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Manuscript Title: TaylUR, an arbitrary-order diagonal automatic differentiation package for Fortran 95
Authors: G.M. von Hippel
Program title: TaylUR
Catalogue identifier: ADXR_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 174(2006)569
Programming language: Fortran 95.
Computer: Any computer with a conforming Fortran 95 compiler.
Operating system: Any system with a conforming Fortran 95 compiler.
Keywords: automatic differentiation, higher derivatives, Fortran 95.
PACS: 02.60.Jh, 02.30.Mv.
Classification: 4.12, 4.14.

Nature of problem:
Problems that require potentially high orders of derivatives with respect to some variables, such as e.g. expansions of Feynman diagrams in particle masses in perturbative Quantum Field Theory, and which cannot be treated using existing Fortran modules for automatic differentiation [1-2].

Solution method:
Arithmetic operators and Fortran intrinsics are overloaded to act correctly on objects of a defined type taylor, which encodes a function along with its first few derivatives with respect to the user-defined independent variables. Derivatives of products and composite functions are computed using Leibniz's rule and Fàa di Bruno's formula.

Restrictions:
Memory and CPU time constraints may restrict the number of variables and Taylor expansion order that can be achieved. Loss of numerical accuracy due to cancellation may become an issue at very high orders.

Unusual features:
No mixed higher-order derivatives are computed. The complex conjugation operation assumes all independent variables to be real.

Running time:
The running time of TaylUR operations depends linearly on the number of variables. Its dependence on the Taylor expansion order varies from linear (for linear operations) through quadratic (for multiplication) to exponential (for elementary function calls).

References:
[1] C. W. Straka, ADF95: Tool for automatic differentiation of a FORTRAN code designed for large numbers of independent variables, Comput. Phys. Commun. 168 (2005) 123-139 [arXiv:cs.MS/0503014].
[2] S. Stamatiadis, R. Prosmiti, S. C. Farantos, AUTO_DERIV: Tool for automatic differentiation of a FORTRAN code, Comput. Phys. Commun. 127 (2000) 343-355.