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

Abstract

When two or more branches of a function merge, the Chebyshev series of u ( λ ) will converge very poorly with coefficients a n of T n ( λ ) falling as O ( 1 / n α ) for some small positive exponent α . 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 ( λ ) and u 2 ( λ ) , then both branches can be written without approximation as the roots of ( u − u 1 ( λ ) ) ( u − u 2 ( λ ) ) = u 2 + β ( λ ) u + γ ( λ ) . By expanding the nonsingular coefficients of the quadratic, β ( λ ) and γ ( λ ) , 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 ( − σ N ) for some constant σ > 0 where N is the truncation of the Chebyshev series. This is dubbed the “Chebyshev–Shafer” or “Chebyshev–Hermite–Padé” method because it substitutes Chebyshev series for power series in the generalized Padé approximants known variously as “Shafer” or “Hermite–Padé” 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 ; λ , ϵ ) ≡ u − ϵ sin ( u ) − λ = 0 . Only one of the merging roots is real-valued and the root depends on two parameters ( λ , ϵ ) rather than one. Nonetheless, the Chebyshev–Hermite–Padé 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.