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Clenshaw algorithm - Definition and Overview |
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In the mathematical subfield of numerical analysis the Clenshaw algorithm is a recursive method to evaluate polynomials in Chebyshev form.
Polynomial in Chebyshev form
A polynomial of degree N in Chebyshev form is a polynomial p(x) of the form
- <math>p(x) = \sum_{n=0}^{N} a_n T_n(x)<math>
where Tn is the nth Chebyshev polynomial.
Clenshaw algorithm
The Clenshaw algorithm can be used to evaluate a polynomial in the Chebyshev form. Given
- <math>p(x) = \sum_{n=0}^{N} a_n T_n(x)<math>
we define
| <math>b_{N} \,\!<math>
| <math>:= a_{N} \,<math>
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| <math>b_{N-1} \,\!<math>
| <math>:= 2 x b_{N} + a_{N-1} \,<math>
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| <math>b_{N-n} \,\!<math>
| <math>:= 2 x b_{N-n+1} + a_{N-n} + b_{N-n+2} \,,\; n=2,\ldots,N-1 \,<math>
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| <math>b_{0} \,\!<math>
| <math>:= x b_{1} + a_{0} - b_{2} \,<math>
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then
- <math>p(x) = \sum_{n=0}^{N} a_n T_n(x) = b_{0}<math>
See also
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Example Usage of algorithm |
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SanFran_TechJob: New #job: algorithm Testing Engineer #jobs #tech http://bit.ly/6yowY0 |
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kohlerm: @williamlouth yes indexes help. but #mat pioneered using the Lengauer-T. http://bit.ly/7eMUOH algorithm which is almost O(n) vs O(n^2) |
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niseema: @AnnykaV He does algorithm work, writes code, pretty much exclusively with audio/video stuffs. |
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