# lebesgue

lebesgue, a FORTRAN90 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, and which uses gnuplot to make plots of the Lebesgue function.

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.

### 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:

gnuplot_test, FORTRAN90 codes which write data and command files so that gnuplot() can create plots.

INTERP, a FORTRAN90 code which can be used for parameterizing and interpolating data;

QUADRULE, a FORTRAN90 code which defines quadrature rules for approximating an integral over a 1D domain.

### Reference:

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

### Source Code:

Last revised on 25 July 2020.