# VANDERMONDE_INTERP_1D Polynomial Interpolation with the Vandermonde Matrix

VANDERMONDE_INTERP_1D is a MATLAB library 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.

### Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

### Languages:

VANDERMONDE_INTERP_1D is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version.

### Related Data and Programs:

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

DIVDIF, a MATLAB library which uses divided differences to compute the polynomial interpolant to a given set of data.

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

LAGRANGE_INTERP_1D, a MATLAB library 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 MATLAB library which interpolates a set of data using a piecewise constant interpolant defined by the nearest neighbor criterion.

NEWTON_INTERP_1D, a MATLAB library which finds a polynomial interpolant to data using Newton divided differences.

PWL_INTERP_1D, a MATLAB library which interpolates a set of data using a piecewise linear interpolant.

R8LIB, a MATLAB library which contains many utility routines, using double precision real (R8) arithmetic.

RBF_INTERP_1D, a MATLAB library which defines and evaluates radial basis function (RBF) interpolants to 1D data.

SHEPARD_INTERP_1D, a MATLAB library which defines and evaluates Shepard interpolants to 1D data, which are based on inverse distance weighting.

SPLINE, a MATLAB library which constructs and evaluates spline interpolants and approximants.

TEST_INTERP_1D, a MATLAB library which defines test problems for interpolation of data y(x), depending on a 2D argument.

VANDERMONDE_APPROX_1D, a MATLAB library 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, a MATLAB library 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.

### Examples and Tests:

The test code requires the test_interp library as well. If this library is available in a separate folder at the same "level" as this library, then a Matlab command such as "addpath ( '../test_interp')" will make that library accessible for a run of the test program.

The code generates some plots of the data and approximants.

• p01_data.png, a plot of the data and piecewise linear interpolant for problem p01;
• p01_poly.png, a plot of the polynomial interpolant for problem p01;
• p02_data.png, a plot of the data and piecewise linear interpolant for problem p02;
• p02_poly.png, a plot of the polynomial interpolant for problem p02;
• p03_data.png, a plot of the data and piecewise linear interpolant for problem p03;
• p03_poly.png, a plot of the polynomial interpolant for problem p03;
• p04_data.png, a plot of the data and piecewise linear interpolant for problem p04;
• p04_poly.png, a plot of the polynomial interpolant for problem p04;
• p05_data.png, a plot of the data and piecewise linear interpolant for problem p05;
• p05_poly.png, a plot of the polynomial interpolant for problem p05;
• p06_data.png, a plot of the data and piecewise linear interpolant for problem p06;
• p06_poly.png, a plot of the polynomial interpolant for problem p06;
• p07_data.png, a plot of the data and piecewise linear interpolant for problem p07;
• p07_poly.png, a plot of the polynomial interpolant for problem p07;
• p08_data.png, a plot of the data and piecewise linear interpolant for problem p08;
• p08_poly.png, a plot of the polynomial interpolant for problem p08;

You can go up one level to the MATLAB source codes.

Last modified on 29 July 2012.