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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.

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