line_fekete_rule, a C code which approximates the location of Fekete points in an interval [A,B]. A family of sets of Fekete points, indexed by size N, represents an excellent choice for defining a polynomial interpolant.
Given a desired number of points N, the best choice for abscissas is a set of Lebesgue points, which minimize the Lebesgue constant, which describes the error in polynomial interpolation. Sets of Lebesgue points are difficult to define mathematically. Fekete points are a related, computable set, defined as those sets maximizing the magnitude of the determinant of the Vandermonde matrix associated with the points. Analytic definitions of these points are known for a few cases, but there is a general computational procedure for approximating them, which is demonstrated here.
The information on this web page is distributed under the MIT license.
line_fekete_rule is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and a Octave version.
c_rule, a C code which computes a quadrature rule which estimates the integral of a function f(x), which might be defined over a one dimensional region (a line) or more complex shapes such as a circle, a triangle, a quadrilateral, a polygon, or a higher dimensional region, and which might include an associated weight function w(x).
line_grid, a C code which computes a grid of points over the interior of a line segment in 1D.
qr_solve, a C code which computes the least squares solution of a linear system A*x=b.
quadrature_weights_vandermonde, a C code which computes the weights of a quadrature rule using the Vandermonde matrix, assuming that the points have been specified.
vandermonde, a C code which carries out certain operations associated with the Vandermonde matrix.