The downward Clenshaw recurrence formula evaluates a sum of products of indexed coefficients by functions which obey a recurrence relation. If

(1)

and

(2)

where the s are known, then define

			(3)
			(4)

for and solve backwards to obtain and .

(5)

			(6)
			(7)
			(8)
			(9)
			(10)

The upward Clenshaw recurrence formula is

(11)

(12)

for .

(13)

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.

CITE THIS AS:

Weisstein, Eric W. "Clenshaw Recurrence Formula." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ClenshawRecurrenceFormula.html