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Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55)
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25. Numerical Interpolation, Differentiation, and Integration
Numerical analysts have a tendency to ac-
cumulate a multiplicity of tools each designed for
highly specialized operations and each requiring
special knowledge to use properly. From the
vast stock of formulas available we have culled
the present selection. We hope that it will be
useful. As with all such compendia, the reader
may miss his favorites and find others whose
utility he thinks is marginal.
We would have liked to give examples to
illuminate the formulas, but this has not been
feasible. Numerical analysis is partially a science
and partially an art, and short of writing a text-
book on the subject it has been impossible to
indicate where and under what circumstances the
various formulas are useful or accurate, or to
elucidate the numerical difficulties to which one
might be led by uncritical use. The formulas are
therefore issued together with a caveat against
their blind application.
Notation: Abscissas: xo<xl< . . .; functions:
f, g! . . .; values: f ( x t ) = f t , f ' ( x t ) = f ; . j', jC2), . . .
indicate lst. 2d. . . . derivatives. If abscissas
are equally spaced,. X ~ + ~ - X ( = ~ and f,=f(xo+ph)
( p not necessardy integral). R , Rn indicate re-
G+,,-A++k) k - k if 12 and k are of same parity.
Forward Differences Central Diferences
G f o
25.1.3 CC( j n > = 3 C f n + t + f n - t >
25.1.4 [aa,aI=-- j0-f xo- 21 '-[ XI, G]
Divided Differences in Terms of Functional Values
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