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Optimized higher-order automatic differentiation for the Faddeeva function. (English) Zbl 1378.65067
Summary: 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 \(w(z(t)) = w \circ z(t)\) 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 \(w \circ z(t)\) in an efficient manner. A thorough evaluation is carried out considering a nontrivial case study in optics to support this assertion.

65D25 Numerical differentiation
65Q30 Numerical aspects of recurrence relations
78A60 Lasers, masers, optical bistability, nonlinear optics
65Y15 Packaged methods for numerical algorithms
dcc; HOAD_MathFun
Full Text: DOI
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