In this short not, examples are constructed to show that a recent algorithm given by Soliman, Christensen and Rouhi [1] may give a non-optimal solution.

In this paper, the numerical solution of the matrix problems over a ring of integers is discussed. The rank-$k$ updating algorithm for the exact inversion of a matrix is proposed. This algorithm is generally more effective than Jordan elimination. The common divisor of the numbers involved is reduced to avoid over-swelling of intermediate numbers.

A class of three-level six-point explicit schemes $L_3$ with two parameters $s, p$ and accuracy $O(\tau h+h^2)$ for a dispersion equation $U_1=aU_{xxx}$ is established. The stability condition $|R|\leq 1.35756176$ $(s=3/2,p=1.184153684)$ for $L_3$ is better than $|R|$ ＜ 1.1851 in [1] and seems to be the best for schemes of the same type.  

The equilibrium strategy for $N$-person differential games can be obtained from a min-max problem subject to differential constraints. The differential constraints are treated here by the duality and penalty methods.
We first formulate the duality theory. This involves the introduction of $N+1$ Lagrange multipliers: one for each player and one commonly shared by all players. The primal min-max problem thus results in a dual problem, which is a max-min problem with no differential constraints.
We develop the penalty theory by penalizing $N+1$ differential constraints. We give a convergence proof which generalizes a theorem due to B.T. Polyak.

The equilibrium strategy for $N$-person differential games can be obtained from a min-max problem subject to differential constraints. The differential constraints can be treated by the duality and penalty methods and then an unconstrained problem can be obtained. In this paper we develop methods applying the finite element methods to compute solutions of linear-quadratic $N$-person games using duality and penalty formulations.
The calculations are efficient and accurate. When a (4,1)-system of Hermite cubic splines are used, our numerical results agree well with the theoretical predicted rate of convergence for the Lagrangian. Graphs and numerical data are included for illustration.  

Optimal interpolation problems of scattered data on a circular domain with two different types of boundary value conditions are studied in this paper. Closed-form optimal solutions, a new type of spline functions defined by partial differential operators, are obtained. This type of new splines is a generalization of the well-known $L_g$-splines and thin-plate splines. The standard reproducing kernel structure of the optimal solutions is demonstrated. The new idea and technique developed in this paper are finally generalized to solve the same interpolation problems involving a more general class of partial differential operators on a general region.  

It is shown that the error corresponding to certain spline collocation approximations for nonlinear Volterra integral equations of the second kind is the solution of a nonlinearly perturbed linear Volterra integral equation. On the basis of this result it is possible to derive general estimates for the order of convergence of the spline solution at the underlying mesh points. Extensions of these techniques to other types of Volterra equations are indicated.  

In this paper we study the MSOR method with fixed parameters, when applied to a linear system of equations $Ax=b(1)$, where $A$ is consistently ordered and all the eigenvalues of the iteration matrix of the Jacobi method for (1) are purely imaginary. The optimum parameters and the optimum virtual spectral radius of the MSOR method are also obtained by an analysis similar to that of [5, pp. 277-281] for the real case. Finally, a comparison of the optimum MSOR method with the optimum SOR and AOR methods is presented, showing the superiority of the MSOR one.

This paper deals with the superconvergence phenomena for Galerkin approximations of solutions of Biot's dynamic equations describing elastic wave propagation in fluid-saturated porous media. An asymptotic expansion to high order of Galerkin solutions is used to derive these results.  

In this paper we discuss the numerical methods with second-order accuracy for solving stochastic differential equations. An unbiased sample approximation method for $I_n=\int ^{t_{n+1}}_{t_n}(B_u-B_{t_n})^2du$ is proposed, where {$B_u$} is a Brownian motion. Then second-order schemes are derived both for scalar cases and for system cases. The errors are measured in the mean square sense. Several numerical examples are included, and numerical results indicate that second-order schemes compare favorably with Euler's schemes and 1.5th-order schemes.