The Progressive GMRES algorithm, introduced by Beckermann and Reichel in 2008, is a residual-minimizing short-recurrence Krylov subspace method for solving a linear system in which the coefficient matrix has a low-rank skew-Hermitian part. We analyze this algorithm, observing a critical instability that makes the method unsuitable for some problems. To work around this issue we introduce a diff...

Recently, in (M. Masoudi, D.K. Salkuyeh, An extension of positive-definite and skew-Hermitian splitting method for preconditioning generalized saddle point problems, Computers \& Mathematics with Application, https://doi.org/10.1016/j.camwa.2019.10.030, 2019) an the positive definite (EPSS) iteration nonsingular problems has been presented. In this article, we study semi-convergence EPSS singul...

The Hermitian and skew-Hermitian splitting (HSS) method is an unconditionally convergent iteration method for solving large sparse non-Hermitian positive definite system of linear equations. By making use of the HSS iteration as the inner solver for the Newton method, we establish a class of Newton-HSS methods for solving large sparse systems of nonlinear equations with positive definite Jacobi...

We begin with the definition of a skew-Hermitian form and the corresponding Hermitian symplectic group. We motivate these definitions with a discussion of their relevance to self-adjoint extensions of Hamiltonian operators. In doing so, we introduce the basics of von Neumann’s extension theory. Next, we develop the necessary tools from Hermitian symplectic linear algebra to study self-adjoint e...

We consider large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of skew-shift $\binom{j}{2} \omega+jy+x \mod 1$ for irrational $\omega$. prove that eigenvalue distribution these converges to corresponding from random matrix theory on global scale, namely, Wigner semicircle law square and Marchenko-Pastur rectangular matrices. The results eviden...

We consider the real eigenvalues of an $(N \times N)$ elliptic Ginibre matrix whose entries are correlated through a non-Hermiticity parameter $\tau_N\in [0,1]$. In almost-Hermitian regime where $1-\tau_N=\Theta(N^{-1})$, we obtain large-$N$ expansion mean and variance number eigenvalues. Furthermore, derive limiting empirical distributions eigenvalues, which interpolate Wigner semicircle law u...

Abstract In this paper, we consider skew-Hermitian solution of coupled generalized Sylvester matrix equations encompassing $$*$$ ∗ -hermicity over complex field. The compact formula the general system is presented in terms inverses when some necessary and sufficient conditi...

In this note, we discuss symmetric brackets on skew-symmetric algebroids associated with metric or symplectic structures. Given a pseudo-Riemannian structure, describe the induced by connections totally torsion in language of Lie derivatives and differentials functions. We formulate generalization fundamental theorem Riemannian geometry. particular, obtain an explicit formula Levi-Civita connec...