Related papers: Generalized eigenvalue methods for Gaussian quadra…
We consider the approximation of the inverse square root of regularly accretive operators in Hilbert spaces. The approximation is of rational type and comes from the use of the Gauss-Legendre rule applied to a special integral formulation…
Solving the generalized eigenvalue problem is a useful method for finding energy eigenstates of large quantum systems. It uses projection onto a set of basis states which are typically not orthogonal. One needs to invert a matrix whose…
In this paper we bring to light an unprecedented property of the eigenvalues of a matrix A with the eigenvalues and eigenvectors of a submatrix of A. This property can be used, through the technique developed here, to determine some of…
We use a generalization of Vinogradov's mean value theorem of S. Parsell, S. Prendiville and T. Wooley and ideas of W. Schmidt to give nontrivial bounds for the number of solutions to polynomial congruences, for arbitrary polynomials, when…
Positive semi-definite matrices commonly occur as normal matrices of least squares problems in statistics or as kernel matrices in machine learning and approximation theory. They are typically large and dense. Thus algorithms to solve…
We explore the block nature of the matrix representation of multiplex networks, introducing a new formalism to deal with its spectral properties as a function of the inter-layer coupling parameter. This approach allows us to derive…
We consider the bifurcation problem $u'' + \lambda u = N(u)$ with two point boundary conditions where $N(u)$ is a general nonlinear term which may also depend on the eigenvalue $\lambda$. We give a variational characterization of the…
In this paper, using combinatorial and analytic methods, we prove an exact calculating formula on the $2m$-th power mean value of the generalized quadratic Gauss sums for $m\geq 2$. This solves a conjecture of He and Zhang [`On the $2k$-th…
We present novel fully-symmetric quadrature rules with positive weights and strictly interior nodes of degrees up to 84 on triangles and 40 on tetrahedra. Initial guesses for solving the nonlinear systems of equations needed to derive…
We address the count of isolated and embedded eigenvalues in a generalized eigenvalue problem defined by two self-adjoint operators with a positive essential spectrum and a finite number of isolated eigenvalues. The generalized eigenvalue…
We study multivariate normal models that are described by linear constraints on the inverse of the covariance matrix. Maximum likelihood estimation for such models leads to the problem of maximizing the determinant function over a…
In this paper, a new approach is presented to determine common eigenvalues of two matrices. It is based on Gerschgorin theorem and Bisection method. The proposed approach is simple and can be useful in image processing and noise estimation.
We investigate a Gaussian quadrature rule and the corresponding orthogonal polynomials for the oscillatory weight function $e^{i\omega x}$ on the interval $[-1,1]$. We show that such a rule attains high asymptotic order, in the sense that…
We consider polynomial approximation over the interval $[-1,1]$ by regularized weighted discrete least squares methods with $\ell_2-$ or $\ell_1-$regularization, respectively. As the set of nodes we use Gauss quadrature points (which are…
We construct the general permutation invariant Gaussian 2-matrix model for matrices of arbitrary size $D$. The parameters of the model are given in terms of variables defined using the representation theory of the symmetric group $S_D$. A…
The problem of finding out the global minimum of a multiextremal functional is discussed. One frequently faces with such a functional in various applications. We propose a procedure, which depends on the dimensionality of the problem…
Given a quadratic two-parameter matrix polynomial Q, we develop a systematic approach to generating a vector space of linear two-parameter matrix polynomials. We identify a set of linearizations of Q that lie in the vector space. Finally,…
We derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular Gaussian random matrices in the limit of large matrix dimensions. We show that…
A real quadratic matrix is generalized doubly stochastic (g.d.s.) if all of its row sums and column sums equal one. We propose numerically stable methods for generating such matrices having possibly orthogonality property or/and satisfying…
In this paper we give a new and simple algorithm to put any multivariate polynomial into a normal determinant form in which each entry has the form , and in each column the same variable appears. We also apply the algorithm to obtain a…