# lagrange_interp_2d

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).

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 code 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 code 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 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.

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, based on inverse distance weighting.

TEST_INTERP_2D, a C code which defines test problems for interpolation of data z(x,y)), depending on 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_INTERP_2D, a C code 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.