# VANDERMONDE_INTERP_1D Polynomial Interpolation with the Vandermonde Matrix

VANDERMONDE_INTERP_1D is a FORTRAN90 library which finds a polynomial interpolant to data by setting up and solving a linear system involving the Vandermonde matrix, writing 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.

VANDERMONDE_INTERP_1D needs access to the QR_SOLVE and R8LIB libraries. The test code also needs access to the CONDITION and TEST_INTERP libraries.

### Licensing:

The computer code and data files 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 FORTRAN90 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 FORTRAN90 library which determines the combination of Chebyshev polynomials which interpolates a set of data, so that p(x(i)) = y(i).

CONDITION, a FORTRAN90 library which implements methods of computing or estimating the condition number of a matrix.

DIVDIF, a FORTRAN90 library which uses divided differences to compute the polynomial interpolant to a given set of data.

GNUPLOT, FORTRAN90 programs which illustrate how a program can write data and command files so that gnuplot can create plots of the program results.

HERMITE, a FORTRAN90 library which computes the Hermite interpolant, a polynomial that matches function values and derivatives.

LAGRANGE_INTERP_1D, a FORTRAN90 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 FORTRAN90 library which interpolates a set of data using a piecewise constant interpolant defined by the nearest neighbor criterion.

NEWTON_INTERP_1D, a FORTRAN90 library which finds a polynomial interpolant to data using Newton divided differences.

PWL_INTERP_1D, a FORTRAN90 library which interpolates a set of data using a piecewise linear interpolant.

QR_SOLVE, a FORTRAN90 library which computes the least squares solution of a linear system A*x=b.

R8LIB, a FORTRAN90 library which contains many utility routines, using double precision real (R8) arithmetic.

RBF_INTERP_1D, a FORTRAN90 library which defines and evaluates radial basis function (RBF) interpolants to 1D data.

SHEPARD_INTERP_1D, a FORTRAN90 library which defines and evaluates Shepard interpolants to 1D data, based on inverse distance weighting.

SPLINE, a FORTRAN90 library which constructs and evaluates spline interpolants and approximants.

TEST_INTERP, a FORTRAN90 library which defines a number of test problems for interpolation, provided as a set of (x,y) data.

TEST_INTERP_1D, a FORTRAN90 library which defines test problems for interpolation of data y(x), depending on a 2D argument.

VANDERMONDE_APPROX_1D, a FORTRAN90 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 FORTRAN90 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 files which can be used by GNUPLOT to create PNG plots:

### 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 FORTRAN90 source codes.

Last revised on 01 June 2013.