# barycentric_interp_1d

barycentric_interp_1d, a MATLAB 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;
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;
```
```          exact(xdiff==0) = j;
```

barycentric_interp_1d() requires the R8LIB library. The test also requires the TEST_INTERP_1D library.

### Languages:

barycentric_interp_1d is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

### Related Data and Programs:

chebyshev_interp_1d, a MATLAB code 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 code 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 code which displays the basis functions associated with a given set of nodes used with the Lagrange interpolation scheme.

lagrange_interp_1d, a MATLAB 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 MATLAB code which interpolates a set of data using a piecewise constant interpolant defined by the nearest neighbor criterion.

newton_interp_1d, a MATLAB code which finds a polynomial interpolant to data using Newton divided differences.

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

r8lib, a MATLAB code which contains many utility routines using double precision real (R8) arithmetic.

rbf_interp_1d, a MATLAB code which defines and evaluates radial basis function (RBF) interpolants to 1D data.

shepard_interp_1d, a MATLAB code which defines and evaluates Shepard interpolants to 1D data, which are based on inverse distance weighting.

spline, a MATLAB code which constructs and evaluates spline interpolants and approximants.

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

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

vandermonde_interp_1d, a MATLAB 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:

• lagcheby1_interp_1d.m, evaluates the barycentric Lagrange interpolant in cases where the data is evaluated at the Chebyshev type 1 points.
• lagcheby2_interp_1d.m, evaluates the barycentric Lagrange interpolant in cases where the data is evaluated at the Chebyshev type 2 points.
• lageven_interp_1d.m, evaluates the barycentric Lagrange interpolant in cases where the data is evaluated at evenly-spaced midpoints.