New version announcement for TaylUR, an arbitrary-order diagonal automatic differentiation package for Fortran 95

arXiv:0704.0274v1 [physics.comp-ph] 2 Apr 2007 New version announcement for TaylUR, an arbitrary-order diagonal automatic differentiation package for Fortran 95 G.M. von Hippel 1 Department of Physics, University of Regina, Regina, Saskatchewan, S4S 0A2, Canada Abstract We present a new version of TaylUR, a Fortran 95 module to automatically compute the numerical values of a complex-valued function’s derivatives with respect to sev- eral variables up to an arbitrary order in each variable, but excluding mixed deriva- tives. The new version fixes a potentially serious bug in the code for exponential- related functions that could corrupt the imaginary parts of derivatives, as well as being compatible with a wider range of compilers. Key words: automatic differentiation, higher derivatives, Fortran 95 PACS: 02.60.Jh, 02.30.Mv 1991 MSC: 41-04, 41A58, 65D25 NEW VERSION PROGRAM SUMMARY Manuscript Title: New version announcement for TaylUR, an arbitrary-order diag- onal automatic differentiation package for Fortran 95 Authors: G.M. von Hippel Program Title: TaylUR Journal Reference: Catalogue identifier: Licensing provisions: none 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 Email address: vonhippg@uregina.ca (G.M. von Hippel). URL: http://uregina.ca/~vonhippg/ (G.M. von Hippel). 1 Corresponding author Preprint submitted to Elsevier Science 4 November 2018 Classification: 4.12 Other Numerical Methods, 4.14 Utility Catalogue identifier of previous version: ADXR v1 0 Journal reference of previous version: Comput. Phys. Commun. 174 (2006) 569-576 Does the new version supersede the previous version?: yes Nature of problem: Problems that require potentially high orders of derivatives with respect to some variables or derivatives of complex-valued functions, such as e.g. expansions of Feyn- man diagrams in particle masses in perturbative Quantum Field Theory. Solution method: Arithmetic operators and Fortran intrinsics are overloaded to act correctly on ob- jects 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 F`aa di Bruno’s formula. Reasons for the new version: The previous version [1] contained a potentially serious bug in the functions over- loading the exponential-related intrinsics (EXP, LOG, SIN, COS, TAN, SINH, COSH, TANH), which could corrupt the imaginary parts of derivatives. It also contained some features which caused it to crash when compiled with certain compilers (no- tably the NAG and Lahey/Fujitsu compilers). Summary of revisions: The bug in the exponential-related intrinsics has been corrected. A number of ad- ditional changes have been made to the code to enable better compatibility with a greater range of compilers, including the NAG and Lahey/Fujitsu compilers. Users of some of these compilers may have to define useintrinsic as a preprocessor sym- bol when compiling TaylUR. 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 cancella- tion may become an issue at very high orders. Unusual features: No mixed higher-order derivatives are computed. The complex conjugation opera- tion assumes all independent variables to be real. Running time: The running time of TaylUR operations depends linearly on the number of vari- ables. Its dependence on the Taylor expansion order varies from linear (for linear operations) through quadratic (for multiplication) to exponential (for elementary function calls). References: 2 [1] G. M. von Hippel, TaylUR, an arbitrary-order diagonal automatic differentiation package for Fortran 95, Comput. Phys. Commun. 174 (2006) 569-576. 3

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