Chebyshev expansion on intervals with branch points with application to the root of Kepler's equation: A Chebyshev-Hermite-Padé method

• Authors: John P. Boyd
• Venue: Journal of Computational and Applied Mathematics
• Year: 2009

When two or more branches of a function merge, the Chebyshev series of u(@l) will converge very poorly with coefficients a"n of T"n(@l) falling as O(1/n^@a) for some small positive exponent @a. However, as shown in [J.P. Boyd, Chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one-dimensional Bratu equation, Appl. Math. Comput. 143 (2002) 189-200], it is possible to obtain approximations that converge exponentially fast in n. If the roots that merge are denoted as u"1(@l) and u"2(@l), then both branches can be written without approximation as the roots of (u-u"1(@l))(u-u"2(@l))=u^2+@b(@l)u+@c(@l). By expanding the nonsingular coefficients of the quadratic, @b(@l) and @c(@l), as Chebyshev series and then applying the usual roots-of-a-quadratic formula, we can approximate both branches simultaneously with error that decreases proportional to exp(-@sN) for some constant @s0 where N is the truncation of the Chebyshev series. This is dubbed the ''Chebyshev-Shafer'' or ''Chebyshev-Hermite-Pade'' method because it substitutes Chebyshev series for power series in the generalized Pade approximants known variously as ''Shafer'' or ''Hermite-Pade'' approximants. Here we extend these ideas. First, we explore square roots with branches that are both real-valued and complex-valued in the domain of interest, illustrated by meteorological baroclinic instability. Second, we illustrate triply branched functions via roots of the Kepler equation, f(u;@l,@e)=u-@esin(u)-@l=0. Only one of the merging roots is real-valued and the root depends on two parameters (@l,@e) rather than one. Nonetheless, the Chebyshev-Hermite-Pade scheme is successful over the whole two-dimensional parameter plane. We also discuss how to cope with poles and logarithmic singularities that arise in our examples at the extremes of the expansion domain.