# vandermonde_approx_1d

vandermonde_approx_1d, a Fortran90 code 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.

### Languages:

vandermonde_approx_1d is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

### Related Data and Programs:

bernstein, a Fortran90 code which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

chebyshev, a Fortran90 code which computes the Chebyshev interpolant/approximant to a given function over an interval.

CONDITION, a Fortran90 code which implements methods of computing or estimating the condition number of a matrix.

LAGRANGE_APPROX_1D, a Fortran90 code 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 Fortran90 code which approximates a set of data using a piecewise linear function.

QR_SOLVE, a Fortran90 code which computes the least squares solution of a linear system A*x=b.

R8LIB, a Fortran90 code which contains many utility routines using double precision real (R8) arithmetic.

SPLINE, a Fortran90 code which constructs and evaluates spline interpolants and approximants.

TEST_APPROX, a Fortran90 code which defines test problems for approximation, provided as a set of (x,y) data.

VANDERMONDE_APPROX_2D, a Fortran90 code 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 Fortran90 code 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:

Last revised on 10 September 2020.