SIAM Journal on Numerical Analysis

We investigate the validity of the process of deferred approach to the limit in the approximation of eigenvalues and eigenfunctions of integral equations, and we indicate the practical significance of the theory which we develop.

In this paper we derive cubature error bounds for the product rule generated by two Gauss-Legendre quadrature rules. An analogue to the fundamental theorem of algebra, for monosplines, is used to characterize the kernels in the two-dimensional form of ...

In this paper, we apply a 2-dimensional extension of Peano’s theorem, due to Sard, to derive error bounds for a fully symmetric cubature rule. The weight function is assumed to be unity and the region of integration is the square $[ { - 1,1} ] \times [ { -...

A procedure for obtaining low order, high accuracy spline function approximations for solutions of the initial value problem in ordinary differential equations is presented. The method is a single step method of order m. The spline is of class $C^1 $, but ...

It is shown that if $x \in C^4 [a,b]$ is approximated by a natural cubic spline, then the error is $O(h^4 )$ in a closed interval which is asymptotic to $[a,b]$ as h, the maximum interval length, decreases to zero. A by-product of the technique used is ...

An existence and uniqueness theorem is proved for an implicit free boundary problem for the heat equation—denoted as “implicit” due to the absence of the derivative of the free boundary in a free boundary condition.

Apth order iterative method for computing $A^\dag B$ is studied, where $p \geqq 2$, A and B are arbitrary complex matrices with equal number of rows and $A^\dag $ is the Moore–Penrose generalized inverse of A. The rate of convergence and the ...

Two fundamental theorems concerning quadratures are given here. The first theorem gives a constructive criterion for determining if a quadrature formula is convergent in the space of continuous functions on $[0,1]$ with maximum norm. The second theorem ...

In this paper we shall give an extension of Ostrowski's point of attraction theorem to multistep iterative procedures for finding a zero of a nonljnear function defined on $R^n $. We shall also obtain a rate of convergence statement in terms of the ...

The Dirichlet problem is formulated in terms of an integral equation. The integral equation is solved in an approximate manner and a system of linear algebraic equations is obtained. It is shown that an iteration process converges if any tangent to the ...

Multistep predictor-corrector methods are commonly used for the numerical solution of ordinary differential equations. In its simplest form, a k-step method with accuracy of order exceeding $k+2$ is unstable. Methods such as those of Gragg and Stetter [7] ...