Chebyshev polynomials

The Chebyshev polynomials of the first (T_n  and second (U_n ) kinds can be defined by the trigonometric identity

T_n (x) = cos(n arccos x)
U_n (x) = sin((n+1) arccos x) / sin( arccos x)

or can be constructed using the three term recurrence relation:

T₀ (x) = 1
T₁ (x) = x
T_n+1 (x) = 2xT_n (x) - T_n-1 (x)
U₀ (x) = 1
U₁ (x) = 2x
U_n+1 (x) = 2xT_n (x) - T_n-1 (x)

Unit description

The recurrence relation given above is the most efficient way to calculate the Chebyshev polynomial. The ChebyshevCalculate subroutine uses this relation to calculate T_n (x) and/or U_n (x) for any given x.

The ChebyshevSum subroutine calculates the sum of Chebyshev polynomials c₀ T₀ (x) + c₁ T₁ (x) + ... + c_n T_n (x) using Clenshaw's recurrence formula.

The ChebyshevCoefficients subroutine can represent T_n (x) as a sum of powers of x: c₀  + c₁ x + ... + c_n x ⁿ.

The FromChebyshev subroutine can perform a conversion of a series of Chebyshev polynomials to a power series.

This article is intended for personal use only.

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