SIAM Journal on Numerical Analysis

The problem is, given A, B, C, to find D such that $\left\| ( {\begin{array}{*{20}c} A & C \\ B & D \\ \end{array} )} \right\| \leqq \mu $; here we deal with Hilbert-space operators, A, B, and C are given, and $\mu $ is a given positive number. ...

For nonnormal matrices the norms of the residuals of approximate eigenvectors are not by themselves sufficient information to bound the error in the approximate eigenvalue. It is sufficient however to give a bound on the distance to the nearest matrix for ...

Many powerful methods for solving systems of equations can be regarded as projection methods. Most of the projection methods known for solving linear systems are orthogonal projection methods but little attention has been given to the class of ...

The Birkhoff interpolation polynomial $P(X,t)$, $0 \leqq t \leqq 1$ of a given function depends upon the knots of interpolation, $X:0 \leqq x_1 < x_2 < \cdots < x_m \leqq 1$. There is a natural extension of $P(X,t)$ to the case when some of the knots ...

The main result of this paper is a formula relating a multivariate B-spline and a certain directional derivative to a lower order B-spline.

The structure of discontinuous B-splines is given. A formula relates the discontinuous k-variate B-spline to a $(k - ...

While there is now an extensive literature on the approximation of the eigenvalue problem for compact operators, these theories do not include the classical finite-difference approximation. In the self-adjoint case the problem is resolved through the ...

In this paper the Neumann difference problem for Poisson's equation and the difference problemm for the boundary value problem of the theory of elasticity, when the stresses are given in the boundary of the domain, are considered. This paper investigates ...

The main objective of this paper is to treat the correction cycle of multigrid as a Newton-like method and to analyze it together with relaxation via a natural decomposition of the grid function space. The purpose is to provide a simplified view of ...

A semidiscrete finite element method for a two-point boundary value problem of a version of the Boussinesq equation is analyzed. The method is a combination of the $H^1 $-Galerkin method and the standard Galerkin method. Global convergence estimates and ...

For parabolic equations in one space variable with a strongly coercive self-adjoint $2m$th order spatial operator, a kth degree Faedo–Galerkin method is developed which has local convergence of order $2(K + 1 - m)$ at the knots for the first $m - 1$ ...

In a recent paper, Gourlay (in Advances in Computer Methods for Partial Differential Equations II, IMACS, 1977) has considered several block hopscotch methods to solve parabolic and elliptic partial differential equations. In this paper, a new block ...

The backward Euler method is applied for the discretization in time of a general homogeneous parabolic equation in weak form. A short proof is given that, with k the time step, the norm of the error at time t is bounded by $Ckt^{ - 1} $ times the norm of ...

We define general Runge–Kutta approximations for the solution of stochastic differential equations (sde). These approximations are proved to converge in quadratic mean to the solution of an sde with a corrected drift. The explicit form of the correction ...

We present a method for solving the linear integral equation \[ \int _0^T {R(\exp ( -| x - y| )f(y)dy = S(\exp ( - | t - x| )),\quad 0 \leqq x \leqq T} \] when R and S are given by power series about the origin. The method has a simple derivation and is ...

A convergence theorem for Marti’s method for solving the Fredholm first kind equation $Kf = g$ is corrected. Additional convergence results are given. Marti’s method is shown to be well-posed under perturbations in g and under certain types of ...

The linear one-dimensional advection equation with variable coefficients and nonperiodic boundary conditions is considered. We investigate some spectral and pseudospectral (collocation) approximations of Jacobi and Legendre type (in the latter case the ...

We describe an algorithm for the approximation of a zero $z^ * $ of an M-function. The essential ingredient is a combinatorial search process for optimal sub- and supersolutions of $z^ * $in a set of equidistant gridpoints. We discuss computational ...

The study of various equilibrium phenomena leads to nonlinear equations (1) $F(y,u) = 0$, where $y \in R^n $ is a vector of behavior or state variables, $u \in R^p $ a vector of $p \geqq 2$ parameters or controls and $F:D \subset R^n \times R^p \to R^n $...