the extension of classical Lagrange interpolation in one real or complex variable to "polynomials of one quaternionic variable". To do this we develop some aspects of the theory of such polynomials. We then give a…
Vandermonde matrices are well known. They have a number of interesting properties and play a role in (Lagrange) interpolation problems, partial fraction expansions, and finding solutions to linear ordinary differential equations, to mention…
The second author recently suggested to identify the generating matrices of a digital $(t,m,s)$-net over the finite field $F_q$ with an $s \times m$ matrix $C$ over $F_{q^m}$. More exactly, the entries of $C$ are determined by interpreting…
Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. In this paper, an effective algorithm is presented for computing the…
Inverse Vandermonde matrix calculation is a long-standing problem to solve nonsingular linear system $Vc=b$ where the rows of a square matrix $V$ are constructed by progression of the power polynomials. It has many applications in…
In multicentric representation of piecewise holomorphic functions one combines Lagrange interpolation at roots of a polynomial $p$ with convergent power series of $p$ as the "coefficients" multiplying the Lagrange basis polynomials. When…
The Hermite interpolation formulas are based on the interpretation of interpolation nodes as roots of suitable polynomials. Therefore, such formulas belong to the class of algebraic interpolations. The article considers a multidimensional…
The theory of the column-row determinants has been considered for matrices over a non-split quaternion algebra. In this paper the concepts of column-row determinants are extending to a split quaternion algebra. New definitions of the column…
We study matrix forms of quaternionic versions of the Fourier Transform and Convolution operations. Quaternions offer a powerful representation unit, however they are related to difficulties in their use that stem foremost from…
We collect here elementary properties of differentiation matrices for univariate polynomials expressed in various bases, including orthogonal polynomial bases and non-degree-graded bases such as Bernstein bases and Lagrange \& Hermite…
In this paper, we give several matrix representations for the Horadam quaternions. We derive several identities related to these quaternions by using the matrix method. Since quaternion multiplication is not commutative, some of our results…