Research On Iterative Algorithms For Several Classes Of Matrix Equations

The solution of matrix equation is one of the most important research topics in the field of scientific and engineering computation,especially in the field of numerical algebra and its application.Matrix equation solving problems have a wide range of sources,and different practical application backgrounds can abstract various types of matrix equation solving problems.This paper studies the iterative algorithms of several types of matrix equations,and the main contents are as follows:In Chapter 2,the biconjugate residual(BCR)algorithm for the reflexive or anti-reflexive solutions of a class of generalized coupled Sylvester matrix equations is proposed.The convergence of the algorithm is proved theoretically.By constructing a special form of initial matrix,without considering rounding errors,the minimal norm solution of the equation can be obtained within a finite number of iterative steps.Finally,several numerical examples are given to illustrate the feasibility and effectiveness of the proposed algorithm.In Chapter 3,a modified conjugate gradient(MCG)algorithm for solving generalized Hamiltonian solutions of a class of generalized coupled Sylvester-conjugate matrix equations is proposed.The convergence of the algorithm is proved theoretically.By constructing special initial matrices,the minimal norm solutions of the equation can be obtained within finite iterative steps without considering rounding errors.Finally,several numerical examples are given to illustrate the feasibility and effectiveness of the proposed algorithm.In Chapter 4,an adaptive parameter alternating direction(APAD)algorithm for solving the centrosymmetric solution of a class of generalized coupled Sylvestertransposed matrix equations is proposed.Under appropriate constraints,it is proved that the algorithm is convergent.Finally,several numerical examples are given to illustrate the feasibility and effectiveness of the proposed algorithm.In Chapter 5,a finite iterative algorithm for solving periodic solutions of a class of discrete-time periodic Sylvester matrix equations is proposed.Theoretical analysis proves that the algorithm can obtain the exact solution of the periodic Sylvester matrix equations in finite steps without considering the rounding error.In addition,when the discrete periodic Sylvester matrix equations are consistent,its unique minimal norm solution can be obtained by choosing appropriate initial periodic matrices.In Chapter 6,a class of SOR-based alternately linearized implicit(SORALI)iteration algorithm for computing the minimal nonnegative solution of nonsymmetric algebraic Riccati equations(NARE)is proposed.Under appropriate constraints,it is proved that the algorithm is convergent.Finally,several numerical examples are given to illustrate the feasibility and effectiveness of the proposed algorithm.