Chebyshev Polynomials and Related Functions

Functions to compute Chebyshev polynomials of the first and second kind.

Usage

chebyshev_next(x, Tn, Tn_1)

chebyshev_t(n, x)

chebyshev_u(n, x)

chebyshev_t_prime(n, x)

chebyshev_clenshaw_recurrence(c, x)

chebyshev_clenshaw_recurrence_ab(c, a, b, x)

Arguments

x

Argument of the polynomial

Tn

Value of the Chebyshev polynomial \((T_n(x))\)

Tn_1

Value of the Chebyshev polynomial \((T_{n-1}(x))\)

n

Degree of the polynomial

c

Coefficients of the Chebyshev polynomial

a

Lower bound of the interval

b

Upper bound of the interval

Value

A single numeric value with the computed Chebyshev polynomial, its derivative, or related functions.

See also

Boost Documentation for more details on the mathematical background.

Examples

# Chebyshev polynomial of the first kind T_2(0.5)
chebyshev_t(2, 0.5)
#> [1] -0.5
# Chebyshev polynomial of the second kind U_2(0.5)
chebyshev_u(2, 0.5)
#> [1] 0
# Derivative of the Chebyshev polynomial of the first kind T_2'(0.5)
chebyshev_t_prime(2, 0.5)
#> [1] 2
# Next Chebyshev polynomial of the first kind T_3(0.5) using T_2(0.5) and T_1(0.5)
chebyshev_next(0.5, chebyshev_t(2, 0.5), chebyshev_t(1, 0.5))
#> [1] -1
# Chebyshev polynomial of the first kind using Clenshaw's recurrence with coefficients
# c = c(1, 0, -1) at x = 0.5
chebyshev_clenshaw_recurrence(c(1, 0, -1), 0.5)
#> [1] 1
# Chebyshev polynomial of the first kind using Clenshaw's recurrence with interval [0, 1]
chebyshev_clenshaw_recurrence_ab(c(1, 0, -1), 0, 1, 0.5)
#> [1] 1.5