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Evaluates the Chebyshev polynomial series

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Interface

#include <codecogs/maths/approximation/polynomial/cheb_eval.h>

using namespace Maths::Approximation::Polynomial;

double chebEval(double x, const double coef[], int N)

Evaluates the Chebyshev polynomial series of the First Kind:

$f(x) = \sum_{i=0}^{N-1} c_i T_i(\frac{x}{2})$

(1)

where c are the coefficient, and $T-i$ are the Chebyshev polynomials evaluated at x/2,

$T_0(n) = 1$

(2)

$T_1(n) = x$

(3)

$T_2(n) = 2n^2 -1$

(4)

$T_3(n) = 4n^3 - 3n$

(5)

$T_4(n) = 8n^4 - 8n^2 +1$

(6)

$T_5(n) = 16n^5 - 20n^3 + 5n$

(7)

$T_6(n) = 32n^6 = 48n^4 + 18n^2 -1$

(8)

The Chebyshev polynomials of the first kind are a set of orthogonal polynomials defined as the solutions to the Chebyshev differential equation. They are also used as an approximation to a least squares fit and are intimately connected with trigonometric multiple-angle formulas.

If coefficients are for the interval a to b, x must be transformed to

$x \rightarrow \frac{2 (2x - b - a)}{b-a}$

(9)

before entering the routine. This maps x from (a, b) to (-1, 1), over which the Chebyshev polynomials are defined.

If the coefficients are for the inverted interval, in which (a, b) is mapped to (1/b, 1/a), the transformation required is

$x \rightarrow \frac{2 (2ab/x - b - a)}{b-a}$

(10)

If b is infinity, this becomes

$x \rightarrow \frac{4a}{x} - 1$

(11)

Speed:

Taking advantage of the recurrence properties of the Chebyshev polynomials, the routine requires one more addition per loop than evaluating a nested polynomial of the same degree.

Example:

The following code computes solutions to the polynomial

$f(x) = 1 + 2\frac{x}{2} + 3 \left ( 2\frac{x}{2}^2 - 1 \right )$

(12)

#include <stdio.h>
#include <codecogs/maths/approximation/polynomial/cheb_eval.h>

int main()
{
  using namespace Maths::Algebra::Polynomial;
  static double C[] = { 3,2,1 };
  for(int x=2;x<=5;x++)
    printf("\n chebEval(%d, A, 2)=%.1lf", x, chebEval(x, C, 2));

  return 0;
}

Output:

chebEval(2, A, 2)=4.0
chebEval(3, A, 2)=5.5
chebEval(4, A, 2)=7.0
chebEval(5, A, 2)=8.5

References

Cephes Math Library Release 2.0: April, 1987

Parameters

value to evaluate

coef

coefficients from [0..N-1], stored in reverse order.

number of coefficients, not the order. Must be 2 or more

The provided coefficients are stored in reverse order, i.e.

$coef[0] = C_{N-1}, ..., coef[N-1]=C_0$

(13)

Cheb_Eval