On Some Numerical Methods for Solving Large Differential Nonsymmetric Stein Matrix Equations

Abstract

In this paper, we propose a new numerical method based on the extended block Arnoldi algorithm for solving large-scale differential nonsymmetric Stein matrix equations with low-rank right-hand sides. This algorithm is based on projecting the initial problem on the extended block Krylov subspace to obtain a low-dimensional differential Stein matrix equation. The obtained reduced-order problem is solved by the backward differentiation formula (BDF) method or the Rosenbrock (Ros) method, the obtained solution is used to build the low-rank approximate solution of the original problem. We give some theoretical results and report some numerical experiments.

The general Bernstein function: Application to χ‐fractional differential equations

“…Bernstein polynomials [18, 19], delta functions [20], Mittag‐Leffler [21], block pulse functions [22], Legendre [23], Chebyshev [24] and Bernoulli functions [25], Jacobi operational matrix method [26], Taylor collocation method [27], Legendre functions method [28], variational iteration method [29], and Tau methods [30] are the most frequently used special functions. Karimi et al [31] proposed a new effective basis to solve partial FDEs, while Sadek et al [32] applied the Bernstein polynomials bases for solving Sylvester matrix equations. In earlier studies [33, 34], the fractional Riccati differential equations have been solved by ADM.…”

Efficient iterative schemes based on Newton's method and fixed-point iteration for solving nonlinear matrix equation Xp = Q±A(X−1+B)−1AT

“…(2020), a new numerical method for solving large-scale differential Sylvester matrix equations with low-rank right-hand sides was presented. Fractional backward differentiation formulas methods were presented for the numerical solution of fractional differential matrix equations in the fractional sense, for example, Sylvester, Lyapunov, Riccati and Stein for the first time (Sadek, 2022b; Sadek et al ., 2022). In Sadek and Talibi Alaoui (2021), two new methods to solve large-scale systems of differential equations, which are based on the Krylov method, were proposed.…”

“…Sadek and Alaoui proposed two new approaches to solve the large linear system of ordinary differential equations, which are based on the projection technique on the extended global Krylov or global Krylov subspaces if are not invertible (Sadek and Alaoui, 2022). Also, some methods for solving large-scale differential Lyapunov and T-Lyapunov equations were introduced (Sadek and Alaoui, 2021; Sadek and Talibi Alaoui, 2022). In Sadek et al .…”