Alan Feldstein

Convergence Estimates for the Distribution of Trailing Digits

This paper analyzes the distribution of trailing digits (tail end digits) of positive real floating-point numbers represented in arbitrary base <italic>β</italic> and randomly chosen from a logarithmic distribution. The analysis shows that the <italic>n</italic>th digit for <italic>n</italic> ≥ 2 is actually approximately uniformly distributed. The approximation depends upon both <italic>n</italic> and the base<italic>β</italic>. It becomes better as <italic>n</italic> increases, and it is exact in the limit as <italic>n</italic> ⇒ ∞. A table of this distribution is presented for various β and <italic>n</italic>, along with a table of the maximum digit by digit deviation Δ of the logarithmic distribution from the uniform distribution. Various asymptotic results for Δ are included. These results have application in resolving open questions of Henrici, of Kaneko and Liu, and of Tsao.

Boundedness of Solutions of a Nonlinear Nonautonomous Neutral Delay Equation

Abstract Sufficient conditions are obtained for the boundedness of solutions of the non-linear nonautonomous neutral equation x (t) = r(t) x(t)(a(t) − x(t − 1) − c(t) x (t − 1)) , which arise in a “food-limited” population model. This partially answers a recent open question proposed by K. Gopalsamy and B. G. Zhang.

High Order Methods for State-Dependent Delay Differential Equations with Nonsmooth Solutions

This work presents a theoretical basis for high order numerical methods to solve state-dependent delay differential equations of the form: \[\begin{gathered} \dot x(t) = f(t,x(t),x(\alpha (t,x(t))))\quad {\text{for }}t \in [a,b], \hfill \\ \alpha (t,x(t)) \leqq t, \hfill \\ x(t) = \phi (t)\quad {\text{for }}t \in [\bar a,a] \hfill \\ \end{gathered} \] where

Characterization of jump discontinuities for state dependent delay differential equations

\bar a = \min \alpha (t,x(t))

Numerical solution of ordinary and retarded differntial equations with discontinuou derivatives

for

Round-off error in products

t \in [a,b]

Effect of guard digits and normalization options on floating point multiplication

. The solutions to such equations typically have derivative jump discontinuities (jump points) which propagate from the initial jump point

Relative error in floating-point multiplication

t = a

Some Aspects of Floating Point Computation

. Thus, high order methods require an accurate determination of the location of jump discontinuities in lower order derivatives of the solution

Round-off error for retarded ordinary differential equations: A priori bounds and estimates

x(t)