Some problems in optimally stable Lagrangian differentiation
Author:
Herbert E. Salzer
Journal:
Math. Comp. 28 (1974), 11051115
MSC:
Primary 65D25
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 npoint 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:
http://dx.doi.org/10.1090/S00255718197403683910
PII:
S 00255718(1974)03683910
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
