VANDERMONDE_APPROX_1D
Data Approximation with Polynomials using the Vandermonde Matrix

VANDERMONDE_APPROX_1D is a MATLAB library which finds a polynomial approximant to 1D data 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 approximatino 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_INTERP_1D needs access to the R8LIB library. The test code also needs access to the TEST_INTERP 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_1D 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:

BERNSTEIN, a MATLAB library which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

CHEBYSHEV, a MATLAB library which computes the Chebyshev interpolant/approximant to a given function over an interval.

LAGRANGE_APPROX_1D, a MATLAB library which defines and evaluates the Lagrange polynomial p(x) of degree m which approximates a set of nd data points (x(i),y(i)).

PWL_APPROX_1D, a MATLAB library which approximates a set of data using a piecewise linear function.

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

SPLINE, a MATLAB library which constructs and evaluates spline interpolants and approximants.

TEST_APPROX, a MATLAB library which defines test problems for approximation, provided as a set of (x,y) data.

VANDERMONDE_APPROX_2D, a MATLAB library which finds a polynomial approximant p(x,y) to a function of 2D data 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:

• r8mat_transpose_print.m, prints an R8MAT, transposed.
• r8mat_transpose_print_some.m, prints some of an R8MAT, transposed.
• r8poly_value.m, evaluates a polynomial, using the coefficient ordering that is the opposite of that which Matlab's polyval() assumes.
• timestamp.m, prints the current YMDHMS date as a time stamp.
• vandermonde_approx_1d_coef.m, solves the Vandermonde system of equations for the coefficients of the polynomial that interpolates a given set of (x,y) data.
• vandermonde_approx_1d_matrix.m, returns the Vandermonde matrix associated with a given set of nodes x.

Examples and Tests:

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

• vandermonde_approx_1d_test.m, calls all the tests;
• vandermonde_approx_1d_test_output.txt, the output file.
• vandermonde_approx_1d_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 data and piecewise linear interpolant 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_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_poly05.png, a plot of the polynomial interpolant for problem p01, degree 5;
• p01_poly07.png, a plot of the polynomial interpolant for problem p01, degree 7;
• p01_poly09.png, a plot of the polynomial interpolant for problem p01, degree 9;
• p01_poly12.png, a plot of the polynomial interpolant for problem p01, degree 12;
• p02_data.png, a plot of the data and piecewise linear interpolant 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_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_poly05.png, a plot of the polynomial interpolant for problem p02, degree 5;
• p02_poly07.png, a plot of the polynomial interpolant for problem p02, degree 7;
• p02_poly09.png, a plot of the polynomial interpolant for problem p02, degree 9;
• p02_poly12.png, a plot of the polynomial interpolant for problem p02, degree 12;
• p03_data.png, a plot of the data and piecewise linear interpolant 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_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_poly05.png, a plot of the polynomial interpolant for problem p03, degree 5;
• p03_poly07.png, a plot of the polynomial interpolant for problem p03, degree 7;
• p03_poly09.png, a plot of the polynomial interpolant for problem p03, degree 9;
• p03_poly12.png, a plot of the polynomial interpolant for problem p03, degree 12;
• p04_data.png, a plot of the data and piecewise linear interpolant 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_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_poly05.png, a plot of the polynomial interpolant for problem p04, degree 5;
• p04_poly07.png, a plot of the polynomial interpolant for problem p04, degree 7;
• p04_poly09.png, a plot of the polynomial interpolant for problem p04, degree 9;
• p04_poly12.png, a plot of the polynomial interpolant for problem p04, degree 12;
• p05_data.png, a plot of the data and piecewise linear interpolant 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_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_poly05.png, a plot of the polynomial interpolant for problem p05, degree 5;
• p05_poly07.png, a plot of the polynomial interpolant for problem p05, degree 7;
• p05_poly09.png, a plot of the polynomial interpolant for problem p05, degree 9;
• p05_poly12.png, a plot of the polynomial interpolant for problem p05, degree 12;
• p06_data.png, a plot of the data and piecewise linear interpolant 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_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_poly05.png, a plot of the polynomial interpolant for problem p06, degree 5;
• p06_poly07.png, a plot of the polynomial interpolant for problem p06, degree 7;
• p06_poly09.png, a plot of the polynomial interpolant for problem p06, degree 9;
• p06_poly12.png, a plot of the polynomial interpolant for problem p06, degree 12;
• p07_data.png, a plot of the data and piecewise linear interpolant 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_poly03.png, a plot of the polynomial interpolant for problem p07, degree 3;
• p07_poly04.png, a plot of the polynomial interpolant for problem p07, degree 4;
• p07_poly05.png, a plot of the polynomial interpolant for problem p07, degree 5;
• p07_poly07.png, a plot of the polynomial interpolant for problem p07, degree 7;
• p07_poly09.png, a plot of the polynomial interpolant for problem p07, degree 9;
• p07_poly12.png, a plot of the polynomial interpolant for problem p07, degree 12;
• p08_data.png, a plot of the data and piecewise linear interpolant 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_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_poly05.png, a plot of the polynomial interpolant for problem p08, degree 5;
• p08_poly07.png, a plot of the polynomial interpolant for problem p08, degree 7;
• p08_poly09.png, a plot of the polynomial interpolant for problem p08, degree 9;
• p08_poly12.png, a plot of the polynomial interpolant for problem p08, degree 12;

You can go up one level to the MATLAB source codes.