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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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A necessary condition for $A$-stability of multistep multiderivative methods
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by Rolf Jeltsch PDF
Math. Comp. 30 (1976), 739-746 Request permission

Abstract:

The region of absolute stability of multistep multiderivative methods is studied in a neighborhood of the origin. This leads to a necessary condition for A-stability. For methods where $\rho (\zeta )/(\zeta - 1)$ has no roots of modulus 1 this condition can be checked very easily. For Hermite interpolatory and Adams type methods a necessary condition for A-stability is found which depends only on the error order and the number of derivatives used at $({x_{n + k}},{y_{n + k}})$.
References
    R. L. BROWN, "Multi-derivative numerical methods for the solution of stiff ordinary differential equations," Dept. of Comput. Sci., Univ. of Illinois, Report UIUCDCS-R-74-672, 1974.
  • S. D. Conte, Elementary numerical analysis: An algorithmic approach, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1965. MR 0202267
    • B. L. EHLE, On Padé Approximations to the Exponential Function and A-stable methods for the Numerical Solution of Initial Value Problems, Res. Rep. CSRR 2010, Dept. of Appl. Anal. and Comput. Sci., Univ. of Waterloo, Canada, 1969.
  • W. H. Enright, Second derivative multistep methods for stiff ordinary differential equations, SIAM J. Numer. Anal. 11 (1974), 321–331. MR 351083, DOI 10.1137/0711029
  • C. William Gear, Numerical initial value problems in ordinary differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0315898
  • Eberhard Griepentrog, Mehrschrittverfahren zur numerischen Integration von gewöhnlichen Differentialgleichungssystemen und asymptotische Exaktheit, Wiss. Z. Humboldt-Univ. Berlin Math.-Natur. Reihe 19 (1970), 637–653 (German, with Russian, English and French summaries). MR 0321300
  • Rolf Jeltsch, Stiff stability and its relation to $A_{0}$- and $A(0)$-stability, SIAM J. Numer. Anal. 13 (1976), no. 1, 8–17. MR 411174, DOI 10.1137/0713002
  • Rolf Jeltsch, Multistep methods using higher derivatives and damping at infinity, Math. Comp. 31 (1977), no. 137, 124–138. MR 428716, DOI 10.1090/S0025-5718-1977-0428716-7
  • Rolf Jeltsch, Multistep multiderivative methods and Hermite-Birkhoff interpolation, Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) Congressus Numerantium, No. XVI, Utilitas Math. Publ., Winnipeg, Man., 1976, pp. 417–428. MR 0436600
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Math. Comp. 30 (1976), 739-746
  • MSC: Primary 65L05
  • DOI: https://doi.org/10.1090/S0025-5718-1976-0431690-X
  • MathSciNet review: 0431690