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 x^2 - 1. For further information see Abramowitz & Stegun, Chapter 22.
GSL::Cheb classGSL::Cheb.alloc(n)GSL::Cheb#init(f, a, b)ex: Approximate a step function defined in (0, 1) by a Chebyshev series of order 40.
f = GSL::Function.alloc { |x|
if x < 0.5
0.25
else
0.75
end
}
cs = GSL::Cheb.alloc(40)
cs.init(f, 0, 1)GSL::Cheb#eval(x)GSL::Cheb#eval_n(n, x)GSL::Cheb#calc_deriv()GSL::Cheb#deriv()GSL::Cheb#calc_integ()GSL::Cheb#integ()#!/usr/bin/env ruby
require("gsl")
f = GSL::Function.alloc { |x|
if x < 0.5
0.25
else
0.75
end
}
n = 1000
order = 40
cs = GSL::Cheb.alloc(order)
cs.init(f, 0, 1)
x = Vector.linspace(0, 1, n)
ff = f.eval(x)
r10 = cs.eval_n(10, x)
r40 = cs.eval(x)
GSL::graph(x, ff, r10, r40)
See also the example scripts in examples/cheb/.