Considerable research efforts have been directed at implementing the Faddeeva function w(z)w(z) and its derivatives with respect to zz, but these did not consider the key computing issue of a possible dependence of zz on some variable tt. The general case is to differentiate the compound function w(z(t))=w∘z(t)w(z(t))=w∘z(t) with respect to tt by applying the chain rule for a first order derivative, or Faa di Bruno’s formula for higher-order ones. Higher-order automatic differentiation (HOAD) is an efficient and accurate technique for derivative calculation along scientific computing codes. Although codes are available for w(z)w(z), a special symbolic HOAD is required to compute accurate higher-order derivatives for w∘z(t)w∘z(t) in an efficient manner. A thorough evaluation is carried out considering a nontrivial case study in optics to support this assertion.#R##N#Program summary#R##N#Program title: HOAD_MathFun#R##N##R##N#Catalogue identifier: AFAG_v1_0#R##N##R##N#Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AFAG_v1_0.html#R##N##R##N#Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland#R##N##R##N#Licensing provisions: yes#R##N##R##N#No. of lines in distributed program, including test data, etc.: 4737#R##N##R##N#No. of bytes in distributed program, including test data, etc.: 103450#R##N##R##N#Distribution format: tar.gz#R##N##R##N#Programming language: Fortran 90.#R##N##R##N#Computer: Non-specific.#R##N##R##N#Operating system: Non-specific.#R##N##R##N#RAM: 1 megabyte#R##N##R##N#Classification: 4.12.#R##N##R##N#External routines: Algorithm 680 [8], Mathematical functions (included)#R##N##R##N#Nature of problem:#R##N##R##N#General optimized higher-order automatic differentiation of mathematical functions. Complex refractive index as a case study.#R##N##R##N#Solution method:#R##N##R##N#Higher-order differentiation for the general second order ordinary equation defining the mathematical functions. Optimized operator overloading recurrence formula.#R##N##R##N#Unusual features:#R##N##R##N#Automatic differentiation, Quadratic complexity.#R##N##R##N#Running time:#R##N##R##N#1 second.
['Isabelle Charpentier']
Optimized higher-order automatic differentiation for the Faddeeva function ☆