Numerical Methods For Several Classes Of Nonlinear Equations

This thesis focuses on fast numerical iterative algorithms for solving the following equations:the nonsymmetric algebraic Riccati equation,the spatial fractional Ginzburg-Landau equation,and the fractional Schr(?)dinger equation.For the nonsymmetric algebraic Riccati equation,a two-step Newton-type iterative algorithm with third-order convergence is proposed,and the monotonic convergence of the vector sequence generated by this algorithm is proved.Numerical experiments show that this algorithm exhibits better convergence behavior than two other Newton-type iterative algorithms.For the linear system with a Toeplitz matrix obtained by discretizing the spatial fractional Ginzburg-Landau equation,a new preconditioner is proposed based on approximating the Toeplitz matrix with a circulant matrix and applying a preconditioning technique.The eigenvalue distribution of the preconditioning matrix is also analyzed.To avoid the inverse of the coefficient matrix,an effective block-splitting iterative algorithm is proposed.The convergence of this algorithm is shown,and numerical results show that both numerical algorithms are superior to other methods.For the linear system with a block Toeplitz matrix obtained by discretizing the twodimensional spatial fractional Ginzburg-Landau equation,a complex diagonal-plus-Toeplitzlike splitting iteration algorithm is proposed.The convergence of this algorithm is proved.Additionally,the circulant preconditioning technique and inverse fast Fourier transform are adopted to improve the convergence property of the proposed method.Numerical results verify its effectiveness.For a series of dense complex linear systems obtained by discretizing the coupled spatial fractional Schr(?)dinger equation,a fast block Gauss-Seidel iterative algorithm is proposed.In each iteration,only two tridiagonal linear systems need to be solved,which greatly reduces the computational complexity.The convergence of this algorithm is proved,and numerical results confirm that the proposed algorithm outperforms some existing algorithms.