Numerical Methods For Two Nonlinear Matrix Equations

This thesis focuses on efficient numerical algorithms for two class of nonlinear matrix equations.The first one is a nonsymmetric algebraic Riccati equation from transport theory,and the other one arises in Green's function calculations in nano research.For solving the nonsymmetric algebraic Riccati equation,a superquadratically convergent modified Newton's method is proposed.Then we present the monotone convergence anaiysis of this method.Numerical experiments show that the proposed method is effective,and outperforms the Newton method,especially in the near-to-singular case.Furthermore,for solving the nonsymmetric algebraic Riccati equation,we apply Anderson acceleration technique to the fixed-point iterative method.Based on QR factorizations involving unconstrained least-squares problem,a new efficient fixed-point type iterative algorithm is proposed.Numerical experiments confirm our proposed algorithm brings significant improvements on the computational performance.In particular,the algorithm has greater improvements in the near-to-singular case.For the nonlinear matrix equations arising in nano research,we present a fast convergent two-step Newton method.To solve the Stein equation at each Newton step in the implementation,by using the Schur decomposition instead of the Kronecker inner product method,we obtain a new improved algorithm.Some numerical examples illustrate the efficiency of the proposed method.