In this paper, the isoparametric element of 2-degree Lagrange type for second order elliptic P.D.E, with nonhomogeneous Dirichlet boundary value is considered. We prove $\|u-u_h\|_{1,\Omega}=O(h^2)$, which is the same as $\|u-u_h\|_{1,\Omega_h}=O(h^2)$ where $\Omega$ and $\Omega_h$ in $R^2$ are the domain of the boundary value problem and isoparametric triangulation domain respectively.

Starting from the canonical boundary reduction, this paper studies an approximate differential boundary condition and an approximate integral boundary condition on an artificial boundary for the exterior problem of a harmonic equation, and gives an error estimate for the latter. This estimate reveals the relationship between the error and the approximate grade boundary conditions as well as the radius of the artificial boundary.

The solvability of the rational Hermite-interpolation problem is investigated through an approach similar to that developed in an earlier paper [1] for the ordinary case. However, the subsequent deduction of analogous results involves much complications. The Quasi-Rational Hermite Interpolant $r_{mn}^{*}$ is introduced. In the case of $r_{mn}^{*}$ being nondegenerate, its explicit expression is given. Working with the notion of l-fold unattainable point and using algebraic elaboration, we have successively established several theorems concerning interpolating properties of $r_{mn}^{*}$ and, in particular, obtained existence theorems for the solution of the proposed problem.  

The convergence of diagonal elements of an irreducible symmetric triadiagonal matrix under QL algorithm with some kinds of shift is discussed. It is proved that if $\alpha_1-\sigma$→0 and $\beta_j$→0, j=1,2,...,m, then $\alpha_j$→$λ_j$ where $λ_j$ are m eigenvalues of the matrix, and $\sigma$ is the origin shift. The asymptotic convergence rates of three kinds of shift, Rayleigh quotient shift, Wilkinson's shift and RW shift, are analysed.  

Perturbation theorems for the spectrum of a regular matrix pencil $λA-B$ are given. As it may include points near or at infinity the Euclidean distance is not appropriate. We use the chordal metric and the distances. For those purpose we develop here an algebraic treatment of matrix pairs, with special reference to diagonable and definite pairs, using ideas from the theory of matrix polynomials.

The classical Eneström-Kakeya Theorem, which provides an upper bound for the moduli of zeros of any polynomial with positive coefficients, has been recently extended by Anderson, Saff and Varga to the case of any complex polynomial having no zeros on the ray [0,$+∞$). Their extension is sharp in the sense that, given such a complex polynomials $p_n(z)$ of degree $n≥1$, a sequence of multiplier polynomial can be found for which the Eneström-Kakeya upper bound, applied to the products $Q_{mi}(z)$ · $p_n(z)$, converges, in the limit as $i$ tends to $∞$, to the maximum of the moduli of the zeros of $p_n(z)$. Here, the rate of convergence of these upper bounds is studied. It is shown that the obtained rate of convergence is best possible.