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Some relations between the values of a function and its first derivative at $ n$ abscissa points


Author: Robert E. Huddleston
Journal: Math. Comp. 25 (1971), 553-558
MSC: Primary 65D25
DOI: https://doi.org/10.1090/S0025-5718-1971-0297097-9
Remarks: Math. Comp. 26, no. 120 (1972), pp. 901-902.
MathSciNet review: 0297097
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Abstract: For a polynomial, P, of degree $ 2n - 2$, there exists a relation between the values of P and the values of its first derivative, $ P'$, at the n abscissa points $ {x_1}, \cdots ,{x_n}$,

$\displaystyle \sum\limits_{i = 1}^n {[{a_i}P({x_i}) + {b_i}P'({x_i})] = 0.} $

Replacing P by a differentiable function y yields

$\displaystyle \sum\limits_{i = 1}^n {[{a_i}y({x_i}) + {b_i}y'({x_i})] = E(y,x).} $

These relations are obtained and the error function $ E(y,x)$ is given explicitly.

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

  • [1] F. Ceschino & J. Kuntzmann, Numerical Solution of Initial Value Problems, Prentice-Hall, Englewood Cliffs, N. J., 1966. MR 33 #3465. MR 0195262 (33:3465)
  • [2] R. E. Huddleston, Variable-Step Truncation Error Estimates for Runge-Kutta Methods of Order 4 or Less, Report #DC-70-261, Sandia Laboratories, Livermore, California.
  • [3] E. Isaacson & H. B. Keller, Analysis of Numerical Methods, Wiley, New York, 1966. MR 34 #924. MR 0201039 (34:924)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1971-0297097-9
Keywords: Ordinary differential equations, Runge-Kutta, error estimation for Runge-Kutta, one-step methods for ordinary differential equations, polynomials
Article copyright: © Copyright 1971 American Mathematical Society

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