Mathematica notebooks: OrthoPIter and cell ChebyU

C and Fortran-90 function: ChebyU

Background

By comparison with the Chebyshev polynomial of the first kind, T_n(x), the polynomial of the second kind, U_n(x), is relatively little used. One reason for this is that near the endpoints x = ±1 the polynomial changes rapidly, as seen in Visualization below.

 

 

 

Function Definition

Chebyshev's polynomial of the second kind can be defined in terms of the differential equation

 

 

or its explicit expression in trigonometric form

 

 

from which follows the reflection symmetry U_n(-x) = (-1)ⁿU_n(x). Alternatively, one can use the recurrence relation

 

to build up the U_n , given U₀(x)  = 1 and U₁(x) = 2x. From the recurrence equation and these values one also derives that U_n (1) = n+1.

    Formulas for Chebyshev polynomials of the second kind, from n = 0 up to a chosen maximum value n = maxn, can be generated iteratively by using cell ChebyU in notebook OrthoPIter, described in the chapter Overview of Orthogonal Polynomials. For example, with maxn = 4 one obtains:

 

 

 

 

 

 

Visualization

A plot of the Chebyshev polynomials of the second kind as functions of n and x shows a very interesting surface, arising from their relation to sine and cosine functions. We start the surface at n = 2 and run up to n = 8. (The functions exist for noninteger n but are not orthogonal.) The range of x is limited to [-0.8, 0.8] so that values for small n are not dwarfed by those for large n.

 

 

This graphic is obtained by running Mathematica cell ChebyU.

 

Algorithm and Program

For computing numerical values of U_n(x) we use the explicit polynomial expressions for n < 4, and for most x values the trigonometric form for larger n. For x near ±1 this form becomes indeterminate, therefore a suitable approximation—which requires accuracy parameter eps—is needed. Two considerations affect the programming of function ChebyU for U_n(x). First, the optimum switchover n value between the two methods depends upon the relative efficiency of computing polynomial, trigonometric, and inverse trigonometric functions in your computer system. Second is the indeterminate form of the trigonometric form as x approaches ±1. To handle this danger zone—for x near 1, say—we expand the numerator and denominator of this form into Taylor series about their x = 1 values, keeping terms through the squares of the angles. In the numerator, if the first neglected term is to contribute a fractional accuracy e, then x and n must be related by

 

Within this range of x we use the approximation

 

ChebyU therefore requires the additional input parameter eps to control the accuracy of this approximation near the endpoints.   

 

C driver and function for ChebyU

 

 

Fortran-90 driver and function for ChebyU