A few remarks on "On certain Vandermonde determinants whose variables separate"

In the recent paper \u201cOn certain Vandermonde determinants whose variables separate\u201d [Linear Algebra and its Applications 449 (2014) pp. 17\u201327], there was established a factorized formula for some bivariate Vandermonde determinants (associated to almost square grids) whose basis functions are formed by Hadamard products of some univariate polynomials. That formula was crucial for proving a conjecture on the Vandermonde determinant associated to Padua-like points. In this note we show that the same formula holds when those polynomials are replaced by arbitrary functions and we extend this formula to general rectangular grids. We also show that the Vandermonde determinants associated to Padua-like points are nonvanishing

Accurate and Efficient Expression Evaluation and Linear Algebra

We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By "accurate" we mean that the computed answer has relative error less than 1, i.e., has some correct leading digits. We also address efficiency, by which we mean algorithms that run in polynomial time in the size of the input. Our results will depend strongly on the model of arithmetic: Most of our results will use the so-called Traditional Model (TM). We give a set of necessary and sufficient conditions to decide whether a high accuracy algorithm exists in the TM, and describe progress toward a decision procedure that will take any problem and provide either a high accuracy algorithm or a proof that none exists. When no accurate algorithm exists in the TM, it is natural to extend the set of available accurate operations by a library of additional operations, such as $x+y+z$, dot products, or indeed any enumerable set which could then be used to build further accurate algorithms. We show how our accurate algorithms and decision procedure for finding them extend to this case. Finally, we address other models of arithmetic, and the relationship between (im)possibility in the TM and (in)efficient algorithms operating on numbers represented as bit strings.Comment: 49 pages, 6 figures, 1 tabl

Derivation of determinantal structures for random matrix ensembles in a new way

There are several methods to treat ensembles of random matrices in symmetric spaces, circular matrices, chiral matrices and others. Orthogonal polynomials and the supersymmetry method are particular powerful techniques. Here, we present a new approach to calculate averages over ratios of characteristic polynomials. At first sight paradoxically, one can coin our approach "supersymmetry without supersymmetry" because we use structures from supersymmetry without actually mapping onto superspaces. We address two kinds of integrals which cover a wide range of applications for random matrix ensembles. For probability densities factorizing in the eigenvalues we find determinantal structures in a unifying way. As a new application we derive an expression for the k-point correlation function of an arbitrary rotation invariant probability density over the Hermitian matrices in the presence of an external field.Comment: 36 pages; 2 table

Asymptotic Behaviour of Random Vandermonde Matrices with Entries on the Unit Circle

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, 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.Comment: 28 pages. To appear in IEEE Transactions on Information Theor

Toward accurate polynomial evaluation in rounded arithmetic

Given a multivariate real (or complex) polynomial $p$ and a domain $\cal D$, we would like to decide whether an algorithm exists to evaluate $p(x)$ accurately for all $x \in {\cal D}$ using rounded real (or complex) arithmetic. Here ``accurately'' means with relative error less than 1, i.e., with some correct leading digits. The answer depends on the model of rounded arithmetic: We assume that for any arithmetic operator $op(a,b)$, for example $a+b$ or $a \cdot b$, its computed value is $op(a,b) \cdot (1 + \delta)$, where $| \delta |$ is bounded by some constant $\epsilon$ where $0 < \epsilon \ll 1$, but $\delta$ is otherwise arbitrary. This model is the traditional one used to analyze the accuracy of floating point algorithms.Our ultimate goal is to establish a decision procedure that, for any $p$ and $\cal D$, either exhibits an accurate algorithm or proves that none exists. In contrast to the case where numbers are stored and manipulated as finite bit strings (e.g., as floating point numbers or rational numbers) we show that some polynomials $p$ are impossible to evaluate accurately. The existence of an accurate algorithm will depend not just on $p$ and $\cal D$, but on which arithmetic operators and which constants are are available and whether branching is permitted. Toward this goal, we present necessary conditions on $p$ for it to be accurately evaluable on open real or complex domains ${\cal D}$. We also give sufficient conditions, and describe progress toward a complete decision procedure. We do present a complete decision procedure for homogeneous polynomials $p$ with integer coefficients, {\cal D} = \C^n, and using only the arithmetic operations $+$, $-$ and $\cdot$.Comment: 54 pages, 6 figures; refereed version; to appear in Foundations of Computational Mathematics: Santander 2005, Cambridge University Press, March 200

Lyapunov exponents for products of complex Gaussian random matrices

The exact value of the Lyapunov exponents for the random matrix product $P_N = A_N A_{N-1}...A_1$ with each $A_i = \Sigma^{1/2} G_i^{\rm c}$, where $\Sigma$ is a fixed $d \times d$ positive definite matrix and $G_i^{\rm c}$ a $d \times d$ complex Gaussian matrix with entries standard complex normals, are calculated. Also obtained is an exact expression for the sum of the Lyapunov exponents in both the complex and real cases, and the Lyapunov exponents for diffusing complex matrices.Comment: 15 page