# VANDERMONDE_INTERP_1D Polynomial Interpolation with the Vandermonde Matrix

VANDERMONDE_INTERP_1D is a C++ library 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++ 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 C++ library which determines the combination of Chebyshev polynomials which interpolates a set of data, so that p(x(i)) = y(i).

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

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

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

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

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

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

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

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

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

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

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

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

TEST_INTERP, a C++ library which defines a number of test problems for interpolation, provided as a set of (x,y) data.

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

VANDERMONDE_APPROX_1D, a C++ library 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++ 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 program makes data files that can be used by GNUPLOT to create graphics:

### List of Routines:

• VANDERMONDE_INTERP_1D_COEF computes a 1D polynomial interpolant.
• VANDERMONDE_INTERP_1D_MATRIX computes a Vandermonde 1D interpolation matrix.

You can go up one level to the C++ source codes.

Last revised on 02 June 2013.