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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
MathSciNet review: 0307441
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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] Germund G. Dahlquist, A special stability problem for linear multistep methods, Nordisk Tidskr. Informations-Behandling 3 (1963), 27–43. MR 0170477
  • [2] George E. Forsythe, Generation and use of orthogonal polynomials for data-fitting with a digital computer, J. Soc. Indust. Appl. Math. 5 (1957), 74–88. MR 0092208
  • [3] C. W. Gear, The automatic integration of stiff ordinary differential equations., Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968) North-Holland, Amsterdam, 1969, pp. 187–193. MR 0260180
  • [4] Peter Henrici, Discrete variable methods in ordinary differential equations, John Wiley & Sons, Inc., New York-London, 1962. MR 0135729
  • [5] R. W. Klopfenstein, Numerical differentiation formulas for stiff systems of ordinary differential equations, RCA Rev. 32 (1971), 447–462. MR 0293854
  • [6] Gabor Szegö, Orthogonal polynomials, American Mathematical Society Colloquium Publications, Vol. 23. Revised ed, American Mathematical Society, Providence, R.I., 1959. MR 0106295
  • [7] E. C. Titchmarsh, Han-shu lun, Translated from the English by Wu Chin, Science Press, Peking, 1964 (Chinese). MR 0197687

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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