Chebyshev polynomials

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

T(x) = cos(n arccos x)
U(x) = sin((n+1) arccos x) / sin( arccos x)

or can be constructed using the three term recurrence relation:

T(x) = 1
T(x) = x
Tn+1 (x) = 2xT(x) - Tn-1 (x)
U(x) = 1
U(x) = 2x
Un+1 (x) = 2xT(x) - Tn-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(x) and/or U(x) for any given x.

The ChebyshevSum subroutine calculates the sum of Chebyshev polynomials cT(x) + cT(x) + ... + cT(x) using Clenshaw's recurrence formula.

The ChebyshevCoefficients subroutine can represent T(x) as a sum of powers of x: c + cx + ... + cx n.

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

This article is licensed for personal use only.

Download ALGLIB for C++ / C# / ...

ALGLIB Project offers you two editions of ALGLIB:

ALGLIB Free Edition:
delivered for free
offers full set of numerical functionality
extensive algorithmic optimizations
no low level optimizations
non-commercial license

ALGLIB Commercial Edition:
flexible pricing
offers full set of numerical functionality
extensive algorithmic optimizations
high performance (SMP, SIMD)
commercial license with support plan

Links to download sections for Free and Commercial editions can be found below:

ALGLIB 3.12.0 for C++

C++ library.
Delivered with sources.
Monolithic design.
Extreme portability.
Editions:   FREE   COMMERCIAL

ALGLIB 3.12.0 for C#

C# library with native kernels.
Delivered with sources.
VB.NET and IronPython wrappers.
Extreme portability.
Editions:   FREE   COMMERCIAL

ALGLIB 3.12.0 for Delphi

Delphi wrapper around C core.
Delivered as precompiled binary.
Compatible with FreePascal.
Editions:   FREE   COMMERCIAL

ALGLIB 3.12.0 for CPython

CPython wrapper around C core.
Delivered as precompiled binary.
Editions:   FREE   COMMERCIAL