VANDERMONDE_INTERP_2D
Data Interpolation with Polynomials using the Vandermonde Matrix

VANDERMONDE_INTERP_2D is a MATLAB library 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.

VANDERMONDE_INTERP_2D needs access to the QR_SOLVE and R8LIB libraries. The test code also needs access to 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:

VANDERMONDE_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_2D, 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).

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) of a 2D argument.

VANDERMONDE_APPROX_2D, a MATLAB library 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 MATLAB library 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:

• triangle_num.m, computes the N-th triangular number.
• vandermonde_interp_2d_matrix.m, returns the Vandermonde matrix associated with a given set of 2D nodes (x,y).

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 vandermonde_interp_2d library, then a Matlab command such as "addpath ( '../test_interp_2d')" will make that library accessible for a run of the test program.

• vandermonde_interp_2d_test.m, calls all the tests;
• vandermonde_interp_2d_test_output.txt, the output file.
• vandermonde_interp_2d_test01.m, approximates a given test problem with a polynomial of given order.

The code generates some plots of the data and approximants.

• 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_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_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_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_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_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_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_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_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_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_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_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_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_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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