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 illconditioned 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:

Kendall Atkinson,
An Introduction to Numerical Analysis,
Prentice Hall, 1989,
ISBN: 0471624896,
LC: QA297.A94.1989.

Philip Davis,
Interpolation and Approximation,
Dover, 1975,
ISBN: 0486624951,
LC: QA221.D33

David Kahaner, Cleve Moler, Steven Nash,
Numerical Methods and Software,
Prentice Hall, 1989,
ISBN: 0136272584,
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:
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.
Last modified on 04 July 2015.