In this paper we present a redesign of a linear algebra kernel of an interior point method to avoid the explicit use of problem matrices. The only access to the original problem data needed are the matrix-vector multiplications with the Hessian and Jacobian matrices. Such a redesign requires the use of suitably preconditioned iterative methods and imposes restrictions on the way the preconditio...

In this work, we perform a spectral analysis of flipped multilevel Toeplitz sequences, i.e., study the asymptotic behavior $\{Y_{{n}} T_{{n}} (f)\}_{{n}}$, where $T_{{n}}(f)$ is real, square matrix generated by function $f\in L^1([-\pi,\pi]^d)$ and $Y_n$ exchange matrix, which has 1's on main antidiagonal. line with what have shown for unilevel spectrum determined 2 x matrix-valued whose eigenv...

We propose a method for preconditioner construction and parallel implementations of the Preconditioned Conjugate Gradient algorithm on GPU platforms. The preconditioning matrix is an approximate inverse derived from an algorithm for the iterative improvement of a solution to linear equations. Using a sparse matrix-vector product, our preconditioner is well suited for massively parallel GPU arch...

The stationary Navier{Stokes equations are solved in 2D with preconditioned Krylov subspace methods, where the preconditioning matrix is derived from a semi-implicit Runge{Kutta scheme. By this approach the spectrum of the coeecient matrix is improved. Numerical experiments for the ow over a semi-innnite at plate show that the preconditioning substantially improves the convergence rate. The num...

In this paper, we consider a block Jacobi preconditioner and various deflation techniques applied in the Deflated Preconditioned Conjugate Gradient (DPCG) method for solving sparse system of linear equations derived from statistical mixed model that analyses simultaneously phenotypic pedigree information genotyped ungenotyped animals with Single Polymorphism Nucleotide genotypes animals. livest...

We study the performance of a new block preconditioner for class \(3\times 3\) saddle point problems which arise from finite-element methods solving time-dependent Maxwell equations and some other practical problems. also estimate lower upper bounds eigenvalues preconditioned matrix. Finally, we examine our to accelerate convergence speed GMRES method shows effectiveness preconditioner.

In this paper, the modified Gauss-Seidel method with the new preconditioner for solving the linear system Ax = b, where A is a nonsingular M -matrix with unit diagonal, is considered. The convergence property and the comparison theorems of the proposed method are established. Two examples are given to show the efficiency and effectiveness of the modified Gauss-Seidel method with the presented n...

Numerical solution of extremely large and ill conditioned eigenvalue problems is attracting a growing attention recently as such problems are of major importance in applications. They arise typically as discretization of continuous models described by systems of partial differential equations (PDE’s). For such problems, preconditioned matrix-free eigensolvers are especially effective as the sti...

In this paper, we propose approximate inverse-free preconditioners for solving Toeplitz systems. The preconditioners are constructed based on the famous Gohberg-Sememcul formula. We show that if a Toepltiz matrix is generated by a positive bounded function and its entries enjoys the off-diagonal decay property, then the eigenvalues of the preconditioned matrix are clustered around one. Experime...

For the discretized linear systems of spatial fractional diffusion equations, we construct a class modified DTS iteration method and give its asymptotic convergence conditions. Then, design fast preconditioner by replacing Toeplitz matrix T with τ to accelerate rates GMRES method. Theoretically, show that spectrum preconditioned is clustered around one. Numerical experiments verify validity con...