The downward Clenshaw recurrence formula evaluates a sum of products of indexed coefficients by functions which obey a recurrence
relation. If
![f(x)=sum_(k=0)^Nc_kF_k(x)](/images/equations/ClenshawRecurrenceFormula/NumberedEquation1.svg) |
(1)
|
and
![F_(n+1)(x)=alpha(n,x)F_n(x)+beta(n,x)F_(n-1)(x),](/images/equations/ClenshawRecurrenceFormula/NumberedEquation2.svg) |
(2)
|
where the
s
are known, then define
for
and solve backwards to obtain
and
.
![c_k=y_k-alpha(k,x)y_(k+1)-beta(k+1,x)y_(k+2)](/images/equations/ClenshawRecurrenceFormula/NumberedEquation3.svg) |
(5)
|
The upward Clenshaw recurrence formula is
![y_(-2)=y_(-1)=0](/images/equations/ClenshawRecurrenceFormula/NumberedEquation4.svg) |
(11)
|
![y_k=1/(beta(k+1,x))[y_(k-2)-alpha(k,x)y_(k-1)-c_k]](/images/equations/ClenshawRecurrenceFormula/NumberedEquation5.svg) |
(12)
|
for
.
![f(x)=c_NF_N(x)-beta(N,x)F_(N-1)(x)y_(N-1)-F_N(x)y_(N-2).](/images/equations/ClenshawRecurrenceFormula/NumberedEquation6.svg) |
(13)
|
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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.Referenced on Wolfram|Alpha
Clenshaw Recurrence Formula
Cite this as:
Weisstein, Eric W. "Clenshaw Recurrence Formula."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ClenshawRecurrenceFormula.html
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