The Library

NUMAL

of Algol 60 Procedures in Numerical Mathematics

Mathematical Centre, Amsterdam

4th Revision, 1980

Note added in April, 1996

The Stichting Mathematisch Centrum at Amsterdam has granted permission to incorporate the complete NUMAL library manual into this CDROM. It may be freely used. It may be copied, provided that the name NUMAL and the attribution to the Stichting Mathematisch Centrum at Amsterdam are retained.

Introduction

At the request of the Academic Computing Centre of Amsterdam (SARA) the Mathematical Centre started in 1973 the adaptation of its collection of numerical procedures for use with the CD Cyber 70 system. The resulting numerical library is called "NUMAL" (NUMerical procedures in ALgol 60).

The documentation appeared in 1974 in a loose leaf manual. It was revised and updated in 1975, 1977 and 1979. In this period the library grew to a collection of about 430 routines in all different fields of numerical mathematics. Although no attempts have been made to create a collection that gives a complete overview of all numerical procedures available, most areas of numerical mathematics are covered and most programs for which a need was felt have been implemented.

The aim of NUMAL was to provide Algol 60 programmers with a high level numerical library which contains a set of validated numerical procedures together with supporting documentation.

The library has been constructed in a modular way: Most routines refer to and rely on other routines that perform a well-defined sub-task. the more experienced user can avail himself of these more elementary routines in the same way as the authors of the library do. The novice, it is better to use the more comprehensive procedures which perform complete mathematical computations.

In 1976 and 1977 the Numerical Mathematics department organized a seminar on numerical software. In this seminar the use of some Algol procedures in the NUMAL library was explained in detail and their algorithmic background was clarified. Also, the numerical libraries IMSL (Fortran), NAG (Algol 60 and Fortran) and NUMAL (Algol 60) were compared. The proceedings of this seminar appeared (in Dutch) in the MC-syllabus series of the Mathematical Centre as:

   COLLOQUIUM NUMERIEKE PROGRAMMATUUR, DEEL 1
   J.C.P.BUS  ED., MCS 29.1,
   MATHEMATISCH CENTRUM, AMSTERDAM,1976,

and

   COLLOQUIUM NUMERIEKE PROGRAMMATUUR, DEEL 2
   H.J.J.TE RIELE  ED., MCS 29.2,
   MATHEMATISCH CENTRUM, AMSTERDAM,1977.

The Language Algol 60 and the Modular Structure

Two important characteristics of NUMAL are : its modular structure, and the consistent use of standard Algol 60 (in the sense of the revised report on Algol 60, see ref.).

Input and output routines, not being defined in standard Algol 60, have not been used in the library source texts. They only appear in the documentation where examples of use of the library routines are given.

Since double precision is also not defined in Algol 60, a small number of double precision arithmetic routines could not be coded in Algol 60 and, hence, a few (8) double precision modules were coded in assembly language.

Except for this small number of double precision arithmetic routines all the source texts are written in Algol 60 and, hence, they are in principle independent of the computer/compiler used.

In its practical implementation on the CDC-Cyber system most elementary routines in the modular structure of the library (i.e. the matrix and vector operations) were re-coded by hand in assembly language. This has accelerated the operation of most linear algebra routines by a factor 2.5. Thus, one of the major disadvantages of the use of Algol 60, the relatively long execution times (which first were approximately 4 times longer than for an equivalent Fortran program) was circumvented to a large extent. This refinement was possible because of the consistent application of the modularity principle.

References

      P.NAUR (ED.)
      REVISED REPORT ON THE ALGORITHMIC LANGUAGE ALGOL 60
      A/S REGNECENTRALEN, COPENHAGEN, 1964.

      P.W.HEMKER
      CRITERIA FOR TRANSPORTABLE ALGOL LIBRARIES.
      IN: PORTABILITY OF NUMERICAL SOFTWARE (W.COWELL ED.)
          LECTURE NOTES IN COMP.SC. 57,SPRINGER VERLAG, 1977.

Organization of the Library

Each routine in the library is identified by a name and a code number. The code number can be used in an Algol 60 program when reference is made to a pre-compiled procedure in the object code library. All procedures in NUMAL are classified according to subject. The subjects are identified by a section number. The manual is ordered by these section numbers.

In order to find a particular procedure, there is a systematic index in which all procedures (their names and their code numbers) are recorded and classified by their section number (i.e. by subject).

For cross referencing there is an index by code number, which has references to procedure name and section number, and there is also a KWIC index in which keywords and procedure names have been ordered alphabetically.

The Status of NUMAL

In 1979 the interest at the Mathematical Centre in the creation of general numerical software in Algol 60 decreased and the drafting-committee decided to conclude the NUMAL project with the publication of a final revision of the library in book form. In its present form the library NUMAL can be seen as a description of the state-of-the-art of numerical Algol 60 programming at the Mathematical Centre at the end of the 1970s. We think that it contains a valuable collection of routines in a language that still can describe numerical procedures better than many other programming languages currently in use.

Imperfections and Responsibility

Although the Numerical Mathematics department of the Mathematical Centre assumes the responsibility for imperfections both in programs and in documentation, neither the Mathematical Centre nor the authors/contributors accept responsibility for the consequences of such imperfections. Although much effort has been spent to keep the number of errors to a minimum, it is possible that some minor errors still remain. Therefore the Numerical Mathematics department will keep a list of all errors in the documentation and/or the programs that become known after publication and this list will be made available upon request.

NUMAL in Fortran

A translation of NUMAL into a Fortran version suitable for use on a mini-computor is carried out under the supervision of P. Wynn by H.T. Lau in the School of Computer Science, McGill University, Montreal, Canada. Further research on the Fortran version is also being carried out at IIMAS (Instituto de Investigaciones en Matematicas Applicadas y en Systemas), Universidad Nacional Autonoma de Mexico.

Origin of the Programs

The major part of the library consists of procedures that have been developed at the Mathematical Centre. However, some procedures are adapted versions of procedures published in the open literature. In particular a number of programs are derived from procedures published by A. Bjoerck, R. Bulirsch, J.R. Bunch, G.H. Golub, L. Kaufman, B. Lindberg, B.N. Parlett, C. Reinsch and J. Stoer.

List of Contributors and Authors of Procedures in NUMAL

J.J.G. ADMIRAAL
M. BAKKER                        J.KOOPMAN
P.A. BEENTJES                    G.J. LAAN
J.C.P. BUS                       C.G. VAN DER LAAN
C.H. CONVENT                     R. MONTIJN
T.M.T. COOLEN                    R. PLOEGER
J. DANIEL                        T.H.P. REYMER
K. DEKKER                        H.J.J. TE RIELE
TH.J. DEKKER                     C.J. ROOTHART
B. VAN DOMSELAAR                 H.G.J. ROZENHART
H. FIOLET                        E. SLAGT
H.N. GLORIE                      N.M. TEMME
P.J. HARINGHUIZEN                J.G. VERWER
P.W. HEMKER                      M. VOORINTHOLT
W. HOFFMANN                      D.T. WINTER
P.J. VAN DER HOUWEN              J. WOLLESWINKEL
S.P.N. VAN KAMPEN                A.C. IJSSELSTEIN
J. KOK                           J.A. ZONNEVELD

ACKNOWLEDGEMENTS

The library NUMAL has been developed by the joint efforts of the members of the Library Group of the Numerical Mathematics department of the Mathematical Centre. In particular, however, we want to acknowledge the members F. Groen, K. van 't Hoff, R. Piscaer, B.P. Sommeijer, G.J.F. Vinkesteyn, who took care of file manipulation, editing of the documentation files and adaption and running of the KWIC index program.

Further we want to acknowledge the members of the drafting comittee: C. den Heyer, P.J. van der Houwen, J. Kok, N.M. Temme and D.T. Winter, and the external advisors: Th. J. Dekker, W. Hoffmann (University of Amsterdam) and C.G. van der Laan (University of Groningen).

P.W. HEMKER
GENERAL EDITOR