Abstract
This paper proposes a new effective pseudo-spectral approximation to solve the Sylvester and Lyapunov matrix differential equations. The properties of the Chebyshev basis operational matrix of derivative are applied to convert the main equation to the matrix equations. Afterwards, an iterative algorithm is examined for solving the obtained equations. Also, the error analysis of the propounded method is presented, which reveals the spectral rate of convergence. To illustrate the effectiveness of the proposed framework, several numerical examples are given.
Highlights
For study on the dynamical systems, filtering, model reduction, image restoration, etc., many phenomena can be modeled more efficiently by the matrix differential equations [1,2,3,4,5]
The reported results reveal the superior convergence properties of their algorithm in comparison to the algorithms derived via the extension of the conjugate gradient method [29], which was presented in the literature for solving different types of coupled linear matrix equations
4 Numerical simulations we show the application of the Chebyshev collocation method to solve (1)
Summary
For study on the dynamical systems, filtering, model reduction, image restoration, etc., many phenomena can be modeled more efficiently by the matrix differential equations [1,2,3,4,5]. We propose an iterative algorithm based on Paige’s algorithm [28] to solve the obtained coupled matrix equations. Paige’s algorithm has been extended to find the bisymmetric minimum norm solution of the coupled linear matrix equations [27]. The reported results reveal the superior convergence properties of their algorithm in comparison to the algorithms derived via the extension of the conjugate gradient method [29], which was presented in the literature for solving different types of coupled linear matrix equations. This motivates us to generalize Paige’s algorithm to resolve (10). If G – M(Xi) ≤ Stop; Otherwise go to 2
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