Chebyshev Polynomials
The Chebyshev polynomials are two sequences of polynomials, $T_n$ and $U_n$. The Chebyshev polynomials of the first kind, $T_n$, can be defined by the recurrence relation:
\[T_0(x)=1,\ T_1(x)=x\]
\[T_{n+1}(x) = 2xT_n(x)-T_{n-1}(x)\]
The Chebyshev polynomioals of the second kind, $U_n(x)$, can be defined by
\[U_0(x)=1,\ U_1(x)=2x\]
\[U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x)\]
Both $T_n$ and $U_n$ have degree $n$, and any polynomial of degree $n$ may be uniquely written as a linear combination of the polynomials $T_0$, $T_1$, ..., $T_n$ (similarly with $U_n$).
First Kind
Polynomials.ChebyshevT — Type
ChebyshevT{T, X}(coeffs::AbstractVector)Chebyshev polynomial of the first kind.
Construct a polynomial from its coefficients coeffs, lowest order first, optionally in terms of the given variable var, which can be a character, symbol, or string.
ChebyshevT is not axis-aware, and it treats coeffs simply as a list of coefficients with the first index always corresponding to the coefficient of T_0(x).
Examples
julia> using Polynomialsjulia> p = ChebyshevT([1, 0, 3, 4])ChebyshevT(1⋅T_0(x) + 3⋅T_2(x) + 4⋅T_3(x))julia> ChebyshevT([1, 2, 3, 0], :s)ChebyshevT(1⋅T_0(s) + 2⋅T_1(s) + 3⋅T_2(s))julia> one(ChebyshevT)ChebyshevT(1.0⋅T_0(x))julia> p(0.5)-4.5julia> evalpoly(5.0, p, false) # bypasses the domain check done in p(5.0)2088.0The latter shows how to evaluate a ChebyshevT polynomial outside of its domain, which is [-1,1].
The ChebyshevT type holds coefficients representing the polynomial $a_0 T_0 + a_1 T_1 + ... + a_n T_n$.
For example, the basis polynomial $T_4$ can be represented with ChebyshevT([0,0,0,0,1]).
Conversion
ChebyshevT can be converted to Polynomial and vice-versa.
julia> c = ChebyshevT([1, 0, 3, 4])ChebyshevT(1⋅T_0(x) + 3⋅T_2(x) + 4⋅T_3(x))julia> p = convert(Polynomial, c)Polynomial(-2.0 - 12.0*x + 6.0*x^2 + 16.0*x^3)julia> convert(ChebyshevT, p)ChebyshevT(1.0⋅T_0(x) + 3.0⋅T_2(x) + 4.0⋅T_3(x))