BARYCENTRIC_INTERP_1D
Barycentric Lagrange Polynomial Interpolation in 1D


BARYCENTRIC_INTERP_1D is a MATLAB library 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

and any linear mapping of these points to an arbitary interval [A,B].

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;
        
but it should read
          exact(xdiff==0) = j;
        

BARYCENTRIC_INTERP_1D requires the R8LIB library. The test also requires the TEST_INTERP_1D library.

Licensing:

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

Languages:

BARYCENTRIC_INTERP_1D is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CHEBYSHEV_INTERP_1D, a MATLAB library which determines the combination of Chebyshev polynomials which interpolates a set of data, so that p(x(i)) = y(i).

LAGRANGE_APPROX_1D, a MATLAB library which defines and evaluates the Lagrange polynomial p(x) of degree m which approximates a set of nd data points (x(i),y(i)).

LAGRANGE_BASIS_DISPLAY, a MATLAB library which displays the basis functions associated with a given set of nodes used with the Lagrange interpolation scheme.

LAGRANGE_INTERP_1D, a MATLAB library 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 MATLAB library which interpolates a set of data using a piecewise constant interpolant defined by the nearest neighbor criterion.

PWL_INTERP_1D, a MATLAB library which interpolates a set of data using a piecewise linear interpolant.

R8LIB, a MATLAB library which contains many utility routines using double precision real (R8) arithmetic.

RBF_INTERP_1D, a MATLAB library which defines and evaluates radial basis function (RBF) interpolants to 1D data.

SHEPARD_INTERP_1D, a MATLAB library which defines and evaluates Shepard interpolants to 1D data, which are based on inverse distance weighting.

SPLINE, a MATLAB library which constructs and evaluates spline interpolants and approximants.

TEST_INTERP, a MATLAB library which defines a number of test problems for interpolation, provided as a set of (x,y) data.

TEST_INTERP_1D, a MATLAB library which defines test problems for interpolation of data y(x), depending on a 2D argument.

VANDERMONDE_INTERP_1D, a MATLAB library 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:

Examples and Tests:

The test code requires the test_interp_1d library. If this library is available in a separate folder at the same "level", then a Matlab command such as "addpath ( '../test_interp_1d')" will make it accessible for a run of the test program.

barycentric_interp_1d_test01() samples a function at Chebyshev Type 1 points and determines and plots the barycentric Lagrange polynomial interpolant.

barycentric_interp_1d_test02() samples a function at Chebyshev Type 2 points and determines and plots the barycentric Lagrange polynomial interpolant.

You can go up one level to the MATLAB source codes.


Last modified on 19 August 2012.