# chebyshev

chebyshev, a Python code which constructs the Chebyshev interpolant to a function.

Note that the user is not free to choose the interpolation points. Instead, the function f(x) will be evaluated at points chosen by the algorithm. In the standard case, in which the interpolation interval is [-1,+1], these points will be the zeros of the Chebyshev polynomial of order N. However, the algorithm can also be applied to an interval of the form [a,b], in which case the evaluation points are linearly mapped from [-1,+1].

The resulting interpolant is defined by a set of N coefficients c(), and has the form:

```        C(f)(x) = sum ( 1 <= i <= n ) c(i) T(i-1,x) - 0.5 * c(1)
```
where T(i-1,x) is the (i-1)-th Chebyshev polynomial.

Within the interval [-1,+1], or the generalized interval [a,b], the interpolant actually remains bounded by the sum of the absolute values of the coefficients c(). It is therefore common to use Chebyshev interpolants as approximating functions over a given interval.

### Languages:

chebyshev is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

### Related Data and Programs:

bernstein_polynomial, a Python code which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

chebyshev_polynomial, a Python code which evaluates the Chebyshev polynomial and associated functions.

clausen, a Python code which evaluates a Chebyshev approximant to the Clausen function Cl2(x).

lagrange_interp_1d, a Python code which defines and evaluates the Lagrange polynomial p(x) which interpolates a set of data, so that p(x(i)) = y(i).

rbf_interp_1d, a Python code which defines and evaluates radial basis function (RBF) interpolants to 1D data.

shepard_interp_1d, a Python code which defines and evaluates Shepard interpolants to 1D data, which are based on inverse distance weighting.

test_interp_1d, a Python code which defines test problems for interpolation of data y(x), depending on a 1D argument.

vandermonde_interp_1d, a Python code which finds a polynomial interpolant to a function of 1D data by setting up and solving a linear system for the polynomial coefficients, involving the Vandermonde matrix.

### Reference:

1. Roger Broucke,
Algorithm 446: Ten Subroutines for the Manipulation of Chebyshev Series,
Communications of the ACM,
Volume 16, Number 4, April 1973, pages 254-256.
2. Philip Davis,
Interpolation and Approximation,
Dover, 1975,
ISBN: 0-486-62495-1,
LC: QA221.D33

### Source Code:

Last revised on 10 June 2022