Some problems in optimally stable Lagrangian differentiation

Author:
Herbert E. Salzer

Journal:
Math. Comp. **28** (1974), 1105-1115

MSC:
Primary 65D25

DOI:
https://doi.org/10.1090/S0025-5718-1974-0368391-0

MathSciNet review:
0368391

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Abstract: In many practical problems in numerical differentiation of a function that is known, observed, measured, or found experimentally to limited accuracy, the computing error is often much more significant than the truncating error. In numerical differentiation of the *n*-point Lagrangian interpolation polynomial, i.e., , a criterion for optimal stability is minimization of . Let . For and *fixed* in , one problem is to find the *n* 's to give . When the truncation error is negligible for any within , a second problem is to find to obtain . A third much simpler problem, for equally spaced, , is to find to give . For lower values of *n*, some results were obtained on and when , and on when and 2 by direct calculation from available tables of . The relation of and to equally spaced points, Chebyshev points, Chebyshev polynomials for , minimax solutions, and central difference formulas, considering also larger values of *n*, is indicated sketchily.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1974-0368391-0

Keywords:
Numerical differentiation,
Lagrangian interpolation,
optimally stable numerical differentiation,
criteria for optimal stability,
minimin and minimax solutions,
Chebyshev points,
Chebyshev polynomials,
central difference formulas

Article copyright:
© Copyright 1974
American Mathematical Society