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 look at the real positive (semi)definite matrix completion problem from the relative entropy minimization viewpoint. After the problem is transformed into the standard maxdet from, conditions are sought for existence of positive (semi)definite completions. Using basic tools of convex analysis results previously established using graph-theoretic or functional-analytic techniques are recovered...

The power method for computing the dominant eigenvector of a positive definite matrix will converge slowly when the dominant eigenvalue is poorly separated from the next largest eigenvalue. In this note it is shown that in spite of this slow convergence, the Rayleigh quotient will often give a good approximation to the dominant eigenvalue after a very few iterations-even when the order of the m...

In this paper, we consider a broad class of interpolation problems, for both scalarand vector-valued multivariate functions subject to linear side conditions, such as being divergence-free, where the data are generated via integration against compactly supported distributions. We show that, by using certain families of matrix-valued conditionally positive definite functions, such interpolation ...

Based on the elegant properties of the Thompson metric, we prove that the following two kinds of nonlinear matrix equations X = m ∑ i=1 AiX iAi and X = m ∑ i=1 (AiXAi) δi , (0 < |δi| < 1) always have a unique positive definite solution. Iterative methods are proposed to compute the unique positive definite solution. We show that the iterative methods are more effective as δ = max{|δi|, i = 1, 2...

The notion of a <7-continuous matrix-valued positive definite distribution on 5/v(2a) x RM x G is introduced, where G is an abelian separable locally compact group and where Sp/(2a) is an open ball around zero in RA' with radius la > 0 . This notion generalizes that one of strongly continuous positive definite operator-valued functions. For these objects, a Bochner-type theorem gives a suitable...

Some perturbation estimates for the unique positive definite solution of a nonlinear matrix equation connected to the interpolation theory are derived. The considered estimations are modification of some existing one. They are obtained by similar transformations of the matrix coefficients with a positive definite matrix. The theoretical results are illustrated by several numerical examples.