arXiv subjects

Mehdi Najafi Kalyani

Publications and source records attributed to Mehdi Najafi Kalyani.

3 recordsLinked to original sources

arXiv2026

A parameterized block LU preconditioning for double saddle point problems

We present a biparametric preconditioning technique for large, sparse, non-symmetric, and non-singular double saddle point problems. The preconditioner is induced using a stationary iteration method. It is also based on a two-parameterized block LU factorization of the coefficient matrix. To begin with, some convergence results of the iteration method are derived. Moreover, the spectral features exhibited by the preconditioned matrix are given. Finally, the presented preconditioner is conjuncted with the GMRES method. Numerical results demonstrate the satisfactory performance of the preconditioner.

math.NA↗

arXiv2026

The mmatrix toolbox: componentwise accurate algorithms for M-matrices with triplet representation

We introduce the mmatrix toolbox, a Matlab software package for componentwise accurate computations with M-matrices described through left or right triplet representations. The core of the toolbox is a Fortran implementation, in the Lapack style, of the unblocked, recursive, and blocked versions of the GTH algorithm and its applications for the accurate computation of the solution of linear systems with M-matrix coefficient and nonnegative right-hand side, the LU factorization of an M-matrix and its inverse. These algorithms avoid subtractive cancellation; this property ensures high componentwise accuracy even for ill-conditioned problems. The toolbox contains also accurate algorithms for related problems, such as computing the Schur complement, the singular values, the square root of an M-matrix, and the solution of nonsymmetric algebraic Riccati equations associated with M-matrices. The Matlab interface is based on an object-oriented implementation allowing one to use standard Matlab operations on M-matrices with triplet representation.

cs.MS↗

arXiv2025

Accuracy and componentwise accuracy in multilinear PageRank

We study the stability with respect to perturbations and the accuracy of numerical algorithms for computing solutions to the multilinear PageRank problem $\mathbf{x} = (1-α)\mathbf{v} + α\mathcal{P} \mathbf{x}^2$. Our results reveal that the solution can be more stable with respect to perturbations and numerical errors with respect to the classical bounds for nonlinear systems of equations (based on the norm of the Jacobian). In detail, one can obtain bounds for the minimal solution which ignore the singularity of the problem for $α=1/2$, and one can show that the limiting accuracy of the Newton method depends not on the norm of the Jacobian but on a quantity that can be much smaller thanks to the nonnegativity structure of the equation. For the minimal solution, we also suggest subtraction-free modifications to the existing algorithms to achieve componentwise stability. Some of the theoretical results we obtain are interesting even outside the scope of this problem: bounds for more general quadratic vector equations, and a partial inverse for M-matrices which remains bounded when the matrix to invert approaches singularity.

math.NA↗