Real Structure-preserving Algorithms For Quaternion Sylvester Matrix Equation And Its Application

Matrix equation has been one of the most popular research topics in computational mathematics and is actively used in image restoration,structural dynamics,automatics control theory,and many other fields.There are considerable kinds of literature on solving matrix equations in real and complex fields.Due to the complex form of quaternions and non-commutative multiplication,there are no effective iterative methods for solving large-scale quaternion matrix equations.This dissertation,based on the idea of quaternion real structure-preserving,dedicates to extend the existing iterative methods to the quaternion field to solve quaternion linear systems with multiple right-hand sides AX = B and the quaternion Sylvester matrix equation AX + X B = C.In this dissertation,we establish the quaternion global conjugate gradient method,and the inexact quaternion Hermitian and skew-Hermitian split iteration method to solve large-scale quaternion matrix equations.First,we analyze and introduce the quaternion global conjugate gradient method,utilizing two real inner products in the quaternion field,for solving the quaternion positive definite linear systems with multiple right-hand sides AX = B.Secondly,based on the structure-preserving tridiagonalization algorithm and the incomplete inverse triangular decomposition,the preconditioned quaternion global conjugate gradient method is proposed.Thirdly,by defining the quaternion linear matrix operator,the quaternion global conjugate gradient method is extended to solve the quaternion Hermitian positive definite Sylvester matrix equation AX + X B = C,and a preconditioned method is designed for a special class of equations.Besides,we introduce the quaternion global CGW method,put forward the inexact quaternion Hermitian and the skew-Hermitian splitting iteration method for solving the quaternion Sylvester matrix equation AX + X B = C.The original problem is decomposed into two kinds of sub-problems concerning shifted Hermitian positive definite matrices and shifted skew-Hermitian matrices among this method,we solve them by quaternion global conjugate gradient method and quaternion global CGW method respectively.Finally,numerical examples are given to verify the effectiveness of the proposed methods.The color image restoration experiment demonstrates the practical application of the proposed methods.