@Article{Naumann:2015:ADN,
  author =       "Uwe Naumann and Johannes Lotz and Klaus Leppkes
                  and Markus Towara",
  title =        "Algorithmic Differentiation of Numerical Methods:
                  Tangent and Adjoint Solvers for Parameterized
                  Systems of Nonlinear Equations",
  journal =      "{ACM} Transactions on Mathematical Software",
  volume =       "41",
  number =       "4",
  accepted =     "11 December 2014",
  upcoming =     "true", 
  abstract =     "
                  We discuss software tool support for
                  the Algorithmic Differentiation (also
                  known as Automatic Differentiation;
                  AD) of numerical simulation programs
                  that contain calls to solvers for
                  parameterized systems of $n$ nonlinear
                  equations. The local computational
                  overhead as well as the additional
                  memory requirement for the computation
                  of directional derivatives or adjoints of
                  the solution of the nonlinear system with
                  respect to the parameters can quickly
                  become prohibitive for large values
                  of $n$. Both are reduced drastically
                  by analytical (in the following also:
                  symbolic) approaches to differentiation
                  of the underlying numerical
                  methods. Following the discussion of
                  the proposed terminology we develop
                  the algorithmic formalism building on
                  prior work by other colleagues and we
                  present an implementation based on the
                  AD software \texttt{dco/c++}. A representative
                  case study supports the theoretically
                  obtained computational complexity results
                  with practical run time measurements.",
}