Because the nonlinear Schrödinger equation is met in many physical problems and is applied widely, The research on well-posedness for its solution and numerical methods has aroused more and more interest. The self-adjoint case has been considered by many authors. For a class of system of nonlinear and non-self-adjoint Schrodinger equations which refers to excitons occurring in one dimensional molecular crystals and in a spiral biomolecules, Guo Boling studied in [6], the pure initial and periodic initial value problems of this system and obtained the existence and uniqueness of its solution. In [7] we discuss the difference solution of this system and obtained its error estimate. In this paper, we shall study the finite element method for the periodic initial value problem of this system. Just as Guo pointed out in [6], since it has a non-self-adjoint term, it not only brings about trouble in mathematics, but also creates more difficulty in numerical analysis.
Our analysis will show that for this system, in theoretically we can obtain the same results as when it has no non-self-adjoint term.  

A new predictor-corrector difference scheme is described for solving the time-dependent Navier-Stokes equations in three-dimensional spherical coordinates. A boundary condition for the pressure is deduced by auxiliary velocity. A multigrid algorithm is employed in solving equations of the pressure. An example of application of this scheme is computed and its results are presented.

Many types of nonlinear systems can be solved by using ordered iterative methods. These systems are discussed in [2] in a unified form for five different initial conditions. This paper is a continuation of [2]. Under arbitrary initial conditions, some iterative methods are given, and several theorems for the existence and uniqueness of the solution and convergence of the methods are proved.

Some results of the author are used to discuss the sensitivity of zero singular values and multiple singular values of a real matrix analytically dependent on several parameters.

This paper discusses the conditioning of algebraic Riccati equations, i.e. the influence of perturbations in data on the positive semi-definite solution. A perturbation bound for the solution is given.

The parallel arithmetic complexities for computing generalized inverse $A^+$, computing the minimum-norm least-squares solution of $Ax=b$, computing order $m+n-r$ determinants and finding the characteristic polynomials of order $m+n-r$ matrices are shown to have the same grawth rate. Algorithms are given that compute $A^+$ and $A_{MN}^+$ in $O(\log r\dot \log n+\log m)$ and $O(\log^2n+\log m)$ steps using a number of processors which is a polynomial in $m, \ n$ and $r$ $(A\in B_r^{m\times n},r=rank \ A)$.

In this paper, we develop, analyze, and test a new algorithm for nonlinear least-squares problems. The algorithm uses a BFGS update of the Gauss-Newton Hessian when some heuristics indicate that the Gauss-Newton method may not make a good step. Some important elements are that the secant or quasi-Newton equations considered are not the obvious ones, and the method does not build up a Hessian approximation over several steps. The algorithm can be implemented easily as a modification of any Gauss-Newton code, and it seems to be useful for large residual problems.

We give an approach for finding a global minimization with equality and inequality Constraints.
Our approach is to construct an exact penalty function, and prove that the global minimal points of this exact penalty function are the primal constrained global minimal points. Thus we convert the problem of global constrained optimization into a problem of global unconstrained optimization.
Furthermore, the integral approach for finding a global minimization for a class of discontinuous functions is used and an implementable algorithm is given.