The family of iterative methods for static and natural vibration analysis, based on preconditioned conjugate gradient (PCG) method with aggregation multilevel preconditioning, is considered. Both: the element-by-element procedure for assembling of stiffness matrix and sparse direct solver for it factoring and fast forward − backward substitutions ensure the high stability of methods against ill...

We construct, analyze and implement SSOR-like preconditioners for non-Hermitian positive definite system of linear equations when its coefficient matrix possesses either a dominant Hermitian part or a dominant skew-Hermitian part. We derive tight bounds for eigenvalues of the preconditioned matrices and obtain convergence rates of the corresponding SSOR-like iteration methods as well as the cor...

We present the convergence analysis of the inexact infeasible pathfollowing (IIPF) interior-point algorithm. In this algorithm, the preconditioned conjugate gradient method is used to solve the reduced KKT system (the augmented system). The augmented system is preconditioned by using a block triangular matrix. The KKT system is solved approximately. Therefore, it becomes necessary to study the ...

A two-step preconditioned iterative method based on the Hermitian/Skew-Hermitian splitting is applied to the solution of nonsymmetric linear systems arising from the Finite Element approximation of convection-diffusion equations. The theoretical spectral analysis focuses on the case of matrix sequences related to FE approximations on uniform structured meshes, by referring to spectral tools der...

The linear finite element approximation of a general linear diffusion problem with arbitrary anisotropic meshes is considered. The conditioning of the resultant stiffness matrix and the Jacobi preconditioned stiffness matrix is investigated using a density function approach proposed by Fried in 1973. It is shown that the approach can be made mathematically rigorous for general domains and used ...

In this paper, we will present a modification of the preconditioned AOR-type method for solving the linear system. A theorem is given to show the convergence rate of modification of the preconditioned AOR methods that can be enlarged than the convergence AOR method.

The discrete Hartley transforms (DHT) of types I – IV and the related matrix algebras are discussed. We prove that any of these DHTs of length N = 2 can be factorized by means of a divide–and–conquer strategy into a product of sparse, orthogonal matrices where in this context sparse means at most two nonzero entries per row and column. The sparsity joint with orthogonality of the matrix factors...

In the present dissertation we consider three crucial problems of numerical linear algebra: solution of a linear system, an eigenvalue, and a singular value problem. We focus on the solution methods which are iterative by their nature, matrix-free, preconditioned and require a fixed amount of computational work per iteration. In particular, this manuscript aims to contribute to the areas of res...