A comparison of algorithms for rational 𝑙_{∞} approximation

Abstract

Results are reported of a numerical study to compare eight algorithms for obtaining rational <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l Subscript normal infinity"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>l</mml:mi> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{l_\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> approximations. The algorithms investigated are Loeb’s algorithm, the linear inequality algorithm, the Osborne-Watson algorithm, the differential correction algorithms I, II and III, the Remes algorithm and Maehly’s algorithm. The results of the study indicate that the Remes algorithm and the differential correction algorithm III are the most satisfactory methods to use in practice.