# LAGRANGE_INTERP_2D Polynomial Interpolation in 2D using Lagrange Polynomials

LAGRANGE_INTERP_2D, a C++ library 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).

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.

LAGRANGE_INTERP_2D needs access to the R8LIB library. The test also needs the TEST_INTERP_2D library.

### Languages:

LAGRANGE_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_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).

LAGRANGE_INTERP_ND, a C++ library which defines and evaluates the Lagrange polynomial p(x) which interpolates a set of data depending on a multidimensional argument x that was evaluated on a product grid, so that p(x(i)) = z(i).

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

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

RBF_INTERP_2D, a C++ library which defines and evaluates radial basis function (RBF) interpolants to 2D data.

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

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

TOMS886, a C++ library 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_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.