Auto_deriv: Tool for automatic differentiation of a Fortran code

Abstract

auto_deriv is a module comprised of a set of fortran 95 procedures which can be used to calculate the first and second partial derivatives (mixed or not) of any continuous function with many independent variables. The mathematical function should be expressed as one or more fortran 77/90/95 procedures. A new type of variables is defined and the overloading mechanism of functions and operators provided by the fortran 95 language is extensively used to define the differentiation rules. Proper (standard complying) handling of floating-point exceptions is provided by using the IEEE_EXCEPTIONS intrinsic module (Technical Report 15580, incorporated in fortran 2003). New version program summary Program title: AUTO_DERIV Catalogue identifier: ADLS_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADLS_v2_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 2963 No. of bytes in distributed program, including test data, etc.: 10 314 Distribution format: tar.gz Programming language: Fortran 95 + (optionally) TR-15580 (Floating-point exception handling) Computer: all platforms with a Fortran 95 compiler Operating system: Linux, Windows, MacOS Classification: 4.12, 6.2 Catalogue identifier of previous version: ADLS_v1_0 Journal reference of previous version: Comput. Phys. Comm. 127 (2000) 343 Does the new version supersede the previous version?: Yes Nature of problem: The need to calculate accurate derivatives of a multivariate function frequently arises in computational physics and chemistry. The most versatile approach to evaluate them by a computer, automatically and to machine precision, is via user-defined types and operator overloading. AUTO_DERIV is a Fortran 95 implementation of them, designed to evaluate the first and second derivatives of a function of many variables. Solution method: The mathematical rules for differentiation of sums, products, quotients, elementary functions in conjunction with the chain rule for compound functions are applied. The function should be expressed as one or more Fortran 77/90/95 procedures. A new type of variables is defined and the overloading mechanism of functions and operators provided by the Fortran 95 language is extensively used to implement the differentiation rules. Reasons for new version: The new version supports Fortran 95, handles properly the floating-point exceptions, and is faster due to internal reorganization. All discovered bugs are fixed. Summary of revisions: • The code was rewritten extensively to benefit from features introduced in Fortran 95. Additionally, there was a major internal reorganization of the code, resulting in faster execution. The user interface described in the original paper was not changed. The values that the user must or should specify before compilation (essentially, the number of independent variables) were moved into ad_types module. • There were many minor bug fixes. One important bug was found and fixed; the code did not handle correctly the overloading of ∗ ∗ in a ∗ ∗ λ when a = 0 . • The case of division by zero and the discontinuity of the function at the requested point are indicated by standard IEEE exceptions ( IEEE_DIVIDE_BY_ZERO and IEEE_INVALID respectively). If the compiler does not support IEEE exceptions, a module with the appropriate name is provided, imitating the behavior of the ‘standard’ module in the sense that it raises the corresponding exceptions. It is up to the compiler (through certain flags probably) to detect them. Restrictions: None imposed by the program. There are certain limitations that may appear mostly due to the specific implementation chosen in the user code. They can always be overcome by recoding parts of the routines developed by the user or by modifying AUTO_DERIV according to specific instructions given in [1]. The common restrictions of available memory and the capabilities of the compiler are the same as the original version. Additional comments: The program has been tested using the following compilers: Intel ifort, GNU gfortran, NAGWare f95, g95. Running time: The typical running time for the program depends on the compiler and the complexity of the differentiated function. A rough estimate is that AUTO_DERIV is ten times slower than the evaluation of the analytical (‘by hand’) function value and derivatives (if they are available).