Developing CRS iterative methods for periodic Sylvester matrix equation

Abstract

In this paper, by applying Kronecker product and vectorization operator, we extend two mathematical equivalent forms of the conjugate residual squared (CRS) method to solve the periodic Sylvester matrix equationWe give some numerical examples to compare the accuracy and efficiency of the matrix CRS iterative methods with other methods in the literature. Numerical results validate that the proposed methods are superior to some existing methods and that equivalent mathematical methods can show different numerical performance.where the coefficient matrices A j , B j , C j , D j , E j ∈ R m×m and the solutions X j ∈ R m×m are periodic with period λ, that is, A j+λ = A j , B j+λ = B j , C j+λ = C j , D j+λ = D j , E j+λ = E j , and X j+λ = X j . The periodic Sylvester matrix equation (1.1) attracts considerable attention because it comes from a variety of fields of control theory and applied mathematics [1][2][3][4][5][6][7][8][9][10][11][12].In recent years, many efficient iterative methods have been proposed to solve the periodic Sylvester matrix equation (1.1). For example, Hajarian [13,14] developed the conjugate gradient squared (CGS), biconjugate gradient stabilized (BiCGSTAB) and biconjugate residual methods for solving the periodic Sylvester matrix equation (1.1). Lv and Zhang [15] proposed a new kind of iterative algorithm for constructing the least square solution for the periodic Sylvester matrix equation. Hajarian [16] studied the biconjugate A-orthogonal residual and conjugate A-orthogonal residual squared (CORS) methods for solving coupled periodic Sylvester matrix equation, and so forth; see [17-28] and the references therein.As we know, by applying Kronecker product and vectorization operator, some iterative algorithms for solving linear system Ax = b can be extended to solve linear matrix equa-

Matrix iteration algorithms for solving the generalized Lyapunov matrix equation

“…The CRS algorithm is mainly aimed to avoid using the transpose of A in the BiCR algorithm and get faster convergence for the same computational cost [19]. Recently, Ma et al [21] used the matrix CRS iteration method to solve a class of coupled Sylvester-transpose matrix equations. Later, they extended two mathematical equivalent Algorithm 3: Matrix form of the Bi-CGSTAB algorithm step 1: Choose initial matrix X(1); step 2: Compute R(1) = -C -AX(1) -X(1)A Tm j=1 N j X(1)N T j , pick arbitrary matrix R(1) (e.g.R(1) = R(1)); step 3: Compute V (1) = P(1) = 0, ρ(1) = α(1) = ω(1) = 1; step 4: While R(k) > tol do:…”

“…forms of the CRS algorithm to solve the periodic Sylvester matrix equation by applying Kronecker product and vectorization operator [21]. In fact, in many cases, the CRS algorithm converges twice as fast as the BiCR algorithm [22,23].…”

Robust gradient‐based neural networks for solving online the discrete periodic Lyapunov matrix equations

“…In reference [6], an iterative algorithm based on conjugate gradient was developed to solve the DPLME. Moreover, many iterative algorithms for solving other matrix equations can also be used to solve the DPLME [7][8][9][10][11].…”

Two modified least-squares iterative algorithms for the Lyapunov matrix equations

“…To solve equation (1.1) or its special cases or generalized versions, different methods have been developed in the literature [5,7,[9][10][11][12][13][14][15][16][17][18][19], which belong to the category of iterative methods. For example, two conjugate gradient methods are proposed in [7] to solve consistent or inconsistent equation (1.1).…”