Title: 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 x^2 - 1. For further information see Abramowitz & Stegun, Chapter 22.

  1. GSL::Cheb class
  2. Chebyshev Series Evaluation
  3. Derivatives and Integrals
  4. Examples

1 GSL::Cheb class

GSL::Cheb.alloc(n)
This create an instance of the GSL::Cheb class for a Chebyshev series of order n.
GSL::Cheb#init(f, a, b)
This computes the Chebyshev approximation the function f over the range (a,b) to the previously specified order. Where f is a GSL::Function object. The computation of the Chebyshev approximation is an O(n^2) process, and requires n function evaluations.

2 Chebyshev Series Evaluation

GSL::Cheb#eval(x)
This evaluates the Chebyshev series at a given point x.
GSL::Cheb#eval_n(n, x)
This evaluates the Chebyshev series at a given point x, to (at most) the given order n.

3 Derivatives and Integrals

GSL::Cheb#calc_deriv()
GSL::Cheb#deriv()
This computes the derivative of the series, and returns a new GSL::Cheb object which contains the computed derivative. The reciever is not changed.
GSL::Cheb#calc_integ()
GSL::Cheb#integ()
This computes the integral of the series, and returns a new GSL::Cheb object which contains the computed integral coefficients. The reciever is not changed.

4 Example

#!/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/.

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