In this paper, we establish the semi-local convergence theorem of the Brent method with regional estimation. By an in-depth investigation in to the algorithm structure of the method, we convert the Brent method into an approximate Newton method with a special error term. Based on such equivalent variation, under a similar condition of the Newton-Kantorovich theorem of the Newton method, we establish a semi-local convergence theorem of the Brent method. This theorem provides a sufficient theoretical basis for initial choices of the Brent method.  

A new approach for stochastic approximation in real time is developed. A number of processors are simultaneously active to carry out a computing task. All processors work on the same system with different starting time. After each iteration, computed date are passed to the next processor on line. Interacting tasks and iterative instructions are carried through pipelining of computation and communication. Asymptotic properties of the algorithm are developed, and comparisons of the performance between the new algorithm and the classical one are made.

While the numerical solution of one-dimensional Volterra integral equations of the second kind with regular kernels is well understood, there exist no systematic studies of asymptotic error expansion for the approximate solution. In this paper, we analyse the Nystrom solution of  one-dimensional nonlinear Volterra integral equation of the second kind and show that approximate solution admits an asymptotic error expansion in even powers of the step-size $h$, beginning with a term in $h^2$. So that the Richardson's extrapolation can be done. This will increase the accuracy of numerical solution greatly.

The nonlinear singular perturbation problem is solved numerically on non-equidistant meshes which are dense in the boundary layers. The method presented is based on the numerical solution of integral equations [1]. The fourth order uniform accuracy of the scheme is proved. A numerical experiment demonstrates the effectiveness of the method. 

A simple finite element method for the Reissner-Mindlin plate model in the primitive variables is presented and analyzed. The method uses conforming linear finite elements for both the transverse displacement and rotation. It is proved that the method converges with optimal order uniformly with respect to thickness. It is simpler and more economical than the Arnold-Falk element.

To obtain the approximate solution of the nonlinear ordinary differential equations requires the solution to systems of nonlinear equations. The authors study the conditions for the existence and uniqueness of the solutions to the algebraic equations in multiderivative block methods.

An optimal order of the multigrid method is given in energy-norm for the nonconforming finite element for solving the biharmonic equation, by using the nodal interpolation operator as the transfer operator between grids.

A method for construction of a preconditioner for the capacitance matrix on the interface is described. The preconditioner is determined by a subdomain covering the interface, and the condition number of the preconditioned matrix is dependent on the width of the covering subdomain and independent of the discrete mesh size and discontinuity of the coefficients of the differential operator. Some applications of our theory are presented at last.

This paper studies the following two problem:
Problem Ⅰ. Given $X,B∈R^{n×m}$, find $A∈P_{s,n}$, such that $AX=B$, where
$P_{s,n}$={$A∈SR^{n×n}|x^T AX≥0，∀S^Tx=0$ , for given $S∈R^{n×p}_p$}.
Problem Ⅱ. Given $A^*∈R^{n×n}$, find $\hat{A}∈S_E$, such that $||A^*-\hat{A}||$=inf$_{A∈S_E}||A^*-A||$ where $S_E$ denotes the solutions set of  Problem Ⅰ.
The necessary and sufficient conditions for the solvability of Problem Ⅰ, the expression of the general solution of Problem Ⅰ and the solution of Problem Ⅱ are given for two case. For the general case, the equivalent form of conditions for the solvability of Problem  Ⅰ is given.

In this paper, we consider the numerical solution of quasi-parabolic equations of higher order by a spectral method, and propose a computational formula. We give an error estimate of approximate solutions, and prove the convergence of the approximate method and numerical stability on initial values. Under certain conditions, which are much weaker than the conditions in [6], we gain the same convergence rate as in [6].