Tau-method approximations for the Bessel function Y1(z)

Symbolic Computation on Complex Polynomial Solution of Differential Equations

“…However, as Zhang and Belward (1995) pointed out, there are some limitations on the approach that they used. The first limitation is the finite precision computation.…”

“…However, the applications were limited by the finite precision of the computation and by the possible rounding error resulted from the large magnitude of the Faber coefficients as indicated by Zhang and Belward (1995). Extensive accuracy comparison with the Chebyshev series approximations was not possible, as there were only Faber polynomials up to the 15th degree available.…”

“…Several methods for computing Faber polynomials for a region in the complex plane have been developed (Coleman and Smith, 1987;Ellacott, 1983;Gatermann et al, 1992) and found useful applications (Coleman, 1987a, b;Zhang, 1995;Zhang and Belward, 1995). We summarize some defining statements about the Faber polynomials and their symbolic representations.…”

“…They computed the coefficients of the Faber polynomials up to degree 15 on six selected circular sectors S θ = {z : z ∈ C : |z| ≤ 1, | arg z| ≤ θ}; where the half opening angle θ = π/ρ with ρ = 2, 4, 6, 12, 18, 36, respectively. These coefficients have been used by Coleman (1987a, b), Zhang (1995), Zhang and Belward (1995), to obtain good polynomial approximations for some special functions with complex arguments.…”

“…Since the Chebyshev polynomials do not provide an ideal basis for optimal approximation on circular sectors, a more satisfactory approach was proposed by Coleman (1987a) to approximate J ν (z) by the Lanczos τ -method, which makes use of the Faber polynomials for circular sectors computed by Coleman and Smith (1987). Zhang (1995) and Zhang and Belward (1995) also used the τ -method for approximating Y 0 (z) and Y 1 (z), and essentially for Y n (z) of integer orders through recurrence formulae (Magnus et al, 1966).…”