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

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.

  • 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)
    

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.

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.

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/.

prev next

Reference index top