Numeric.Polynomial.Chebyshev

Chebyshev polynomials.

Chebyshev polinomials

A Chebyshev polynomial of the first kind is defined by the following recurrence:

\[\begin{aligned} T_0(x) &= 1 \\ T_1(x) &= x \\ T_{n+1}(x) &= 2xT_n(x) - T_{n-1}(x) \\ \end{aligned} \]

chebyshev

Arguments

:: Vector v Double

=> Double

Parameter of each function.

-> v Double

Coefficients of each polynomial term, in increasing order.

-> Double

Evaluate a Chebyshev polynomial of the first kind. Uses Clenshaw's algorithm.

chebyshevBroucke

Arguments

:: Vector v Double

=> Double

Parameter of each function.

-> v Double

Coefficients of each polynomial term, in increasing order.

-> Double

Evaluate a Chebyshev polynomial of the first kind. Uses Broucke's ECHEB algorithm, and his convention for coefficient handling. It treat 0th coefficient different so

chebyshev x [a0,a1,a2...] == chebyshevBroucke [2*a0,a1,a2...]

References

• Broucke, R. (1973) Algorithm 446: Ten subroutines for the manipulation of Chebyshev series. Communications of the ACM 16(4):254–256. http://doi.acm.org/10.1145/362003.362037
• Clenshaw, C.W. (1962) Chebyshev series for mathematical functions. National Physical Laboratory Mathematical Tables 5, Her Majesty's Stationery Office, London.