Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Minimal error constant numerical differentiation (N.D.) formulas


Authors: A. Pelios and R. W. Klopfenstein
Journal: Math. Comp. 26 (1972), 467-475
MSC: Primary 65D25
DOI: https://doi.org/10.1090/S0025-5718-1972-0307441-2
MathSciNet review: 0307441
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we consider a class of $ k$-step linear multistep methods in the form (1.1) of numerical differentiation (N.D.) formulas. For each $ k$, we have required the property of $ A$-stability which implies at most second order for the associated operator. Among such second-order operators, the parameters of the formulas have been selected to minimize the error constant consistent with the $ A$-stability property.

It is shown that the error constant approaches that of the trapezoidal rule as $ k \to \infty $ and that significant reductions occur for quite modest $ k$. Thus, these results have significance in practical applications.


References [Enhancements On Off] (What's this?)

  • [1] G. Dahlquist, ``A special stability problem for linear multistep methods,'' Nordisk Tidskr. Informationsbehandling, v. 3, 1963, pp. 27-43. MR 30 #715. MR 0170477 (30:715)
  • [2] G. E. Forsythe, ``Generation and use of orthogonal polynomials for data-fitting with a digital computer,'' J. Soc. Indust. Appl. Math., v. 5, 1957, pp. 74-88. MR 19, 1079. MR 0092208 (19:1079e)
  • [3] C. W. Gear, The Automatic Integration of Stiff Ordinary Differential Equations. (With Discussion), Proc. IFIP Congress, Information Processing (Edinburgh, 1968), Vol. 1: Mathematics, Software, North-Holland, Amsterdam, 1969, pp. 187-193. MR 41 #4808. MR 0260180 (41:4808)
  • [4] P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York, 1962, pp. 206-209. MR 24 #B1772. MR 0135729 (24:B1772)
  • [5] R. W. Klopfenstein, ``Numerical differentiation formulas for stiff systems of ordinary differential equations,'' RCA Review, v. 32, 1971, pp. 447-462. MR 0293854 (45:2930)
  • [6] G. Szegö, Orthogonal Polynomials, 2nd rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1959. MR 21 #5029. MR 0106295 (21:5029)
  • [7] E. C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford Univ. Press, Oxford, 1939. MR 0197687 (33:5850)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65D25

Retrieve articles in all journals with MSC: 65D25


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1972-0307441-2
Keywords: Ordinary differential equations, numerical solution, multistep formulas, numerical stability
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society