the star-mesh transformation allows the reduction of nodes in an electrical circuit. the gauss elimination method allows the reduction of unknowns in a system of linear equations . triangular decomposition can be used to find the inverse of a matrix. in this paper the coherence between these methods are discussed. it is shown that the gauss elimination method gives the same formula for star-mes...

In this paper, we develop a new classification method for manifold-valued data in the framework of probabilistic learning vector quantization. many scenarios, can be naturally represented by symmetric positive definite matrices, which are inherently points that live on curved Riemannian manifold. Due to non-Euclidean geometry manifolds, traditional Euclidean machine algorithms yield poor result...

A new preconditioner for symmetric positive definite systems is proposed, analyzed, and tested. The preconditioner, compressed incomplete modified Gram–Schmidt (CIMGS), is based on an incomplete orthogonal factorization. CIMGS is robust both theoretically and empirically, existing (in exact arithmetic) for any full rank matrix. Numerically it is more robust than an incomplete Cholesky factoriza...

This paper presents a proximal point approach for computing the Riemannian or intrinsic Karcher mean of, n×n, symmetric positive definite (SPD) matrices. Our method derives from proximal point algorithm with Schur decomposition developed to compute minimum points of convex functions on SPD matrices set when it is seen as a Hadamard manifold. The main idea of the original method is preserved. Ho...

We propose a new cell-centered diffusion scheme on unstructured meshes. The main feature of this scheme lies in the introduction of two normal fluxes and two temperatures on each edge. A local variational formulation written for each corner cell provides the discretization of the normal fluxes. This discretization yields a linear relation between the normal fluxes and the temperatures defined o...

Recently, a novel flow for computing the eigenvectors associated with the smallest eigenvalues of a symmetric but not necessarily positive definite matrix was introduced. This meant that the eigenvectors associated with the smallest eigenvalues could be found simply by reversing the sign of the matrix. The current paper derives a cost function and the corresponding negative gradient flow which ...

In this work, we study the set of strictly accretive matrices, that is, matrices with positive definite Hermitian part, and show can be interpreted as a smooth manifold. Using recently proposed symmetric polar decomposition for sectorial manifold is diffeomorphic to direct product (Hermitian) unitary matrices. Utilizing decomposition, introduce family Finsler metrics on characterize their geode...

DIRECTION-PRESERVING AND SCHUR-MONOTONIC SEMISEPARABLE APPROXIMATIONS OF SYMMETRIC POSITIVE DEFINITE MATRICES∗ MING GU† , XIAOYE S. LI‡ , AND PANAYOT S. VASSILEVSKI§ Abstract. For a given symmetric positive definite matrix A ∈ RN×N , we develop a fast and backward stable algorithm to approximate A by a symmetric positive definite semiseparable matrix, accurate to a constant multiple of any pres...

We present a detailed proof of the Dovbysh-Sudakov representation for symmetric positive definite weakly exchangeable infinite random arrays, called Gram-de Finetti matrices, which is based on the representation result of Aldous and Hoover for arbitrary (not necessarily positive definite) symmetric weakly exchangeable arrays.

In a number of contexts relevant to control problems, including estimation of robot dynamics, covariance, and smart structure mass and stiffness matrices, we need to solve an over-determined set of linear equations AX ≈ B with the constraint that the matrix X be symmetric and positive definite. In the classical least squares method the measurements of A are assumed to be free of error, hence, a...