On the difference of generalized Vandermonde matrices and difference schemes

Analytical methods for finding moments of random Vandermonde matrices with entries on the unit circle are developed. Vandermonde matrices play an important role in signal processing and wireless applications such as direction of arrival estimation, precoding, and sparse sampling theory, just to name a few. Within this framework, we extend classical freeness results on random matrices with independent and identically distributed (i.i.d.) entries and show that Vandermonde structured matrices can be treated in the same vein with different tools. We focus on various types of matrices, such as Vandermonde matrices with and without uniform phase distributions, as well as generalized Vandermonde matrices. In each case, we provide explicit expressions of the moments of the associated Gram matrix, as well as more advanced models involving the Vandermonde matrix. Comparisons with classical i.i.d. random matrix theory are provided, and deconvolution results are discussed. We review some applications of the results to the fields of signal processing and wireless communications.

The values of the determinant of Vandermonde matrices with real elements are analyzed both visually and analytically over the unit sphere in various dimensions. For three dimensions some generalized Vandermonde matrices are analyzed visually. The extreme points of the ordinary Vandermonde determinant on finite-dimensional unit spheres are given as the roots of rescaled Hermite polynomials and a recursion relation is provided for the polynomial coefficients. Analytical expressions for these roots are also given for dimension three to seven. A transformation of the optimization problem is provided and some relations between the ordinary and generalized Vandermonde matrices involving limits are discussed.

This paper examines various statistical distributions in connection with random N x N Vandermonde matrices and their generalization to d-dimensional phase distributions. Upper and lower bound asymptotics for the maximum eigenvalue are found to be O(log N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> ) and O(log N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> / log log N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> ) respectively. The behavior of the minimum eigenvalue is considered by studying the behavior of the maximum eigenvalue of the inverse matrix. In particular, we prove that the minimum eigenvalue λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> is shown to be at most O(exp(-√NW <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> *)) where W <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> * is a positive random variable converging weakly to a random variable constructed from a realization of the Brownian Bridge on [0, 2π). Additional results for (V * V) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> , a trace log formula for V * V, as well as a some numerical examinations of the size of the atom at 0 for the random Vandermonde eigenvalue distribution are also presented.

In this paper we present a concrete algebraic construction of a novel class of convolutional codes. These codes are built upon generalized Vandermonde matrices and therefore can be seen as a natural extension of Reed–Solomon block codes to the context of convolutional codes. For this reason we call them weighted Reed–Solomon (WRS) convolutional codes. We show that under some constraints on the defining parameters these codes are Maximum Distance Profile (MDP), which means that they have the maximal possible growth in their column distance profile. We study the size of the field needed to obtain WRS convolutional codes which are MDP and compare it with the existing general constructions of MDP convolutional codes in the literature, showing that in many cases WRS convolutional codes require significantly smaller fields.

Factorization of finite rank Hankel and Toeplitz matrices

The approach to solving linear systems with structured matrices by means of the bidiagonal factorization of the inverse of the coefficient matrix is first considered in this review article, the starting point being the classical Björck–Pereyra algorithms for Vandermonde systems, published in 1970 and carefully analyzed by Higham in 1987. The work of Higham briefly considered the role of total positivity in obtaining accurate results, which led to the generalization of this approach to totally positive Cauchy, Cauchy–Vandermonde and generalized Vandermonde matrices. Then, the solution of other linear algebra problems (eigenvalue and singular value computation, least squares problems) is addressed, a fundamental tool being the bidiagonal decomposition of the corresponding matrices. This bidiagonal decomposition is related to the theory of Neville elimination, although for achieving high relative accuracy the algorithm of Neville elimination is not used. Numerical experiments showing the good behavior of these algorithms when compared with algorithms that ignore the matrix structure are also included.

New classes of matrix decompositions

Higher-order symplectic integration techniques for molecular dynamics problems

This article presents a new approach to address the resolution of homogeneous linear recurrences of higher order and interpolation problems. By establishing an explicit formula for the entries of the inverse of generalized Vandermonde matrices, a fresh perspective on these mathematical challenges is introduced. The study primarily focuses on linear recurrence relations and thoroughly investigates cases involving characteristic polynomials with both simple roots and roots of multiplicity. To illustrate the effectiveness and practicality of the proposed method, a comprehensive set of illustrative examples is provided, highlighting its applicability in solving a wide range of instances of linear recurrence relations. Additionally, the limitations of the formula are discussed, particularly in scenarios where its applicability may be restricted. The findings of this study contribute significantly to the existing literature, providing an alternative and promising approach for solving problems that rely on the inverse Vandermonde matrix. In conclusion, this article emphasizes the need for further research to explore the computational advantages of the proposed method and to extend its applicability to cases featuring characteristic polynomials with a single root of multiplicity greater than one. By expanding the knowledge in the field, this study offers valuable insights into the resolution of linear recurrences and interpolation problems, presenting a new perspective and expanding the existing knowledge in the field.

The factorization of block matrices with generalized geometric progression rows

In this text we study the regularity of matrices with special polynomial entries. Barring some mild conditions we show that these matrices are regular if a natural limit size is not exceeded. The proof draws connections to generalized Vandermonde matrices and Schur polynomials that are discussed in detail.

We consider a special class of the generalized Vandermonde matrices and obtain an LU factorization for its member by giving closed-form formulae of the entries of $L$ and $U$. Moreover, we express the matrices $L$ and $U$ as products of 1-banded (bidiagonal) matrices. Our result is applied to give the closed-form formula of the inverse of the considered matrix.

A combinatorial approach for computing the determinants of the generalized Vandermonde matrices

“Vandermonde” matrix is a matrix whose (i,j)th entry is in the form of \(x_i^j\). The matrix has a lot of applications in many fields such as signal processing and polynomial interpolations. This paper generalizes the matrix, and let its (i,j) entry be f j (x i ) where f j (x) is a polynomial of x. We present an efficient algorithm to compute the determinant of the generalized Vandermonde matrix. The algorithm is composed of two sub-algorithms: the one that depends on given polynomials f j (x) and the one that does not. The latter algorithm (the one does not depend on f j (x)) can be performed beforehand, and the former (the one that depends on f j (x)) is mainly composed of the computation of determinants of numerical matrices. Determinants of the generalized Vandermonde matrices can be used, for example, to compute the optimal H ∞ and H 2 norm of a system achievable by a static feedback controller (for details, see [18],[19]).KeywordsComputational ComplexityStatic Feedback ControllerMathematical SoftwareVandermonde MatrixNumerical MatrixThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Ill-conditioned systems of linear algebraic equations (SLAEs) and integral equations of the first kind belonging to the class of ill-posed problems are considered. This also includes the problem of inverting the integral Laplace transform, which is used to solve a wide class of mathematical problems. Integral equations are reduced to SLAEs with special matrices. To obtain a reliable solution, regularization methods are used. The general strategy is to use the Tikhonov stabilizer or its modifications, or to represent the desired solution in the form of an orthogonal the sum of two vectors, one of which is determined stably, and to search for the second requires some kind of stabilization procedure. In this article methods for the numerical solution of SLAEs with positive a certain symmetric matrix or with an oscillatory type matrix using regularization, leading to a SLAE with a reduced condition number. A method of reducing the problem of inversion of the integral Laplace transform to a SLAE with generalized Vandermonde matrices of oscillation type, the regularization of which reduces the ill-conditioning of the system, is indicated.