New version announcement for TaylUR, an arbitrary-order diagonal
  automatic differentiation package for Fortran 95
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
http://arxiv.org/abs/0704.0274v1
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 Fàa 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:
[1] G. M. von Hippel, TaylUR, an arbitrary-order diagonal automatic differentiation
package for Fortran 95, Comput. Phys. Commun. 174 (2006) 569-576.
