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 -step linear multistep methods in the form (1.1) of numerical differentiation (N.D.) formulas. For each , we have required the property of -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 -stability property.

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

**[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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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