# vandermonde_interp_1d

vandermonde_interp_1d, a C++ code which finds a polynomial interpolant to data by setting up and solving a linear system involving the Vandermonde matrix, creating graphics files for processing by gnuplot.

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.

### Languages:

vandermonde_interp_1d 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:

BARYCENTRIC_INTERP_1D, a C++ 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, a C++ code which determines the combination of Chebyshev polynomials which interpolates a set of data, so that p(x(i)) = y(i).

CONDITION, a C++ code which implements methods of computing or estimating the condition number of a matrix.

DIVDIF, a C++ code which uses divided differences to compute the polynomial interpolant to a given set of data.

gnuplot_test, C++ codes which illustrate how a program can write data and command files so that gnuplot can create plots of the program results.

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

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).

NEAREST_INTERP_1D, a C++ code which interpolates a set of data using a piecewise constant interpolant defined by the nearest neighbor criterion.

NEWTON_INTERP_1D, a C++ code which finds a polynomial interpolant to data using Newton divided differences.

PWL_INTERP_1D, a C++ code which interpolates a set of data using a piecewise linear interpolant.

QR_SOLVE, a C++ code which computes the least squares solution of a linear system A*x=b.

R8LIB, a C++ code which contains many utility routines, using double precision real (R8) arithmetic.

RBF_INTERP_1D, a C++ code which defines and evaluates radial basis function (RBF) interpolants to 1D data.

SHEPARD_INTERP_1D, a C++ code which defines and evaluates Shepard interpolants to 1D data, based on inverse distance weighting.

SPLINE, a C++ code which constructs and evaluates spline interpolants and approximants.

TEST_INTERP, a C++ code which defines a number of test problems for interpolation, 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 2D argument.

VANDERMONDE_APPROX_1D, a C++ code which finds a polynomial approximant to data of a 1D argument by setting up and solving an overdetermined linear system for the polynomial coefficients, involving the Vandermonde matrix.

VANDERMONDE_INTERP_2D, a C++ 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.

### Reference:

1. Kendall Atkinson,
An Introduction to Numerical Analysis,
Prentice Hall, 1989,
ISBN: 0471624896,
LC: QA297.A94.1989.
2. Philip Davis,
Interpolation and Approximation,
Dover, 1975,
ISBN: 0-486-62495-1,
LC: QA221.D33
3. David Kahaner, Cleve Moler, Steven Nash,
Numerical Methods and Software,
Prentice Hall, 1989,
ISBN: 0-13-627258-4,
LC: TA345.K34.

### Source Code:

Last revised on 09 April 2020.