Several Effective Algorithm Of Truncated Complex Singular Value Decomposition And Low Rank Approximation Solution Of Lyapunov Equation

The truncated singular value decomposition is very important,which is widely applied in ill-posed model problems.The ill-posed of the model is mainly reflected in the amplification of parameters and their variances by small singular values of the coefficient matrix.The basic method of the decomposition is truncate the small singular values and reconstruct the coefficient matrix to weaken the ill-condition of the model.The Low-rank approximation solution of Lyapunov equation have important applications in many fields,such as numerical algebra,non-linear analysis,systems and control theory,transport theory,signal processing and engineering computing fields.The concrete content is organized as follows:Firstly,we study the two novel and fast Riemannian second-order approaches for solving the truncated complex singular value decomposition.A hybrid Riemannian Newtontype algorithm with globally and quadratically convergent is proposed firstly,in which the involved Newton's equation is converted into a symmetric linear system by Kronceker product and complex matrix straightening operator.A Riemannian trust-region method based on the proposed Newton method is further provided,in which a new trust-region subproblem is proposed basing on our expression of the Riemannian Hessian of the objective function.Some numerical tests are given to demonstrate the efficiency of the proposed methods.Then,we focus on the low rank approximation of generalized Lyapunov equation We propose a new variable in order to reformulate the original problem into a unconstrained optimization.To solve the transformed unconstrained optimization problem,two kinds of second-order optimization algorithms,Newton method and trust region method,are designed.We let the Steepest descent method with the Barzilai-Borwein(BB)step size combine the Newton's method to produce a hybrid algorithm,which is globally and quadratically convergence in practical computations.For the trust region algorithm,the classical truncated conjugate gradient method is used to solve the corresponding trust region subproblems.Numerical experiments show the effectiveness of the algorithm.