Geometrical inverse preconditioning for symmetric positive definite matrices

We focus on inverse preconditioners based on minimizing $F(X) = 1-\cos(XA,I)$, where $XA$ is the preconditioned matrix and $A$ is symmetric and positive definite. We present and analyze gradient-type methods to minimize $F(X)$ on a suitable compact set. For that we use the geometrical properties of the non-polyhedral cone of symmetric and positive definite matrices, and also the special properties of $F(X)$ on the feasible set. Preliminary and encouraging numerical results are also presented in which dense and sparse approximations are included

A Matrix Iteration for Finding Drazin Inverse with Ninth-Order Convergence

The aim of this paper is twofold. First, a matrix iteration for finding approximate inverses of nonsingular square matrices is constructed. Second, how the new method could be applied for computing the Drazin inverse is discussed. It is theoretically proven that the contributed method possesses the convergence rate nine. Numerical studies are brought forward to support the analytical parts

Virtual Morse-Bott index, moduli spaces of pairs, and applications to topology of smooth four-manifolds

In Feehan and Leness (2020), we introduced an approach to Morse-Bott theory, called virtual Morse-Bott theory, for Hamiltonian functions of circle actions on closed, real analytic, almost Hermitian spaces. In the case of Hamiltonian functions of circle actions on closed, smooth, almost Kaehler (symplectic) manifolds, virtual Morse-Bott theory coincides with classical Morse-Bott theory due to Bott (1954) and Frankel (1959). Positivity of virtual Morse-Bott indices implies downward gradient flow in the top stratum of smooth points in the analytic space. In this monograph, we apply our method to the moduli space of SO(3) monopoles over a complex, Kaehler surface, we use the Atiyah-Singer Index Theorem to compute virtual Morse-Bott indices of all critical strata (Seiberg-Witten moduli subspaces), and we prove that these indices are positive in a setting motivated by the conjecture that all closed, smooth four-manifolds of Seiberg-Witten simple type obey the Bogomolov-Miyaoka-Yau inequality.Comment: 167+xii pages, 1 figure. Supporting background material drawn from our monograph arXiv:math/020304