Richard Goodman

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.

Numerical solution of ordinary and retarded differntial equations with discontinuou derivatives

SummaryThis paper studies the propagation of discretization error for discontinuou ordinary and retarded differential equations. Various applications are given including one which extends a fundamental theorem of Henrici concerning round-off error.

Improved Trailing Digits Estimates Applied to Optimal Computer Arithmetic

New results are given on the distribution of trailing digits for logarithmically distributed numbers and on error in floating-point multiplication Some of the results have application to computer design In particular, there are certain values of the base (indeed,/3 = 2, 4, 6, and sometimes 8, but NOT 16) which, when carefully balanced with other design parameters, minimize the mean multiphcative error For these special minimizing situations, it suffices to have only one guard flit provided that posmormahzatlon occurs after symmetric rounding

Round-off error in products

ZusammenfassungSeienA1 undA2 zufällige Gleitkommazahlen zu einer beliebigen Basis β mit einer logarithmischen Verteilung. Seir der Rundungsfehler

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

r = fl(A_1 * A_2 ) - (A_1 * A_2 )

Relative Fehler bei der Gleitkomma-Multiplikation

, wo * die Gleitkommamultiplikation bedeutet undfl(A1*A2) das normalisierteN-stellige Computerresultat für (A1*A2). Die Arbeit analysiert Mittelwert und Varianz des Rundungsfehlers sowohl des Ergebnisses wie auch dessen Mantisse. Die Analyse beruht auf scharfen Ordnungsabschätzungen der Abweichung pro Mantissenstelle zwischen logarithmisch verteilten Zahlen und gleichverteilten Zahlen. Offene Probleme von Kaneko und Liu und von Tsao werden vollständig gelöst. Ferner wird ein wichtiger Rundungsfehler-Satz von Henrici auf beliebige Basis (von der Basis 2) verallgemeinert.AbstractLetA1 andA2 be floating point numbers represented in arbitrary base β and randomly chosen from a logarithmic distribution. Letr denote the round-off error

Einflu von Schutzstellen und Normalisierungsverfahren auf die Gleitkommamultiplikation

r = fl(A_1 * A_2 ) - (A_1 * A_2 )