Extended numerical computations on the “1/9” conjecture in rational approximation theory

Talbot quadratures and rational approximations

“…In 1983 [3]. A few years later, Magnus and independently Gonchar and Rakhmanov found exact expressions for this number, with Gonchar and Rakhmanov proving that the limit is indeed (4.1) [12,13].…”

“…−N as N → ∞. The function r * can be computed by the Remes algorithm as in [3] and [5]; we have also been given very high accuracy computed results in a private communication from Alphonse Magnus. A simple practical alternative is to use the Carathéodory-Fejér method as proposed in the final section of [46] and analyzed at length in [24], based on the singular value decomposition of a Hankel matrix of Chebyshev coefficients of the function e z transplanted from R − to [−1, 1].…”

Analysis of Some Krylov Subspace Approximations to the Matrix Exponential Operator

“…In fact since the roots θ i seem to come generally in complex conjugate pairs we only need about half as many factorizations to be carried out in complex arithmetic. The rational functions that are used in [3] as well as in the numerical experiments section in this paper are based on the Chebyshev approximation on [0, ∞) of the function e −x , see [1] and the references therein. This approximation is remarkably accurate and, in general, a small degree approximation, as small as p = 14, suffices to obtain a good working accuracy.…”

Rational approximation to the Fermi–Dirac function with applications in density functional theory

“…In general, the degree of the numerator need not be the same as the degree of the denominator, but we limit our presentation to this case because it is sufficient for our purposes. This problem does not have a closed form solution, but it has been solved numerically for d = 1, ..., 14, by Cody, Meinardus and Varga [6], and subsequently up to degree d = 30 by Carpenter, Ruttan and Varga [5]. Interestingly, the problem does have a closed form solution if it is instead formulated (in the unit disk) over the extended approximation spacẽ…”

“…While we could interchangeably use this approach (and have tested that it works), we continue our presentation with the ordinary rational Chebyshev approximation for which the coefficients of the best approximants p(x) and q(x) have been computed and listed for d = 1, 2, ..., 30 in [6,5]. Starting therefore with e −x ≈ r d (x), we can derive an approximation to the Fermi-Dirac function as From the rational approximation (5), we can compute the partial fraction expansion…”