A Hessenberg method for the numerical solutions to types of block Sylvester matrix equations

Abstract

a b s t r a c tIn this paper, we propose a new algorithm to solve the Sylvester matrix equation XA + BX = C . The technique consists of orthogonal reduction of the matrix A to a block upper Hessenberg form P T AP = H and then solving the reduced equation, YH + BY = C for Y through recurrence relation, where Y = XP, and C = CP. We then recover the solution of the original problem via the relation X = YP T . The numerical results show the accuracy and the efficiency of the proposed algorithm. In addition, how the technique described can be applied to other matrix equations was shown.

Developing Bi-CG and Bi-CR Methods to Solve Generalized Sylvester-transpose Matrix Equations

“…The solution of linear matrix equations such as Sylvester and Lyapunov matrix equations has been addressed in a large body of literature [8−13] . In [14], a Hessenberg method was proposed for solving the Sylvester matrix equation XA+BX = C. A new Smith accelerative iteration (containing the well-known Smith accelerative iteration as a special case) was established in [2] for solving the Stein matrix equation X = AXB + C. In [15], Manuscript received Feburary 4, 2013; revised April 16, 2013 by using the so-called Kronecker matrix polynomials, closed form solutions to a family of generalized Sylvester matrix equation were given. Li and Huang [16] proposed an iterative method to solve generalized coupled Sylvester matrix equations, based on a matrix form of the least-squares QRfactorization (LSQR) algorithm.…”

Explicit solutions to the matrix equation E X F − AX = C

“…It follows from (25) and (19) that (see (26)) By applying the expression of X in (22), it follows from Lemma 4, the preceding relations (19), (23), (24) and (26) that…”

“…In addition, some numerical approaches have been proposed for these two classes of general matrix equations. In [23], a numerical algorithm was proposed to solve the matrix equation XF − AX = C by orthogonal reduction of the matrix F to a block upper Hessenberg form. By using hierarchical principle, iterative algorithms for solving XF − AX = C were established in [24, 25].…”

Numerical Solution of Sylvester Equation Based on Iterative Predictor‐Corrector Method

“…A numerical Arnoldi-based method for solving Sylvester equation when A is sparse and large for partial pole-assignment problem for large matrices was given in [15]. A method consisting of orthogonal reduction of the coe cient matrix A of the Sylvester matrix equation to a block-upper-Hessenberg form was given in [16].…”