Reply to Brundage and Shanks

On Computations of Shanks and Schmid

In 1966, Shanks and Schmid investigated the asymptotic behavior of the number of positive integers less than or equal to x which are represented by the quadratic form X +nY , n ≥ 1. Based on some numerical computations, they observed that the constant occurring in the main term appears to be the largest for n = 2. In this paper, we prove that in fact this constant is unbounded as one runs throu...

Daniel Shanks (1917-1996)

Daniel Shanks was born on January 17, 1917, in the city of Chicago, where he was raised and where in 1937 he received his B.S. in physics from the University of Chicago. In 1940 he worked as a physicist at the Aberdeen Proving Grounds, moving the following year to the position of physicist at the Naval Ordinance Laboratory, a position he retained until 1950. In 1951 his post at the NOL changed ...

The Authors’ Reply

Shanks Sequence Transformations and Anderson Acceleration

This paper presents a general framework for Shanks transformations of sequences of elements in a vector space. It is shown that the Minimal Polynomial Extrapolation (MPE), the Modified Minimal Polynomial Extrapolation (MMPE), the Reduced Rank Extrapolation (RRE), the Vector Epsilon Algorithm (VEA), the Topological Epsilon Algorithm (TEA), and Anderson Acceleration (AA), which are standard gener...

On a conjecture of Shanks

and this was conjectured, on numerical evidence, by Shanks (1961) [7] to be true for α = 2 . © 2010 Elsevier Inc. All rights reserved.