LAGRANGE_INTERP_2D
Polynomial Interpolation in 2D using Lagrange Polynomials

LAGRANGE_INTERP_2D is a MATLAB 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.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

LAGRANGE_INTERP_2D is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

LAGRANGE_INTERP_1D, a MATLAB 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 MATLAB 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).

PWL_INTERP_2D, a MATLAB library which evaluates a piecewise linear interpolant to data defined on a regular 2D grid.

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

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

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

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

VANDERMONDE_INTERP_2D, a MATLAB 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.

Source Code:

• lagrange_basis_function_1d.m, evaluates a 1D Lagrange basis function.
• lagrange_interp_2d.m, evaluates the Lagrange interpolant for a product grid.

Examples and Tests:

The test code requires the test_interp_2d library as well. If this library is available in a separate folder at the same "level" as the lagrange_interp library, then a Matlab command such as "addpath ( '../test_interp_2d')" will make that library accessible for a run of the test program.

• lagrange_interp_2d_test.m, calls all the tests;
• lagrange_interp_2d_test_output.txt, the output file.
• lagrange_interp_2d_test01.m, tests the interpolation condition, and the approximation power of the interpolant at the midpoints.
• lagrange_interp_2d_test02.m, plots a linear interpolant to a triangulation of the original data, and the Lagrange interpolant.

lagrange_interp_2d_test02() plots a piecewise linear interpolant to the original data, and the Lagrange interpolant.

• p01_data.png, a plot of the piecewise linear interpolant to the data for problem p01;
• p01_poly01.png, a plot of the polynomial interpolant for problem p01, degree 1;
• p01_poly02.png, a plot of the polynomial interpolant for problem p01, degree 2;
• p01_poly03.png, a plot of the polynomial interpolant for problem p01, degree 3;
• p01_poly04.png, a plot of the polynomial interpolant for problem p01, degree 4;
• p01_poly08.png, a plot of the polynomial interpolant for problem p01, degree 8;
• p02_data.png, a plot of the piecewise linear interpolant to the data for problem p02;
• p02_poly01.png, a plot of the polynomial interpolant for problem p02, degree 1;
• p02_poly02.png, a plot of the polynomial interpolant for problem p02, degree 2;
• p02_poly03.png, a plot of the polynomial interpolant for problem p02, degree 3;
• p02_poly04.png, a plot of the polynomial interpolant for problem p02, degree 4;
• p02_poly08.png, a plot of the polynomial interpolant for problem p02, degree 8;
• p03_data.png, a plot of the piecewise linear interpolant to the data for problem p03;
• p03_poly01.png, a plot of the polynomial interpolant for problem p03, degree 1;
• p03_poly02.png, a plot of the polynomial interpolant for problem p03, degree 2;
• p03_poly03.png, a plot of the polynomial interpolant for problem p03, degree 3;
• p03_poly04.png, a plot of the polynomial interpolant for problem p03, degree 4;
• p03_poly08.png, a plot of the polynomial interpolant for problem p03, degree 8;
• p04_data.png, a plot of the piecewise linear interpolant to the data for problem p04;
• p04_poly01.png, a plot of the polynomial interpolant for problem p04, degree 1;
• p04_poly02.png, a plot of the polynomial interpolant for problem p04, degree 2;
• p04_poly03.png, a plot of the polynomial interpolant for problem p04, degree 3;
• p04_poly04.png, a plot of the polynomial interpolant for problem p04, degree 4;
• p04_poly08.png, a plot of the polynomial interpolant for problem p04, degree 8;
• p05_data.png, a plot of the piecewise linear interpolant to the data for problem p05;
• p05_poly01.png, a plot of the polynomial interpolant for problem p05, degree 1;
• p05_poly02.png, a plot of the polynomial interpolant for problem p05, degree 2;
• p05_poly03.png, a plot of the polynomial interpolant for problem p05, degree 3;
• p05_poly04.png, a plot of the polynomial interpolant for problem p05, degree 4;
• p05_poly08.png, a plot of the polynomial interpolant for problem p05, degree 8;
• p06_data.png, a plot of the piecewise linear interpolant to the data for problem p06;
• p06_poly01.png, a plot of the polynomial interpolant for problem p06, degree 1;
• p06_poly02.png, a plot of the polynomial interpolant for problem p06, degree 2;
• p06_poly03.png, a plot of the polynomial interpolant for problem p06, degree 3;
• p06_poly04.png, a plot of the polynomial interpolant for problem p06, degree 4;
• p06_poly08.png, a plot of the polynomial interpolant for problem p06, degree 8;
• p07_data.png, a plot of the piecewise linear interpolant to the data for problem p07;
• p07_poly01.png, a plot of the polynomial interpolant for problem p07, degree 1;
• p07_poly03.png, a plot of the polynomial interpolant for problem p07, degree 3;
• p07_poly02.png, a plot of the polynomial interpolant for problem p07, degree 2;
• p07_poly04.png, a plot of the polynomial interpolant for problem p07, degree 4;
• p07_poly08.png, a plot of the polynomial interpolant for problem p07, degree 8;
• p08_data.png, a plot of the piecewise linear interpolant to the data for problem p08;
• p08_poly01.png, a plot of the polynomial interpolant for problem p08, degree 1;
• p08_poly02.png, a plot of the polynomial interpolant for problem p08, degree 2;
• p08_poly03.png, a plot of the polynomial interpolant for problem p08, degree 3;
• p08_poly04.png, a plot of the polynomial interpolant for problem p08, degree 4;
• p08_poly08.png, a plot of the polynomial interpolant for problem p08, degree 8;
• p09_data.png, a plot of the piecewise linear interpolant to the data for problem p09;
• p09_poly01.png, a plot of the polynomial interpolant for problem p09, degree 1;
• p09_poly02.png, a plot of the polynomial interpolant for problem p09, degree 2;
• p09_poly03.png, a plot of the polynomial interpolant for problem p09, degree 3;
• p09_poly04.png, a plot of the polynomial interpolant for problem p09, degree 4;
• p09_poly08.png, a plot of the polynomial interpolant for problem p09, degree 8;
• p10_data.png, a plot of the piecewise linear interpolant to the data for problem p10;
• p10_poly01.png, a plot of the polynomial interpolant for problem p10, degree 1;
• p10_poly02.png, a plot of the polynomial interpolant for problem p10, degree 2;
• p10_poly03.png, a plot of the polynomial interpolant for problem p10, degree 3;
• p10_poly04.png, a plot of the polynomial interpolant for problem p10, degree 4;
• p10_poly08.png, a plot of the polynomial interpolant for problem p10, degree 8;
• p11_data.png, a plot of the piecewise linear interpolant to the data for problem p11;
• p11_poly01.png, a plot of the polynomial interpolant for problem p11, degree 1;
• p11_poly02.png, a plot of the polynomial interpolant for problem p11, degree 2;
• p11_poly03.png, a plot of the polynomial interpolant for problem p11, degree 3;
• p11_poly04.png, a plot of the polynomial interpolant for problem p11, degree 4;
• p11_poly08.png, a plot of the polynomial interpolant for problem p11, degree 8;
• p12_data.png, a plot of the piecewise linear interpolant to the data for problem p12;
• p12_poly01.png, a plot of the polynomial interpolant for problem p12, degree 1;
• p12_poly02.png, a plot of the polynomial interpolant for problem p12, degree 2;
• p12_poly03.png, a plot of the polynomial interpolant for problem p12, degree 3;
• p12_poly04.png, a plot of the polynomial interpolant for problem p12, degree 4;
• p12_poly08.png, a plot of the polynomial interpolant for problem p12, degree 8;
• p13_data.png, a plot of the piecewise linear interpolant to the data for problem p13;
• p13_poly01.png, a plot of the polynomial interpolant for problem p13, degree 1;
• p13_poly02.png, a plot of the polynomial interpolant for problem p13, degree 2;
• p13_poly03.png, a plot of the polynomial interpolant for problem p13, degree 3;
• p13_poly04.png, a plot of the polynomial interpolant for problem p13, degree 4;
• p13_poly08.png, a plot of the polynomial interpolant for problem p13, degree 8;

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