lebesgue

lebesgue, a Python code 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 MIT 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:

quadrule, a Python code 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:

lebesgue.py, the source code.
lebesgue.sh, runs all the tests;
lebesgue.txt, the output file.

The test programs create plots of the Lebesgue function for specific sizes of the point sets.

Last revised on 21 August 2016.