chebyshev, an Octave 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.
The computer code and data files described and made available on this web page are distributed under the MIT license
chebyshev is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.
bernstein_polynomial, an Octave code which evaluates the Bernstein polynomials, useful for uniform approximation of functions;
chebyshev_polynomial, an Octave code which evaluates the Chebyshev polynomial and associated functions.
chebyshev_series, an Octave 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.
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interp, an Octave code which can be used for parameterizing and interpolating data;
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rbf_interp_1d, an Octave code which defines and evaluates radial basis function (RBF) interpolants to 1D data.
shepard_interp_1d, an Octave code which defines and evaluates Shepard interpolants to 1D data, which are based on inverse distance weighting.
spline, an Octave code which includes many routines to construct and evaluate spline interpolants and approximants.
test_approx, an Octave code which defines test problems for approximation, provided as a set of (x,y) data.
test_interp_1d, an Octave code which defines test problems for interpolation of data y(x), depending on a 1D argument.
vandermonde_approx_1d, an Octave code which finds a polynomial approximant to a function of 1D data by setting up and solving an overdetermined linear system for the polynomial coefficients, involving the Vandermonde matrix.
vandermonde_interp_1d, an Octave 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.