…
►Results similar to these appear in Langhoff et al. (1976) in methods developed for physics applications, and which includes treatments of systems with discontinuities in , using what is referred to as the Stieltjes derivative which may be traced back to Stieltjes, as discussed by Deltour (1968, Eq. 12).
►
DerivativeRule Approach
►An alternate, and highly efficient, approach follows from the derivativerule conjecture, see Yamani and Reinhardt (1975), and references therein, namely that
…
►Comparisons of the precisions of Lagrange and PWCF interpolations to obtain the derivatives, are shown in Figure 18.40.2.
…
►Further, exponential convergence in , via the DerivativeRule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate for these OP systems on and respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a).
…
…
►The trapezoidal rule (§3.5(i)) is then applied.
…
►In consequence of §9.6(i), algorithms for generating Bessel functions, Hankel functions, and modified Bessel functions (§10.74) can also be applied to , , and their derivatives.
…
►Zeros of the Airy functions, and their derivatives, can be computed to high precision via Newton’s rule (§3.8(ii)) or Halley’s rule (§3.8(v)), using values supplied by the asymptotic expansions of §9.9(iv) as initial approximations.
…
►For the computation of the zeros of the Scorer functions and their derivatives see Gil et al. (2003c).
…
►where is the unit vector normal to and whose direction is determined by the right-hand rule; see Figure 1.6.1.
►►►Figure 1.6.1: Vector notation.
Right-hand rule for cross products.
Magnify
…
►
►Newton’s rule (§3.8(i)) or Halley’s rule (§3.8(v)) can be used to compute to arbitrarily high accuracy the real or complex zeros of all the functions treated in this chapter.
Necessary values of the first derivatives of the functions are obtained by the use of (10.6.2), for example.
Newton’s rule is quadratically convergent and Halley’s rule is cubically convergent.
…
►
…
…
►A minimal cubature rule is a numerical integration rule that uses the smallest number of nodes among cubature rules of the same degree.
The nodes of these cubature rules are closely related to common zeros of OPs and they are often good points for polynomial interpolation.
Although Gaussian cubature rules rarely exist and they do not exist for centrally symmetric domains, minimal or near minimal cubature rules on the unit square are known and provide efficient numerical integration rules.
…
M. Razaz and J. L. Schonfelder (1980)High precision Chebyshev expansions for Airy functions and their derivatives.
Technical report
University of Birmingham Computer Centre.
W. P. Reinhardt (2021a)Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivativerule approach to Stieltjes and spectral imaging.
Computing in Science and Engineering23 (4), pp. 91.
W. P. Reinhardt (2021b)Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivativerule approach to Stieltjes and spectral imaging.
Computing in Science and Engineering23 (3), pp. 56–64.