# chebyshev

chebyshev, a C++ code which constructs the Chebyshev interpolant or approximant 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.

### Related Data and Programs:

BERNSTEIN_POLYNOMIAL, a C++ code which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

CHEBYSHEV_POLYNOMIAL, a C++ code which evaluates the Chebyshev polynomial and associated functions.

CHEBYSHEV_SERIES, a C++ code which can evaluate a Chebyshev series approximating a function f(x), while efficiently computing one, two or three derivatives of the series, which approximate f'(x), f''(x), and f'''(x).

CLAUSEN, a C++ code which evaluates a Chebyshev interpolant to the Clausen function Cl2(x).

DIVDIF, a C++ code which computes interpolants by divided differences.

HERMITE, a C++ code which computes the Hermite interpolant, a polynomial that matches function values and derivatives.

HERMITE_CUBIC, a C++ code which can compute the value, derivatives or integral of a Hermite cubic polynomial, or manipulate an interpolating function made up of piecewise Hermite cubic polynomials.

LAGRANGE_INTERP_1D, a C++ 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, a C++ code which defines and evaluates radial basis function (RBF) interpolants to multidimensional data.

SPLINE, a C++ code which includes many routines to construct and evaluate spline interpolants and approximants.

TEST_APPROX, a C++ code which defines test problems for approximation, provided as a set of (x,y) data.

TEST_INTERP_1D, a C++ code which defines test problems for interpolation of data y(x), depending on a 1D argument.

### 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
3. William Press, Brian Flannery, Saul Teukolsky, William Vetterling,
Numerical Recipes in C: The Art of Scientific Computing,
Cambridge University Press, 1988,
ISBN: 0-521-35465-X,
LC: QA76.73.C15N865.

### Source Code:

Last revised on 17 February 2020.