VANDERMONDE_APPROX_2D
Data Approximation with Polynomials using the Vandermonde Matrix

VANDERMONDE_APPROX_2D is a MATLAB library which finds P(X,Y), a polynomial approximant to Z which depends on two independent variables X and Y, by setting up and solving an overdetermined linear system involving the Vandermonde matrix.

This software is primarily intended as an illustration of the problems that can occur when the approximation 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.

VANDERMONDE_APPROX_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_APPROX_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:

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

TEST_INTERP_2D, a MATLAB library which defines test problems for interpolation of data (x,y,z(x,y)), with the data points (x,y) scattered irregularly.

VANDERMONDE_APPROX_1D, a MATLAB library which finds a polynomial approximant to data y(x) of a 1D argument by setting up and solving an overdetermined linear system for the polynomial coefficients, involving the Vandermonde matrix.

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:

• triangle_num.m, computes the N-th triangular number.
• vandermonde_approx_2d_coef.m, solves the Vandermonde system of equations for the coefficients of the polynomial that approximates a given set of z(x,y) data.
• vandermonde_approx_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_approx_2d_test.m, calls all the tests;
• vandermonde_approx_2d_test_output.txt, the output file.
• vandermonde_approx_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_poly00.png, a plot of the polynomial interpolant for problem p01, degree 0;
• 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_poly00.png, a plot of the polynomial interpolant for problem p02, degree 0;
• 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_poly00.png, a plot of the polynomial interpolant for problem p03, degree 0;
• 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_poly00.png, a plot of the polynomial interpolant for problem p04, degree 0;
• 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_poly00.png, a plot of the polynomial interpolant for problem p05, degree 0;
• 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_poly00.png, a plot of the polynomial interpolant for problem p06, degree 0;
• 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_poly00.png, a plot of the polynomial interpolant for problem p07, degree 0;
• 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_poly00.png, a plot of the polynomial interpolant for problem p08, degree 0;
• 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_poly00.png, a plot of the polynomial interpolant for problem p09, degree 0;
• 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_poly00.png, a plot of the polynomial interpolant for problem p10, degree 0;
• 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_poly00.png, a plot of the polynomial interpolant for problem p11, degree 0;
• 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_poly00.png, a plot of the polynomial interpolant for problem p12, degree 0;
• 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_poly00.png, a plot of the polynomial interpolant for problem p13, degree 0;
• 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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