# cheb.rdoc

 Path: rdoc/cheb.rdoc Last Update: Sun Nov 14 14:53:48 -0800 2010

# 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.

## 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.

• 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)