Sparse matrix technology tools in APL

ABSTRACT

We have implemented sparse matrix technology tools in APL. Such tools have been conspicuously scarce, because APL has not been the language of choice for solving boundary value problems governed by partial differential equations. But when carefully coded, APL is able to tackle problems governed by partial differential equations in a way that ads flexibility and, on account of its compactness, maintainability. The main criticism of APL in numerically intensive applications has been execution speed. APL compilation addresses this drawback and shows factors of speed improvement of better than about three. Timings will be presented for some benchmark elliptical boundary value problems, both for interpretive and compiled APL. Examples are given of common tasks that are encountered in conjunction with the finite element method, such as determination of the symbolic form of the stiffness matrix, and the more universal task of solution of a sparse (symmetric) set of equations using the conjugate gradient method.

REFERENCES

• 1: W. Foote , J. Kraemer , G. Foster, APL2 implementation of numerical asset pricing models, Proceedings of the international conference on APL, p.120-125, December 1987, Sydney, Australia
• 2: H.R. Schwarz, 'Finite Element Methods,' Compmational Mathematics and Applications, Academic Press, 1988, Chap. 1.
• 3: G. R. di Brozolo , M. Vitaletti, Conjugate-gradient subroutines for the IBM 3090 vector facility, IBM Journal of Research and Development, v.33 n.2, p.125-135, March 1989
• 4: W.-M. Ching and A. Xu, "A vector Code Back End of the APL370 Compiler on IBM 3090 and Some Performance Comparisons,' APL Quote Quad, Vol. 18, No. 2, Dec. 1987, ACM Press, pp 69-76.

INDEX TERMS

Primary Classification:
  D. Software
  D.3 PROGRAMMING LANGUAGES
      D.3.2 Language Classifications

          Nouns: APL

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.1 Numerical Algorithms and Problems
          Subjects: Computations on matrices

  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.3 Numerical Linear Algebra
          Subjects: Sparse, structured, and very large systems (direct and iterative methods)
      G.1.8 Partial Differential Equations
          Subjects: Finite element methods

General Terms:
Algorithms, Design, Languages