Point-wise calculus

ABSTRACT

The work of Richard Neidinger implementing automatic differentiation in APL as a vector arithmetic is reformulated and extended. For functions of a single variable, an arithmetic is developed for function samples, nested vectors whose components hold the values, at any number of given sample points, of a function and its derivatives up to any specified order. It is argued that, for teaching purposes, this sampling provides a more intuitive introduction to mathematical functions and the rules of calculus than do algebraic formulae and that for certain calculations (such as the computation of polynomial approximations of high degree) the formulation provides superior algorithms for computation. As such, it offers an alternative approach to the teaching of elementary college mathematics.

REFERENCES

• 1: A.Griewank, On Automatic Differentiation In Mathematical Programming: Recent Developments and Applications, M.Iri & K. Tanabe (eds.) Kluwer, Amsterdam (1989) 83-108.
• 2: A.Griewank & G. Corliss (eds.), Automatic Differentiation of Algorithms: Theory, Implementation & Application, SIAM, Philadelphia (1991)
• 3: G.Helzer, Applied Linear Algebra with APL, Little, Brown & Co. Boston (1983)
• 4: R.D.Neidinger, Automatic Differentiation and APL, College Mathematics Journal 20, 3 (May,1989) 238- 251.
• 5: Richard D. Neidinger, Differential equations are recurrence relations in APL, ACM SIGAPL APL Quote Quad, v.23 n.1, p.165-174, July 1992
• 6: Richard D. Neidinger, An efficient method for the numerical evaluation of partial derivatives of arbitrary order, ACM Transactions on Mathematical Software (TOMS), v.18 n.2, p.159-173, June 1992 [doi>10.1145/146847.146924]
• 7: L.B.Rall, The Arithmetic of Differentiation, Mathematics Magazine 59,5 (Dec,1986) 275-282.
• 8: J.Waldvogel, Der Tayloralgorithmus, J. of Applied Maths. and Physics (ZAMP) 35 (Nov,1984) 780-789.

INDEX TERMS

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

          Nouns: APL

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.0 General
          Subjects: Computer arithmetic

  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION

General Terms:
Algorithms, Languages