Clenshaw Recurrence Formula
The downward Clenshaw recurrence formula evaluates a sum of products of indexed Coefficients byfunctions which obey a recurrence relation. If
and
where the
s are known, then define |  |  | |
 | |  |
for
and solve backwards to obtain 
and 
.
 | |  | |
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 |  | ||
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The upward Clenshaw recurrence formula is
for
.
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Recurrence Relations and Clenshaw's Recurrence Formula.'' §5.5 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 172-178, 1992.
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