# vandermonde_approx_2d

vandermonde_approx_2d, a C++ code which finds P(X,Y), a polynomial approximant to Z which depends on two independent variables X and Y, 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_2d is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

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

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

TEST_INTERP_2D, a C++ code which defines test problems for interpolation of data (x,y,z(x,y)), with the data points (x,y) scattered irregularly.

VANDERMONDE_APPROX_1D, a C++ code which finds a polynomial approximant to data y(x) of a 1D argument by setting up and solving an overdetermined linear system for the polynomial coefficients, involving the Vandermonde matrix.

VANDERMONDE_INTERP_2D, a C++ code which finds a polynomial interpolant to data z(x,y) of a 2D argument 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 09 April 2020.