The nonlinear singular problem $f(u)=0$ is considered. Here $f$ is a $C^3$ mapping from $E^n$ to $E^n$. The Jacobian matrix $f'(u)$ is singular at the solution $u^*$ of $f(u)=0$. A new acceleration method in the homotopy Newton's continuation is proposed. The quadratic convergence of the new algorithm is proved. A numerical example is given.

This paper deals with an inverse problem of a system of ordinary differential equations whose coefficient matrix is the so-called compartmental matrix.This problem arises in a variety of areas such as pharmacokinetics, biology, economics, and so on. A numerical method applicable to the cases of non-unique solutions is developed, by which the projection of the starting value of iteration onto the manifold of solution can be obtained. The convergence of the method is proved. A few examples are examined, which shows the effectiveness of the given method.

The technique described in [9] is used to discuss the sensitivity of multiple eigenvalues and corresponding eigenspaces of symmetric eigenproblems analytically dependent to several parameters. The results may be useful for investigating problems of stability and response analysis of linear structures.

Many types of nonlinear systems, which can be solved by ordered iterative methods, are discussed in unified form in the present paper. Under different initial conditions, some generalized ordered iterative methods are given, and the existence and uniqueness of the solution and the convergence of the methods are proved.

The Chebyshev polynomials have good approximation properties which are not affected by boundary values. They have higher resolution near the boundary than in the interior and are suitable for problems in which the solution changes rapidly near the boundary. Also, they can be calculated by FFT. Thus they are used mostly for initial-boundary value problems for P.D.E.'s (see [1, 3-4, 6, 8-11]). Maday and Quarterom discussed the convergence of Legendre and Chebyshev spectral approximations to the steady Burgers equation. In this paper we consider Burgers-like equations. $$\begin{cases} ∂_iu+F(u)_x-vu_{zx}=0, & -1≤x≤1, 0＜t≤T \\ u (-1,t) =u (1,t) =0, & 0≤t≤T  & (0.1) \\ u (x,0) =u_0(x), & -1≤x≤1 \end{cases}$$ where $F\in C(R)$ and there exists a positive function $A\in C(R)$ and a constant $p＞1$ such that $$|F(z+y)-F(z)|\leq A(z)(|y|+|y|^p).$$ We develop a Chebyshev spectral scheme and a pseudospectral scheme for solving (0.1) and establish their generalized stability and convergence.  

This is the first half of an article which develops a theory the hierarchical elements for the $h$-$p$ version of the finite element method in the two-dimensional case. The approximation properties of the hierarchical elements are discussed. The second part will address the convergence rate when geometric meshes are used.

We are interested in solving elliptic problems on bounded convex domains by higher order methods using the Richardson extrapolation. The theoretical basis for the application of the Richardson extrapolation is the asymptotic error expansion with a remainder of higher order. Such an expansion has been derived by the method of finite difference, where, in the neighborhood of the boundary one must reject the elementary difference analogs and adopt complex ones. This plight can be changed if we turn to the method of finite elements, where no additional boundary approximation is needed but an easy triangulation is chosen, i.e. the higher order boundary approximation is replaced by a chosen triangulation. Specifically, a global error expansion with a remainder of fourth order can be derived by the linear finite element discretization over a chosen triangulation, which is obtained by decomposing the domain first and then subdividing each subdomain almost uniformly. A fourth order method can thus be constructed by the simplest linear finite element approximation over the chosen triangulation using the Richardson extrapolation.

The Rayleigh Quotient Iteration (RQI) is a very popular method for computing eigenpairs of symmetric matrices. It is a special kind of inverse iteration method using the Rayleigh Quotient as shifts. Unfortunately, poor initial approximations may render RQI to slow convergence or even to divergence, In this paper we suggest two kinds of numbers each of which can be used instead of the Rayleigh Quotient as a shifts in the RQI. We call the iteration using the new shifts the Modified Rayleigh Quotient Iteration (MRQI). It has been proved that the MRQI always converges and its convergence rate is cubic.

When we study the oscillation of a physical system near its equilibrium and ignore dissipative effects, we may assume it is a linear Hamiltonian system (H-system), which possesses a special symplectic structure. Thus there arises a question: how to take this structure into account in the approximation of the H-system? This question was first answered by Feng Kang for finite dimensional H-systems.
We will in this paper discuss the symplectic difference schemes preserving the symplectic structure and its related properties, with emphasis on the infinite dimensional H-systems.