VANDERMONDE_APPROX_1D
Data Approximation with Polynomials using the Vandermonde Matrix

VANDERMONDE_APPROX_1D is a C++ 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 QR_SOLVE and R8LIB libraries. The test code also needs access to the CONDITION and TEST_INTERP libraries.

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 C++ library which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

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

CONDITION, a C++ library which implements methods of computing or estimating the condition number of a matrix.

LAGRANGE_APPROX_1D, a C++ 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 C++ library which approximates a set of data using a piecewise linear function.

QR_SOLVE, a C++ library which computes the least squares solution of a linear system A*x=b.

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

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

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

VANDERMONDE_APPROX_2D, a C++ 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 C++ 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:

• vandermonde_approx_1d.cpp, the source code.
• vandermonde_approx_1d.hpp, the include file.
• vandermonde_approx_1d.sh, BASH commands to compile the source code.

Examples and Tests:

• vandermonde_approx_1d_prb.cpp, a sample calling program.
• vandermonde_approx_1d_prb.sh, BASH commands to compile and run the sample program.
• vandermonde_approx_1d_prb_output.txt, the output file.

List of Routines:

• VANDERMONDE_APPROX_1D_COEF computes a 1D polynomial approximant.
• VANDERMONDE_APPROX_1D_MATRIX computes a Vandermonde 1D approximation matrix.

You can go up one level to the C++ source codes.