# vandermonde_interp_2d

vandermonde_interp_2d, a C++ code which finds P(X,Y), a polynomial interpolant to data Z(X,Y) which depends on two independent variables X and Y, 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. Unless the data points are well separated, and the degree of the polynomial is low, the linear system will become very difficult to store and solve accurately, because the monomials used as basis vectors by the Vandermonde approach become indistinguishable.

If the data is available on a product grid, then both the LAGRANGE_INTERP_2D and VANDERMONDE_INTERP_2D libraries will be trying to compute the same interpolating function. However, especially for higher degree polynomials, the Lagrange approach will be superior because it avoids the badly conditioned Vandermonde matrix associated with the usage of monomials as the basis. The Lagrange approach uses as a basis a set of Lagrange basis polynomials l(i,j)(x) which are 1 at node (x(i),y(j)) and zero at the other nodes.

### Languages:

vandermonde_interp_2d is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

LAGRANGE_INTERP_2D, a C++ code which defines and evaluates the Lagrange polynomial p(x,y) which interpolates a set of data depending on a 2D argument that was evaluated on a product grid, so that p(x(i),y(j)) = z(i,j).

PADUA, a C++ code which returns the points and weights for Padu sets, useful for interpolation in 2D. GNUPLOT is used to plot the points.

PWL_INTERP_2D, a C++ code which evaluates a piecewise linear interpolant to data defined on a regular 2D grid.

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_2D, a C++ code which defines and evaluates radial basis function (RBF) interpolants to 2D data.

SHEPARD_INTERP_2D, a C++ code which defines and evaluates Shepard interpolants to 2D data, which are based on inverse distance weighting.

TEST_INTERP_2D, a C++ code which defines test problems for interpolation of data z(x,y) of a 2D argument.

TOMS886, a C++ code which defines the Padua points for interpolation in a 2D region, including the rectangle, triangle, and ellipse, by Marco Caliari, Stefano de Marchi, Marco Vianello. This is a C++ version of ACM TOMS algorithm 886.

VANDERMONDE_APPROX_2D, a C++ code which finds a polynomial approximant p(x,y) to data z(x,y) of a 2D argument by setting up and solving an overdetermined linear system for the polynomial coefficients involving the Vandermonde matrix.

VANDERMONDE_INTERP_1D, a C++ code which finds a polynomial interpolant to a function of 1D data 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.