We discuss  some potential advantages of the  orthogonal Symmetric-diagonal reduction in  two main versions of the Schur-QR method  for Symmetric positive definite  generalized eigenvalue problems. We also advise and use the appropriate reductions  as preprocessing on  the solvers, mainly  the Cholesky-QR method, of the  considered  problems. We discuss numerical stability of the  methods via providing upper bound for backward error of the computed eigenpairs and via investigating two kinds of  scaled residual errors. We also propose  and apply  two kinds of symmetrizing  which  improve  the stability and the performance  of the methods. Numerical experiments show that the  implemented versions of the Schur-QR method and the preprocessed versions of the Cholesky-QR  method are  usually more stable than the Cholesky-QR method.

The generalized eigenvalue problem AX=lBX has special properties when (A,B) is a Symmetric and positive definite pair. We have recently developed three methods for computing a few smallest (largest) eigenvalues of a Symmetric positive definite problem AX=lBX [2-3-4]. In this article we compared those methods by some numerical examples.

The aim of the current paper is to study a partially described inverse eigenvalue problem of  a specific Symmetric  matrix, and prove some properties of such matrix. The problem includes the construction of the matrix by  the  minimal eigenvalue of all  leading principal submatrices  and eigenpair $(\lambda_2^{(n)},x)$ such that $ \lambda_2^{(n)}$ is the maximal eigenvalue of the required matrix. We investigate  conditions for the solvability of the problem, and finally an algorithm and  its numerical results are presented.

In this paper we study generalized Symmetric Berwald spaces. We show that if a Berwald space (M, F) admits a parallel s-structure then it is locally Symmetric. For a complete Berwald space which admits a parallel s-structure we show that if the flag curvature of (M, F) is everywhere nonzero, then F is Riemannian.

In this paper, we consider a form of the generalized logistic distribution named Symmetric extended generalized logistic distribution or extended type III generalized logistic distribution. The distribution is derived by compounding a two-parameter generalized Gumbel distribution with a two-parameter generalized gamma distribution. The cumulative distribution and some properties of this distribution like moments and related statistics are established. Some theorems that characterize the distribution are stated and proved. Estimation of the parameters and an application of the distribution are also presented.

In this paper, we represent an inexact inverse subspace iteration method for computing a few eigenpairs of the generalized eigenvalue problem Ax=lBx [Q.Ye and P. Zhang, Inexact inverse subspace iteration for generalized eigenvalue problems, Linear Algebra and its Application, 434 (2011) 1697-1715]. In particular, the linear convergence property of the inverse subspace iteration is preserved.

There are several methods to solve generalized eigenvalue problem, AX=lBX but the number of computed eigenvalues by these methods might be less or more than expected. There are also methods which can lead us to the amount of results needed. In this paper we describe free inverse method for computing the generalized eigenvalue. Then, by combination this method and projection method, we develop two new iterative method for solving the generalized eigenvalue problem. The implementation of the algorithm has been tested by numerical examples, the results show that the algorithm converges fast and works with high accuracy.