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# Cheb Eval

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Evaluates the Chebyshev polynomial series
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Contents

C++

## ChebEval

 doublechebEval( double x const double* coef int N )
Evaluates the Chebyshev polynomial series of the First Kind:

where c are the coefficient, and  are the Chebyshev polynomials evaluated at x/2,








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

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


If b is infinity, this becomes


## 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

#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

### Note

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


### Parameters

 x value to evaluate coef coefficients from [0..N-1], stored in reverse order. N number of coefficients, not the order. Must be 2 or more

### Authors

Stephen L. Moshier Copyright 1985, 1987
Documentation by Will Bateman (August 2005)
##### Source Code

Source code is available when you agree to a GP Licence or buy a Commercial Licence.

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