barycentric_interp_1d

barycentric_interp_1d, a Python code which defines and evaluates the Lagrange polynomial p(x) which interpolates a set of data, so that p(x(i)) = y(i). Because a barycentric formulation is used, polynomials of very high degree can safely be used.

Efficient calculation of the barycentric polynomial interpolant requires that the function to be interpolated be sampled at points from a known family, for which the interpolation weights have been precomputed. Such families include

• evenly spaced points (but this results in an ill-conditioned system);
• Chebyshev Type 1 points;
• Chebyshev Type 2 points;
• Chebyshev Type 3 points;
• Chebyshev Type 4 points;

Note that in the Berrut/Trefethen reference, there is a significant typographical error on page 510, where an adjustment is made in cases where the polynomial is to be evaluated exactly at a data point. The paper reads:

          exact(xdiff==0) = 1;
        

          exact(xdiff==0) = j;
        

Licensing:

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

Languages:

barycentric_interp_1d 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:

chebyshev_interp_1d, a python code which determines the combination of chebyshev polynomials which interpolates a set of data, so that p(x(i)) = y(i).

lagrange_interp_1d, a python code which defines and evaluates the lagrange polynomial p(x) which interpolates a set of data, so that p(x(i)) = y(i).

nearest_interp_1d, a python code which interpolates a set of data using a piecewise constant interpolant defined by the nearest neighbor criterion.

newton_interp_1d, a python code which finds a polynomial interpolant to data using newton divided differences.

pwl_interp_1d, a python code which interpolates a set of data using a piecewise linear interpolant.

rbf_interp_1d, a python code which defines and evaluates radial basis function (rbf) interpolants to 1d data.

shepard_interp_1d, a python code which defines and evaluates shepard interpolants to 1d data, which are based on inverse distance weighting.

test_interp, a python code which defines a number of test problems for interpolation, provided as a set of (x,y) data.

test_interp_1d, a python code which defines test problems for interpolation of data y(x), depending on a 2d argument.

vandermonde_interp_1d, a python code which finds a polynomial interpolant to a function of 1d data 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. Jean-Paul Berrut, Lloyd Trefethen,
   Barycentric Lagrange Interpolation,
   SIAM Review,
   Volume 46, Number 3, September 2004, pages 501-517.
3. Philip Davis,
   Interpolation and Approximation,
   Dover, 1975,
   ISBN: 0-486-62495-1,
   LC: QA221.D33
4. David Kahaner, Cleve Moler, Steven Nash,
   Numerical Methods and Software,
   Prentice Hall, 1989,
   ISBN: 0-13-627258-4,
   LC: TA345.K34.

Source Code:

• barycentric_interp_1d.py, the source code.
• barycentric_interp_1d.sh, runs all the tests;
• barycentric_interp_1d.txt, the output file.

lagcheby1_interp_1d_test() samples a function at Chebyshev Type 1 points and determines and plots the barycentric Lagrange polynomial interpolant.

• p01_lagcheby1_04.png, a plot of the interpolant to 4 Chebyshev 1 points for problem p01;
• p01_lagcheby1_08.png, a plot of the interpolant to 8 Chebyshev 1 points for problem p01;
• p01_lagcheby1_16.png, a plot of the interpolant to 16 Chebyshev 1 points for problem p01;
• p01_lagcheby1_32.png, a plot of the interpolant to 32 Chebyshev 1 points for problem p01;
• p01_lagcheby1_64.png, a plot of the interpolant to 64 Chebyshev 1 points for problem p01;
• p02_lagcheby1_04.png, a plot of the interpolant to 4 Chebyshev 1 points for problem p02;
• p02_lagcheby1_08.png, a plot of the interpolant to 8 Chebyshev 1 points for problem p02;
• p02_lagcheby1_16.png, a plot of the interpolant to 16 Chebyshev 1 points for problem p02;
• p02_lagcheby1_32.png, a plot of the interpolant to 32 Chebyshev 1 points for problem p02;
• p02_lagcheby1_64.png, a plot of the interpolant to 64 Chebyshev 1 points for problem p02;
• p03_lagcheby1_04.png, a plot of the interpolant to 4 Chebyshev 1 points for problem p03;
• p03_lagcheby1_08.png, a plot of the interpolant to 8 Chebyshev 1 points for problem p03;
• p03_lagcheby1_16.png, a plot of the interpolant to 16 Chebyshev 1 points for problem p03;
• p03_lagcheby1_32.png, a plot of the interpolant to 32 Chebyshev 1 points for problem p03;
• p03_lagcheby1_64.png, a plot of the interpolant to 64 Chebyshev 1 points for problem p03;
• p04_lagcheby1_04.png, a plot of the interpolant to 4 Chebyshev 1 points for problem p04;
• p04_lagcheby1_08.png, a plot of the interpolant to 8 Chebyshev 1 points for problem p04;
• p04_lagcheby1_16.png, a plot of the interpolant to 16 Chebyshev 1 points for problem p04;
• p04_lagcheby1_32.png, a plot of the interpolant to 32 Chebyshev 1 points for problem p04;
• p04_lagcheby1_64.png, a plot of the interpolant to 64 Chebyshev 1 points for problem p04;
• p05_lagcheby1_04.png, a plot of the interpolant to 4 Chebyshev 1 points for problem p05;
• p05_lagcheby1_08.png, a plot of the interpolant to 8 Chebyshev 1 points for problem p05;
• p05_lagcheby1_16.png, a plot of the interpolant to 16 Chebyshev 1 points for problem p05;
• p05_lagcheby1_32.png, a plot of the interpolant to 32 Chebyshev 1 points for problem p05;
• p05_lagcheby1_64.png, a plot of the interpolant to 64 Chebyshev 1 points for problem p05;
• p06_lagcheby1_04.png, a plot of the interpolant to 4 Chebyshev 1 points for problem p06;
• p06_lagcheby1_08.png, a plot of the interpolant to 8 Chebyshev 1 points for problem p06;
• p06_lagcheby1_16.png, a plot of the interpolant to 16 Chebyshev 1 points for problem p06;
• p06_lagcheby1_32.png, a plot of the interpolant to 32 Chebyshev 1 points for problem p06;
• p06_lagcheby1_64.png, a plot of the interpolant to 64 Chebyshev 1 points for problem p06;
• p07_lagcheby1_04.png, a plot of the interpolant to 4 Chebyshev 1 points for problem p07;
• p07_lagcheby1_08.png, a plot of the interpolant to 8 Chebyshev 1 points for problem p07;
• p07_lagcheby1_16.png, a plot of the interpolant to 16 Chebyshev 1 points for problem p07;
• p07_lagcheby1_32.png, a plot of the interpolant to 32 Chebyshev 1 points for problem p07;
• p07_lagcheby1_64.png, a plot of the interpolant to 64 Chebyshev 1 points for problem p07;
• p08_lagcheby1_04.png, a plot of the interpolant to 4 Chebyshev 1 points for problem p08;
• p08_lagcheby1_08.png, a plot of the interpolant to 8 Chebyshev 1 points for problem p08;
• p08_lagcheby1_16.png, a plot of the interpolant to 16 Chebyshev 1 points for problem p08;
• p08_lagcheby1_32.png, a plot of the interpolant to 32 Chebyshev 1 points for problem p08;
• p08_lagcheby1_64.png, a plot of the interpolant to 64 Chebyshev 1 points for problem p08;

lagcheby2_interp_1d_test samples a function at Chebyshev Type 2 points and determines and plots the barycentric Lagrange polynomial interpolant.

• p01_lagcheby2_04.png, a plot of the interpolant to 4 Chebyshev 2 points for problem p01;
• p01_lagcheby2_08.png, a plot of the interpolant to 8 Chebyshev 2 points for problem p01;
• p01_lagcheby2_16.png, a plot of the interpolant to 16 Chebyshev 2 points for problem p01;
• p01_lagcheby2_32.png, a plot of the interpolant to 32 Chebyshev 2 points for problem p01;
• p01_lagcheby2_64.png, a plot of the interpolant to 64 Chebyshev 2 points for problem p01;
• p02_lagcheby2_04.png, a plot of the interpolant to 4 Chebyshev 2 points for problem p02;
• p02_lagcheby2_08.png, a plot of the interpolant to 8 Chebyshev 2 points for problem p02;
• p02_lagcheby2_16.png, a plot of the interpolant to 16 Chebyshev 2 points for problem p02;
• p02_lagcheby2_32.png, a plot of the interpolant to 32 Chebyshev 2 points for problem p02;
• p02_lagcheby2_64.png, a plot of the interpolant to 64 Chebyshev 2 points for problem p02;
• p03_lagcheby2_04.png, a plot of the interpolant to 4 Chebyshev 2 points for problem p03;
• p03_lagcheby2_08.png, a plot of the interpolant to 8 Chebyshev 2 points for problem p03;
• p03_lagcheby2_16.png, a plot of the interpolant to 16 Chebyshev 2 points for problem p03;
• p03_lagcheby2_32.png, a plot of the interpolant to 32 Chebyshev 2 points for problem p03;
• p03_lagcheby2_64.png, a plot of the interpolant to 64 Chebyshev 2 points for problem p03;
• p04_lagcheby2_04.png, a plot of the interpolant to 4 Chebyshev 2 points for problem p04;
• p04_lagcheby2_08.png, a plot of the interpolant to 8 Chebyshev 2 points for problem p04;
• p04_lagcheby2_16.png, a plot of the interpolant to 16 Chebyshev 2 points for problem p04;
• p04_lagcheby2_32.png, a plot of the interpolant to 32 Chebyshev 2 points for problem p04;
• p04_lagcheby2_64.png, a plot of the interpolant to 64 Chebyshev 2 points for problem p04;
• p05_lagcheby2_04.png, a plot of the interpolant to 4 Chebyshev 2 points for problem p05;
• p05_lagcheby2_08.png, a plot of the interpolant to 8 Chebyshev 2 points for problem p05;
• p05_lagcheby2_16.png, a plot of the interpolant to 16 Chebyshev 2 points for problem p05;
• p05_lagcheby2_32.png, a plot of the interpolant to 32 Chebyshev 2 points for problem p05;
• p05_lagcheby2_64.png, a plot of the interpolant to 64 Chebyshev 2 points for problem p05;
• p06_lagcheby2_04.png, a plot of the interpolant to 4 Chebyshev 2 points for problem p06;
• p06_lagcheby2_08.png, a plot of the interpolant to 8 Chebyshev 2 points for problem p06;
• p06_lagcheby2_16.png, a plot of the interpolant to 16 Chebyshev 2 points for problem p06;
• p06_lagcheby2_32.png, a plot of the interpolant to 32 Chebyshev 2 points for problem p06;
• p06_lagcheby2_64.png, a plot of the interpolant to 64 Chebyshev 2 points for problem p06;
• p07_lagcheby2_04.png, a plot of the interpolant to 4 Chebyshev 2 points for problem p07;
• p07_lagcheby2_08.png, a plot of the interpolant to 8 Chebyshev 2 points for problem p07;
• p07_lagcheby2_16.png, a plot of the interpolant to 16 Chebyshev 2 points for problem p07;
• p07_lagcheby2_32.png, a plot of the interpolant to 32 Chebyshev 2 points for problem p07;
• p07_lagcheby2_64.png, a plot of the interpolant to 64 Chebyshev 2 points for problem p07;
• p08_lagcheby2_04.png, a plot of the interpolant to 4 Chebyshev 2 points for problem p08;
• p08_lagcheby2_08.png, a plot of the interpolant to 8 Chebyshev 2 points for problem p08;
• p08_lagcheby2_16.png, a plot of the interpolant to 16 Chebyshev 2 points for problem p08;
• p08_lagcheby2_32.png, a plot of the interpolant to 32 Chebyshev 2 points for problem p08;
• p08_lagcheby2_64.png, a plot of the interpolant to 64 Chebyshev 2 points for problem p08;