# chebyshev

chebyshev, a FORTRAN90 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.

### Related Data and Programs:

BERNSTEIN_POLYNOMIAL, a FORTRAN90 code which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

CHEBYSHEV_POLYNOMIAL, a FORTRAN90 code which evaluates the Chebyshev polynomial and associated functions.

CHEBYSHEV_SERIES, a FORTRAN90 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), by Manfred Zimmer.

CLAUSEN, a FORTRAN90 code which evaluates a Chebyshev interpolant to the Clausen function Cl2(x).

DIVDIF, a FORTRAN90 code which computes interpolants by divided differences.

HERMITE, a FORTRAN90 code which computes the Hermite interpolant, a polynomial that matches function values and derivatives.

HERMITE_CUBIC, a FORTRAN90 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 FORTRAN90 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 FORTRAN90 code which defines and evaluates radial basis interpolants to multidimensional data.

SPLINE, a FORTRAN90 code which includes many routines to construct and evaluate spline interpolants and approximants.

TEST_APPROX, a FORTRAN90 code which defines test problems for approximation, provided as a set of (x,y) data.

TEST_INTERP_1D, a FORTRAN90 code which defines test problems for interpolation of data y(x), depending on a 1D argument.

TOMS446, a FORTRAN90 code which manipulates Chebyshev series for interpolation and approximation;
this is a version of ACM TOMS algorithm 446, by Roger Broucke.

VANDERMONDE_INTERP_1D, a FORTRAN90 code which finds a polynomial interpolant to data y(x) of a 1D argument, 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
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 07 September 2021.