vandermonde_interp_1d, an Octave code which finds a polynomial interpolant to data by setting up and solving a linear system involving the Vandermonde matrix.
This software is primarily intended as an illustration of the problems that can occur when the interpolation problem is naively formulated using the Vandermonde matrix. If the underlying interpolating basis is the usual family of monomials, then the Vandermonde matrix will very quickly become ill-conditioned for almost any set of nodes.
If the nodes can be selected, this can provide a small amount of improvement, but, if a polynomial interpolant is desired, a better strategy is to change the basis, which is what is done with the Lagrange interpolation method, in which case, essentially, the linear system to be solved becomes the identity matrix.
VANDERMONDE_INTERP_1D needs access to the QR_SOLVE and R8LIB libraries. The test code also needs access to the TEST_INTERP library.
The computer code and data files described and made available on this web page are distributed under the MIT license
vandermonde_interp_1d 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.
barycentric_interp_1d, an Octave code which defines and evaluates the barycentric Lagrange polynomial p(x) which interpolates a set of data, so that p(x(i)) = y(i). The barycentric approach means that very high degree polynomials can safely be used.
chebyshev_interp_1d, an Octave code which determines the combination of Chebyshev polynomials which interpolates a set of data, so that p(x(i)) = y(i).
divdif, an Octave code which uses divided differences to compute the polynomial interpolant to a given set of data.
hermite_interpolant, an Octave code which computes the Hermite interpolant, a polynomial that matches function values and derivatives.
lagrange_interp_1d, an Octave code which defines and evaluates the Lagrange polynomial p(x) which interpolates a set of data, so that p(x(i)) = y(i).
nearest_interp_1d, an Octave code which interpolates a set of data using a piecewise constant interpolant defined by the nearest neighbor criterion.
newton_interp_1d, an Octave code which finds a polynomial interpolant to data using Newton divided differences.
pwl_interp_1d, an Octave code which interpolates a set of data using a piecewise linear interpolant.
r8lib, an Octave code which contains many utility routines, using double precision real (R8) arithmetic.
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 constructs and evaluates spline interpolants and approximants.
test_interp_1d, an Octave code which defines test problems for interpolation of data y(x), depending on a 2D argument.
vandermonde_approx_1d, an Octave code which finds a polynomial approximant to data z(x,y) of a 1D argument by setting up and solving an overdetermined linear system for the polynomial coefficients, involving the Vandermonde matrix.
vandermonde_interp_2d, an Octave code which finds a polynomial interpolant to data z(x,y) of a 2D argument by setting up and solving a linear system for the polynomial coefficients, involving the Vandermonde matrix.