Optimized higher-order automatic differentiation for the Faddeeva function

Abstract

Considerable research efforts have been directed at implementing the Faddeeva function w(z) and its derivatives with respect to z, but these did not consider the key computing issue of a possible dependence of z on some variable t. The general case is to differentiate the compound function w(z(t))=w∘z(t) with respect to t by applying the chain rule for a first order derivative, or Faà 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), a special symbolic HOAD is required to compute accurate higher-order derivatives for w∘z(t) in an efficient manner. A thorough evaluation is carried out considering a nontrivial case study in optics to support this assertion. Program summaryProgram title: HOAD_MathFunCatalogue identifier: AFAG_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AFAG_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: yesNo. of lines in distributed program, including test data, etc.: 4737No. of bytes in distributed program, including test data, etc.: 103450Distribution format: tar.gzProgramming language: Fortran 90.Computer: Non-specific.Operating system: Non-specific.RAM: 1 megabyteClassification: 4.12.External routines: Algorithm 680 [8], Mathematical functions (included)Nature of problem:General optimized higher-order automatic differentiation of mathematical functions. Complex refractive index as a case study.Solution method:Higher-order differentiation for the general second order ordinary equation defining the mathematical functions. Optimized operator overloading recurrence formula.Unusual features:Automatic differentiation, Quadratic complexity.Running time:1 second.