Study On The Method And Theory Of The Generalized Coupled Sylvester Transpose Matrix Equations

Constraint matrix equations have numerous applications in many scientific and engineering problems.Many problems such as biology,optics,electricity,solid mechanics,cybernetics,image restoration,signal processing,neural networks,model reduction,numerical solutions of differential equations,dynamic systems,structural design,parameter identification,etc.will eventually be transformed into solution constraints matrix equations.Because a large number of questions in these areas have stimulated the rapid development of constraint matrix equation problems,solving constraint matrix equation problems has become one of the most popular and active research topics in the field of numerical algebra.This essay mainly studies the theory and solving method of two types of Sylvester transpose matrix problems,the contents are as follows:1.The modified conjugate gradient method(MCG)of the generalized coupled Sylvestre-transpose matrix equations(?)over generalized reflexive matrix solution,bisymmetric matrix solution and Hermitian reflexive matrix solution is presented.We discuss the convergence of the algorithm.It is assumed that the exact solution can be obtained within finite iterative steps in the absence of roundoff-error.The specific expression of the initial matrix of its least Frobenius norm solution is given,and the least Frobenius norm solutions of the two matrix equations are further solved to obtain the optimal approximation generalized reflexive,bisymmetric and Hermitian reflexive solutions.Finally,numerical examples of corresponding constraint solutions are given to verify the effectiveness of the given algorithm.2.We present a modified conjugate gradient method(MCG)of the generalized coupled Sylvester-transpose linear matrix equations(?)over bisymmetric solutions and Hermitian reflexive solutions.It also proves that the algorithm converges on the complex and real domains.The solution group can be obtained within finite iteration steps in the absence of round-off errors.The least Frobenius norm solution group of the generalized coupled Sylvester-transpose matrix equations can be derived when a suitable initial matrix group is chosen.In addition,for a given bisymmetric or Hermitian reflexive matrix group,the optimal approximation solutions can be obtained by finding the least Frobenius norm solutions of new coupled Sylvester-transpose matrix equations.Finally,numerical experiments are performed to explain the effectiveness of the algorithm.