LEBESGUE
Estimate the Lebesgue Constant

LEBESGUE, a MATLAB library which is given a set of nodes in 1D, and plots the Lebesgue function, and estimates the Lebesgue constant, which measures the maximum magnitude of the potential error of Lagrange polynomial interpolation.

Any set of nodes in the real line X(I), for 1 <= I <= N, defines a corresponding set of Lagrange basis functions:

        L(I)(X) = product ( 1 <= J <= N, J /= I ) ( X    - X(J) ) 
                / product ( 1 <= J <= N, J /= I ) ( X(I) - X(J) )

with the property that

        L(I)(X(J)) = 0 if I /= J
                     1 if I  = J

The Lebesgue function is formed by the sum of the absolute values of these Lagrange basis functions:

        LF(X) = sum ( 1 <= I <= N ) | L(I)(X) |

and the Lebesgue constant LC is the maximum value of LF(X) over the interpolation interval, which is typically X(1) to X(N), or min ( X(*) ), max ( X(*) ), or [-1,+1], or some user-defined interval.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

LEBESGUE 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:

CCL, a MATLAB program which estimates the Lebesgue constants for sets of points in [-1,+1] computed in several ways. The program is probably of limited interest except as an example of an application of the lebesgue_constant() function.

lebesgue_test

QUADRULE, a MATLAB library which defines quadrature rules for approximating an integral over a 1D domain.

Reference:

Jean-Paul Berrut, Lloyd Trefethen,
Barycentric Lagrange Interpolation,
SIAM Review,
Volume 46, Number 3, September 2004, pages 501-517.

Source Code:

chebyshev1.m, evaluates the Chebyshev Type 1 points.
chebyshev2.m, evaluates the Chebyshev Type 2 points.
chebyshev3.m, evaluates the Chebyshev Type 3 points.
chebyshev4.m, evaluates the Chebyshev Type 4 points.
equidistant1.m, evaluates the Equidistant Type 1 points.
equidistant2.m, evaluates the Equidistant Type 2 points.
equidistant3.m, evaluates the Equidistant Type 3 points.
fejer1.m, evaluates the Fejer Type 1 points.
fejer2.m, evaluates the Fejer Type 2 points.
lebesgue_constant.m, estimates the Lebesgue constant.
lebesgue_function.m, evaluates the Lebesgue function.
lebesgue_plot.m, plots the Lebesgue function.
timestamp.m, prints the current YMDHMS date as a timestamp;