SIAM Journal on Numerical Analysis

Any algebraic equation of degree $n \geqq 2$ can be easily transformed into the “reduced form :” \[ f_n (z): = z^n + a_2 z^{n - 2} + \cdots + a_n = 0,\quad \left| {a_v } \right| \leqq 1\qquad (v = 2, \cdots ,n).\]

Denote the zeros of $f(z)$ by $\zeta _...

Methods for solving symmetric indefinite systems are surveyed including a new one which is stable and almost as fast as the Cholesky method.

A backwards error analysis of the diagonal pivoting method for solving symmetric (indefinite) systems of linear equations shows that the elements of the associated error matrix can be bounded in terms of the elements of the reduced matrices. The elements ...

This paper discusses some interpolation formulas \[u(x_{ * 1} , \cdots ,x_{ * n} ) \simeq \sum _{i = 1}^N {A_i u(v_{i1} , \cdots ,v_{in} )} \] which are exact for all harmonic polynomials in n variables of degrees $\leqq 2$ and $\leqq 3$. The points $(v_{...

Truncation error bounds for continued fractions are obtained in terms of general conditions which ensure that the approximants $\{ {w_n } \}$ form a simple sequence; i.e., that $| {w_{n + m} - w_n } | \leqq c| {w_n - w_{n - 1} } |$, where c is a ...

Difference equation problems are studied whose solutions are estimates of the solutions of the eigenvalue problem \[\begin{gathered} Ly \equiv y'' - \frac{1}{x}y' + \frac{{v^2 }}{{x^2 }}y = \lambda y,\quad x \in (0,1), \hfill \\ y(0) = 0,\quad y(1) = 0,...

Let $\{ {T_j ( X )} \}_{j = 0}^n $ be the Chebyshev polynomials of the first kind, where $T_j (x)$ is the polynomial of jth degree. It is shown that there exists a unique best approximation of $T_1 (x)$ with respect to the linear space spanned by ...

There are several very fast direct methods which can be used to solve the discrete Poisson equation on rectangular domains. We show that these methods can also be used to treat problems on irregular regions.

In this paper we adapt a technique of J. Nitsche to obtain new sharp a priori $L^2 $-bounds for the error involved in approximating the solutions to a wide variety of boundary value problems for nonlinear elliptic partial differential equations by the ...

Chebyshev approximation by nonlinear families on a general compact space is studied. Attention is restricted to approximants satisfying a local Haar condition. A necessary and sufficient condition for the approximant to be locally best is given. A linear ...

A technique for modifying a truncated Chebyshev series so that the values of the approximated function at the endpoints are exactly duplicated is presented.

Let $R_n $ denote an n-dimensional region and w a weight function defined on $R_n $. This paper is concerned with the existence and construction of approximations of the type \[ \int_{R_n} {wf \cong \sum\limits_{k = 1}^N {A_k f(\mu _k )} } ,\] where the ...

It is well known that a damped or underrelaxed Newton’s method will sometimes solve a system of nonlinear equations when the full Newton’s method cannot. This happens, for example, when only a poor initial approximation to the solution is known. By ...

This paper presents a class of numerical methods for the approximate solution of ordinary differential equations where the derivatives depend on the history of the solution. These methods, which are self-starting and always stable, are analogues of the ...

Let A be a closed linear operator on a separable Hilbert space $\mathcal{H}$ whose domain is dense in $\mathcal{H}$ Let $\mathcal{X}$ be a subspace of $\mathcal{H}$ contained in the domain of A and let $\mathcal{Y}$ be its orthogonal complement. Let B and ...