# vandermonde_approx_1d

vandermonde_approx_1d, an Octave 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 approximation 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 a Python version.

### Related Data and Programs:

bernstein_polynomial, an Octave code which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

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

vandermonde_interp_1d, an Octave 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:

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;
• poly0.png, a plot of the polynomial interpolant for problem p01, degree 0;
• poly1.png, a plot of the polynomial interpolant for problem p01, degree 1;
• poly2.png, a plot of the polynomial interpolant for problem p01, degree 2;
• poly3.png, a plot of the polynomial interpolant for problem p01, degree 3;
• poly4.png, a plot of the polynomial interpolant for problem p01, degree 4;
• poly5.png, a plot of the polynomial interpolant for problem p01, degree 5;
• poly7.png, a plot of the polynomial interpolant for problem p01, degree 7;
• poly9.png, a plot of the polynomial interpolant for problem p01, degree 9;
• poly12.png, a plot of the polynomial interpolant for problem p01, degree 12;
• poly15.png, a plot of the polynomial interpolant for problem p01, degree 15;
• poly18.png, a plot of the polynomial interpolant for problem p01, degree 18;