## Software Needs in Special Functions (1996)

Venue: | J. Comput. Appl. Math |

Citations: | 7 - 1 self |

### BibTeX

@ARTICLE{Lozier96softwareneeds,

author = {Daniel W. Lozier and Daniel W. Lozier},

title = {Software Needs in Special Functions},

journal = {J. Comput. Appl. Math},

year = {1996},

volume = {66},

pages = {345--358}

}

### OpenURL

### Abstract

. Currently available software for special functions exhibits gaps and defects in comparison to the needs of modern high-performance scientific computing and also, surprisingly, in comparison to what could be constructed from current algorithms. In this paper we expose some of these deficiencies and identify the related need for user-oriented testing software. 1. Introduction A recent article by Lozier and Olver [21] provides a survey of algorithms and software for the numerical evaluation of special functions. Its emphasis is on the generation of function values although selected resources for zeros and integrals are included also. Journals, books, conference proceedings, and software documents were examined and a bibliography of nearly 500 references was constructed. Based on this investigation, the functions were classified and cross-referenced to bibliographic entries and to specific software libraries and systems 1 . The bibliography was prepared using the authors' professional...

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Citation Context ...merical mathematics, with attention paid to uniformity of documentation, style of usage, and handling of errors. Among the many examples are CERN [8], IMSL [2], NAG [16], NSWC [24], Numerical Recipes =-=[27]-=-, NUMPAC 3 , Scientific Desk 4 , and SLATEC [7]. Software packages and libraries are usually written in a standard programming language such as Fortran or C. They are used in the traditional compile-l... |

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Citation Context ...cess tends to be cumbersome. Three examples of interactive systems for symbolic computing with integrated support for graphics and floating-point computing are Macsyma [30], Maple [9] and Mathematica =-=[32]-=-. Three examples of interactive systems for fixed-precision floating-point computing with integrated support for graphics are HiQ [6], Mathcad [22] and Matlab [23]. HiQ and Mathcad, in particular, mak... |

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Citation Context ...quipment were serious issues. Now modern programming languages such as Fortran 90 and C ++ are adequate to support higher-precision operations in a convenient, user-friendly manner; see, for example, =-=[4]-=-. Alternatively, following the suggestion made in the previous section, an interactive system could be used to provide higher precision for testing functions in fixed-precision libraries. And networks... |

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Citation Context ...language but the process tends to be cumbersome. Three examples of interactive systems for symbolic computing with integrated support for graphics and floating-point computing are Macsyma [30], Maple =-=[9]-=- and Mathematica [32]. Three examples of interactive systems for fixed-precision floating-point computing with integrated support for graphics are HiQ [6], Mathcad [22] and Matlab [23]. HiQ and Mathca... |

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Citation Context ... could be answered with the aid of a graphical presentation of the results. Let us take as our example two software packages for the Airy function Ai(z) for complex z. Package A is that of D. E. Amos =-=[3]-=-. This has been incorporated into numerous libraries and systems because of its comprehensive, efficient and accurate coverage of Bessel and Hankel functions. Ai(z) is provided through its representat... |

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Citation Context ...ce function, and it is totally inappropriate in the vicinity of a zero. An alternative measure that overcomes the first deficiency, and that closely approximates relative error, is relative precision =-=[10]-=- rp(x; x) = j ln x \Gamma ln xj , x; x ? 0: Next, in the vicinity of a zero, relative error is often replaced by absolute error. However, this is not entirely satisfactory because the location of the ... |

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Citation Context ...The second approach is verification of functional identities. This has been described and applied to elementary and special functions in a long series of papers by W. J. Cody and co-workers, of which =-=[11, 12, 13]-=- are recent examples. Comparison testing is conceptually simple but requires computing in higher precision. Verification testing is performed entirely in one precision but requires great care in choos... |

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Citation Context ...o meet the practical needs of programmers. Intermediate libraries are limited to a subset of numerical mathematics. Three examples that specialize in mathematical functions are the libraries of Baker =-=[5]-=-, Moshier [25] and United Laboratories, Inc. [31]. Comprehensive libraries strive for complete coverage of numerical mathematics, with attention paid to uniformity of documentation, style of usage, an... |

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Citation Context ... strive for complete coverage of numerical mathematics, with attention paid to uniformity of documentation, style of usage, and handling of errors. Among the many examples are CERN [8], IMSL [2], NAG =-=[16]-=-, NSWC [24], Numerical Recipes [27], NUMPAC 3 , Scientific Desk 4 , and SLATEC [7]. Software packages and libraries are usually written in a standard programming language such as Fortran or C. They ar... |

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Citation Context ...The second approach is verification of functional identities. This has been described and applied to elementary and special functions in a long series of papers by W. J. Cody and co-workers, of which =-=[11, 12, 13]-=- are recent examples. Comparison testing is conceptually simple but requires computing in higher precision. Verification testing is performed entirely in one precision but requires great care in choos... |

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Citation Context ...Hankel functions. Ai(z) is provided through its representation in terms of the modified Bessel function K 1=3 (i) where i = (2=3)z 3=2 . Package B, unpublished as yet, uses the algorithm described in =-=[20]-=-. This algorithm computes Ai(z) directly from its asymptotic expansion and its defining differential equation, which is integrated numerically. Both packages provide an option to evaluate the scaled f... |

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Citation Context ...rpose is to make new algorithms available to programmers in an immediately usable form. Three important series of software packages are the ACM Algorithms in ACM Transactions on Mathematical Software =-=[17]-=-, the AS Algorithms in Applied Statistics [28], and the CPC Programs in Computer Physics Communications [14, 15]. All contributions to these series are refereed before acceptance. Intermediate and com... |

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Citation Context ...syma [30], Maple [9] and Mathematica [32]. Three examples of interactive systems for fixed-precision floating-point computing with integrated support for graphics are HiQ [6], Mathcad [22] and Matlab =-=[23]-=-. HiQ and Mathcad, in particular, make extensive use of menus. Software packages, intermediate and comprehensive libraries, and interactive systems serve different purposes. For special functions in a... |

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Citation Context ...ormity of documentation, style of usage, and handling of errors. Among the many examples are CERN [8], IMSL [2], NAG [16], NSWC [24], Numerical Recipes [27], NUMPAC 3 , Scientific Desk 4 , and SLATEC =-=[7]-=-. Software packages and libraries are usually written in a standard programming language such as Fortran or C. They are used in the traditional compile-link-execute cycle of fixed-precision floating-p... |

2 |
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Citation Context ... complete coverage of numerical mathematics, with attention paid to uniformity of documentation, style of usage, and handling of errors. Among the many examples are CERN [8], IMSL [2], NAG [16], NSWC =-=[24]-=-, Numerical Recipes [27], NUMPAC 3 , Scientific Desk 4 , and SLATEC [7]. Software packages and libraries are usually written in a standard programming language such as Fortran or C. They are used in t... |

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Citation Context ...rogrammers in an immediately usable form. Three important series of software packages are the ACM Algorithms in ACM Transactions on Mathematical Software [17], the AS Algorithms in Applied Statistics =-=[28]-=-, and the CPC Programs in Computer Physics Communications [14, 15]. All contributions to these series are refereed before acceptance. Intermediate and comprehensive libraries consist of software packa... |

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(Show Context)
Citation Context ...rehensive libraries strive for complete coverage of numerical mathematics, with attention paid to uniformity of documentation, style of usage, and handling of errors. Among the many examples are CERN =-=[8]-=-, IMSL [2], NAG [16], NSWC [24], Numerical Recipes [27], NUMPAC 3 , Scientific Desk 4 , and SLATEC [7]. Software packages and libraries are usually written in a standard programming language such as F... |

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Citation Context ...The second approach is verification of functional identities. This has been described and applied to elementary and special functions in a long series of papers by W. J. Cody and co-workers, of which =-=[11, 12, 13]-=- are recent examples. Comparison testing is conceptually simple but requires computing in higher precision. Verification testing is performed entirely in one precision but requires great care in choos... |

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Citation Context ...es of software packages are the ACM Algorithms in ACM Transactions on Mathematical Software [17], the AS Algorithms in Applied Statistics [28], and the CPC Programs in Computer Physics Communications =-=[14, 15]-=-. All contributions to these series are refereed before acceptance. Intermediate and comprehensive libraries consist of software packages that have been collected, developed, organized and unified to ... |

1 |
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(Show Context)
Citation Context ... have avoided the failure. There are two principal approaches to testing. The first is comparison against a standard. More than 20 years ago, this was described and applied to elementary functions in =-=[18, 19, 29]-=-. The second approach is verification of functional identities. This has been described and applied to elementary and special functions in a long series of papers by W. J. Cody and co-workers, of whic... |

1 |
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Citation Context ...actical needs of programmers. Intermediate libraries are limited to a subset of numerical mathematics. Three examples that specialize in mathematical functions are the libraries of Baker [5], Moshier =-=[25]-=- and United Laboratories, Inc. [31]. Comprehensive libraries strive for complete coverage of numerical mathematics, with attention paid to uniformity of documentation, style of usage, and handling of ... |

1 |
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(Show Context)
Citation Context .... It did not raise the question of the quality of the software. This topic has many ramifications but we are interested here only in numerical accuracy. As an example of the need for testing, we cite =-=[26]-=- in which an intermediate library with extensive coverage of special functions was reviewed. It was found that a subroutine for evaluating the Bessel function Js(x), x andsreal, returned highly inaccu... |

1 |
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(Show Context)
Citation Context ... have avoided the failure. There are two principal approaches to testing. The first is comparison against a standard. More than 20 years ago, this was described and applied to elementary functions in =-=[18, 19, 29]-=-. The second approach is verification of functional identities. This has been described and applied to elementary and special functions in a long series of papers by W. J. Cody and co-workers, of whic... |