Study Of The Iterative Method For Some Quaternion Matrix Equations

As the extending of applying scope of quaternion matrices,it is of very important to numerical calculating of quaternion matrix.However,quaternion multiplication is non-exchangeable,which brings a lot of difficulty when deal with numerical calculation. This paper adopts new algebra structure of quaternion matrix,to study how to apply a variety of iterative algorithm to quaternion linear systems.We give different kinds of iterative methods of some quaternion matrix equations,and using structure-preserving property for complexification operator of the quaternion matrices,which analysis and character the convergence of various algorithms.At the same time,the numerical examples indicated the feasibility of the proposed algorithm.The main contents of the dissertation are as follows:1.First of all,the complexification of a quaternion matrix is presented,after-wards, calculation method of Moore-Penrose generalized inverse of a quaternion matrix is discussed by using structure-preserving property of the complexification operator. Meanwhile,numerical method of solving the quaternion matrix equation AXB=C is discussed as well.2.QJ,QGS and QSOR iterative methods is established for quaternion linear systems AX=B,the necessary and sufficient conditions of the three iterative schemes convergent is portrayed by use of the right eigenvalue maximum norm of the quaternion matrix,at the same time,some of the sufficient conditions of the iterative convergence is provided.Whereas,applying the structure preserving property of the complex representation operation of the quaternion matrix,the three iterations are transformed into the iterations in complex field equivalently,and iterative calculation is accomplished.3.Single parameter iterative method of quaternion matrix equation AXB+CXD= F is discussed.Basic thought:by using matrix transform,to construct a iterative formula with parameter X^(k+1)=PX^(k)G+F₀,and from right eigenvalue maximum norm of the quaternion matrix,the necessary and sufficient conditions of the iterative convergence is derived.Meanwhile,the method of selecting parameter is given too.4.Multiple parameter iteration-correction method for solving the generalized mixed-type Lyapunov matrix equation AX+XB+CXD=F is given,which is composed of structuring the iteration formula,the necessary and sufficient conditions of iterative convergence and the method of choose parameters.Furthermore,improve convergence by means of a whole correction model.