VANDERMONDE_INTERP_1D
Polynomial Interpolation with the Vandermonde Matrix

VANDERMONDE_INTERP_1D is a Python library which finds a polynomial interpolant to data 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. If the underlying interpolating basis is the usual family of monomials, then the Vandermonde matrix will very quickly become ill-conditioned for almost any set of nodes.

If the nodes can be selected, this can provide a small amount of improvement, but, if a polynomial interpolant is desired, a better strategy is to change the basis, which is what is done with the Lagrange interpolation method, in which case, essentially, the linear system to be solved becomes the identity matrix.

VANDERMONDE_INTERP_1D needs access to the QR_SOLVE and R8LIB libraries. 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_INTERP_1D is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

BARYCENTRIC_INTERP_1D, a Python library which defines and evaluates the barycentric Lagrange polynomial p(x) which interpolates a set of data, so that p(x(i)) = y(i). The barycentric approach means that very high degree polynomials can safely be used.

CHEBYSHEV_INTERP_1D, a Python library which determines the combination of Chebyshev polynomials which interpolates a set of data, so that p(x(i)) = y(i).

LAGRANGE_INTERP_1D, a Python library which defines and evaluates the Lagrange polynomial p(x) which interpolates a set of data, so that p(x(i)) = y(i).

NEAREST_INTERP_1D, a Python library which interpolates a set of data using a piecewise constant interpolant defined by the nearest neighbor criterion.

NEWTON_INTERP_1D, a Python library which finds a polynomial interpolant to data using Newton divided differences.

PWL_INTERP_1D, a Python library which interpolates a set of data using a piecewise linear interpolant.

RBF_INTERP_1D, a Python library which defines and evaluates radial basis function (RBF) interpolants to 1D data.

SHEPARD_INTERP_1D, a Python library which defines and evaluates Shepard interpolants to 1D data, which are based on inverse distance weighting.

TEST_INTERP_1D, a Python library which defines test problems for interpolation of data y(x), depending on a 2D argument.

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:

• p00_data.py, returns the data for any problem.
• p00_data_num.py, returns the number of data points for any problem.
• p00_dim_num.py, returns the spatial dimension for any problem.
• p00_prob_num.py, returns the number of test problems.
• r8mat_print.py, prints an R8MAT.
• r8mat_print_some.py, prints some of an R8MAT.
• r8mat_transpose_print.py, prints an R8MAT, transposed.
• r8mat_transpose_print_some.py, prints some of an R8MAT, transposed.
• r8poly_print.py, prints an R8POLY.
• r8vec_norm.py, computes the L2 norm of an R8VEC.
• r8vec_norm_affine.py, computes the L2 norm of the difference of two R8VEC's.
• r8vec_print.py, prints an R8VEC.
• r8vec_uniform_ab.py, returns a scaled pseudorandom R8VEC.
• r8vec2_print.py, prints a pair of R8VEC's.
• timestamp.py, prints the YMDHMS date as a timestamp.
• vandermonde_coef_1d.py, solves the Vandermonde system of equations for the coefficients of the polynomial that interpolates a given set of (x,y) data.
• vandermonde_matrix_1d.py, returns the Vandermonde matrix associated with a given set of nodes x.
• vandermonde_value_1d.py, evaluates a Vandermonde interpolant.

Examples and Tests:

• vandermonde_interp_1d_test.py, calls all the tests;
• vandermonde_interp_1d_test.sh, runs all the tests;
• vandermonde_interp_1d_test.txt, the output file.

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_vandermonde.png, a plot of the polynomial interpolant for problem p01;
• p02_data.png, a plot of the data and piecewise linear interpolant for problem p02;
• p02_vandermonde.png, a plot of the polynomial interpolant for problem p02;
• p03_data.png, a plot of the data and piecewise linear interpolant for problem p03;
• p03_vandermonde.png, a plot of the polynomial interpolant for problem p03;
• p04_data.png, a plot of the data and piecewise linear interpolant for problem p04;
• p04_vandermonde.png, a plot of the polynomial interpolant for problem p04;
• p05_data.png, a plot of the data and piecewise linear interpolant for problem p05;
• p05_vandermonde.png, a plot of the polynomial interpolant for problem p05;
• p06_data.png, a plot of the data and piecewise linear interpolant for problem p06;
• p06_vandermonde.png, a plot of the polynomial interpolant for problem p06;
• p07_data.png, a plot of the data and piecewise linear interpolant for problem p07;
• p07_vandermonde.png, a plot of the polynomial interpolant for problem p07;
• p08_data.png, a plot of the data and piecewise linear interpolant for problem p08;
• p08_vandermonde.png, a plot of the polynomial interpolant for problem p08;

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