Module rgsl::types::chebyshev [−] [src]

Chebyshev Approximations

This chapter describes routines for computing Chebyshev approximations to univariate functions. A Chebyshev approximation is a truncation of the series f(x) = \sum c_n T_n(x), where the Chebyshev polynomials T_n(x) = \cos(n \arccos x) provide an orthogonal basis of polynomials on the interval [-1,1] with the weight function 1 / \sqrt{1-x^2}. The first few Chebyshev polynomials are, T_0(x) = 1, T_1(x) = x, T_2(x) = 2 x² - 1.

For further information see Abramowitz & Stegun, Chapter 22.

Definitions

The approximation is made over the range [a,b] using order+1 terms, including the coefficient c[0]. The series is computed using the following convention,

f(x) = (c_0 / 2) + \sum_{n=1} c_n T_n(x)

which is needed when accessing the coefficients directly.

References and Further Reading

The following paper describes the use of Chebyshev series,

R. Broucke, “Ten Subroutines for the Manipulation of Chebyshev Series C1”. Communications of the ACM 16(4), 254–256 (1973)

Structs

ChebSeries