Research On The Iterative Algorithm For Several Kinds Of Constrained Coupled Matrix Equation Problems

Linear matrix equations have widely applied to parameter identification,strue—ture design．1inear system&automatic control theory．vibrational theory,quantummechanics and 0ptoekctronics,ete With regard to the optimum approximation andthe least squares problem of the linear matrix equations with single matrix variant,alot of research results have been achieved However,Many problems also need to bestudied for coupled matrix equations with multi—matrices variant In the thesis,severalkinds of matrix equation problems are studied1．Thematrix equationproblemswithtransposeare studied First of all,we take no account of constrained conditionWhen the matrix equations are consistent,iterative algorithnms are constructed for theCOlIlllion solution and the optimal approximate solution to a given matrix When thematrix equations are inconsistent．an iteration method is proposed for the least squaresproblem；and then,we take account of(R,s)一eonstrainted condition,iterative methodsare presented for the(R,s)一symmetric solution,the(R,s)一anti symmetric solution andthe least squares(R,s)一symmetric solutions,the least squares(R,s)一anti symmetric SO—lutions of the matrix equations Using the proposed iterative method．for any initialvalue,we can obtain the solution of the corresponmng problem within finite iteration steps in the absence of roundoff errors2．The least squares and optimum approximation problems are dieussed on thecoupled matrix equation AXB+CYD=M with the same species matrix pair(X,Y)Using Taylor formula and the properties of convex matrix function,it is obtained thatthe equivent normal equation ofthe least squares problem:min(xyl∈r IAXB+CYD．where F is the set of the same species matrix pair The obtained equivalent normalmatrix equations are coupled matrix equations,then,a conjugate gradient algorithmis proposed for solving the coupled matrix equations Use the iterative algorithm anditerate finite steps,the least squares solutions of the problem are got with the samespecies matrix pair3．The least squares and optimum approximation problems are investigated overgeneralized coupled Sylvester matrix equations with the different species matrix pair(x,Y)After the equivalent normal equations are obtained,an iterative method is con—strueted to find the solutions of the least squares problem with the different speciesmatrix pair For an arbitrary given matrix pair(X0,Y0),by translation,a new gener—alized coupled Sylvester matrix equations can be achieved,and after the least squares minimal normal solution of new matrix equations is acquired,by translation again,the optimal approximat solution to(XO,Y0)of the primary matrix equations can beachieved4．Applying a hierarchical identification principle,a gradient based iterative al-gorithm mr more comprehensive coupled matrix equations:studied．the basic idea iS to regard the unknown matrices X and Y to be solved as theparameters of a system to be identified It is proved that the iterative solution alwaysconverges to the exact solution mr any initial values,and a conservative choice of theconvergence factor is given...