Some relaxed iteration methods for solving matrix equation AXB=C

Highlights：

• Combining the iteration methods in [6] with the tunable parameter ω, some relaxed preconditioned iteration methods are also given.

• The choices of the parameter ω in the relaxed iteration methods are discussed, and the optimal choices of the parameter ω to achieve the fastest convergence rate are also obtained for some special cases.

• Based on the matrix splittings BT=D1−L1−U1,A=D2−L2−U2,three relaxed iteration methods are constructed, which are the relaxed Jacobi-type iteration method, relaxed SSOR-type iteration method and relaxed bilateral Jacobi-type iteration method, respectively. Moreover, the optimal parameter ω can be calculated in these iteration methods.

• For the general matrix equation AXB=C, we consider how to solve it by using the proposed relaxed iteration methods, and investigate how to reduce their computational costs based on the Hessenberg decompositions of the corresponding matrices.

• Numerical examples show that our proposed algorithms are more efficient than GI method [4] HSS method [17] and the iteration methods in [6], respectively.

摘要

•Based on the iteration frameworks [6], by introducing a tunable parameter ω, several relaxed iteration methods are proposed for solving the matrix equation AXB = C, and their convergence properties are analyzed in detail.•Combining the iteration methods in [6] with the tunable parameter ω, some relaxed preconditioned iteration methods are also given.•The choices of the parameter ω in the relaxed iteration methods are discussed, and the optimal choices of the parameter ω to achieve the fastest convergence rate are also obtained for some special cases.•Based on the matrix splittings BT=D1−L1−U1,A=D2−L2−U2,three relaxed iteration methods are constructed, which are the relaxed Jacobi-type iteration method, relaxed SSOR-type iteration method and relaxed bilateral Jacobi-type iteration method, respectively. Moreover, the optimal parameter ω can be calculated in these iteration methods.•For the general matrix equation AXB=C, we consider how to solve it by using the proposed relaxed iteration methods, and investigate how to reduce their computational costs based on the Hessenberg decompositions of the corresponding matrices.•Numerical examples show that our proposed algorithms are more efficient than GI method [4] HSS method [17] and the iteration methods in [6], respectively.