# vandermonde_interp_1d

vandermonde_interp_1d, a Python 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.

### 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 python 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 python code which determines the combination of chebyshev polynomials which interpolates a set of data, so that p(x(i)) = y(i).

lagrange_interp_1d, a python 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 python code which interpolates a set of data using a piecewise constant interpolant defined by the nearest neighbor criterion.

newton_interp_1d, a python code which finds a polynomial interpolant to data using newton divided differences.

pwl_interp_1d, a python code which interpolates a set of data using a piecewise linear interpolant.

rbf_interp_1d, a python code which defines and evaluates radial basis function (rbf) interpolants to 1d data.

shepard_interp_1d, a python code which defines and evaluates shepard interpolants to 1d data, which are based on inverse distance weighting.

test_interp_1d, a python code which defines test problems for interpolation of data y(x), depending on a 2d argument.

### 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:

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_vandermonde.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_vandermonde.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_vandermonde.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_vandermonde.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_vandermonde.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_vandermonde.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_vandermonde.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_vandermonde.png, a plot of the polynomial interpolant for problem p08;