rbf_interp_1d

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

A radial basis interpolant is a useful, but expensive, technique for definining a smooth function which interpolates a set of function values specified at an arbitrary set of data points.

Given nd multidimensional points xd with function values fd, and a basis function phi(r), the form of the interpolant is

       f(x) = sum ( 1 <= i <= nd ) w(i) * phi(||x-xd(i)||)
      

        sum ( 1 <= i <= nd ) w(i) * phi(||xd(j)-xd(i)||) = fd(j)
      

Although the technique is generally applied in a multidimensional setting, in this directory we look specifically at the case involving 1D data. This allows us to easily plot and compare the various results.

Four families of radial basis functions are provided.

• phi1(r) = sqrt ( r^2 + r0^2 ) (multiquadric)
• phi2(r) = 1 / sqrt ( r^2 + r0^2 ) (inverse multiquadric)
• phi3(r) = r^2 * log ( r / r0 ) (thin plate spline)
• phi4(r) = exp ( -0.5 r^2 / r0^2 ) (gaussian)

Licensing:

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

Languages:

rbf_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:

barycentric_interp_1d, a python code which defines and evaluates the barycentric lagrange polynomial p(x) which interpolates a set of data, so that p(x(i)) = y(i). the barycentric approach means that very high degree polynomials can safely be used.

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_2d, a python code which defines and evaluates radial basis function (rbf) interpolants to 2d 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. Richard Franke,
   Scattered Data Interpolation: Tests of Some Methods,
   Mathematics of Computation,
   Volume 38, Number 157, January 1982, pages 181-200.
2. William Press, Brian Flannery, Saul Teukolsky, William Vetterling,
   Numerical Recipes in FORTRAN: The Art of Scientific Computing,
   Third Edition,
   Cambridge University Press, 2007,
   ISBN13: 978-0-521-88068-8,
   LC: QA297.N866.

Source Code:

• rbf_interp_1d.py, the source code.
• rbf_interp_1d.sh, runs all the tests;
• rbf_interp_1d.txt, the output file.

The test program makes a number of plots.

• p01_data.png, the data for problem p01 with a linear interpolant.
• p01_phi1_poly.png, the data for problem p01 with a PHI1 RBF interpolant.
• p01_phi2_poly.png, the data for problem p01 with a PHI2 RBF interpolant.
• p01_phi3_poly.png, the data for problem p01 with a PHI3 RBF interpolant.
• p01_phi4_poly.png, the data for problem p01 with a PHI4 RBF interpolant.
• p02_data.png, the data for problem p02 with a linear interpolant.
• p02_phi1_poly.png, the data for problem p02 with a PHI1 RBF interpolant.
• p02_phi2_poly.png, the data for problem p02 with a PHI2 RBF interpolant.
• p02_phi3_poly.png, the data for problem p02 with a PHI3 RBF interpolant.
• p02_phi4_poly.png, the data for problem p02 with a PHI4 RBF interpolant.
• p03_data.png, the data for problem p03 with a linear interpolant.
• p03_phi1_poly.png, the data for problem p03 with a PHI1 RBF interpolant.
• p03_phi2_poly.png, the data for problem p03 with a PHI2 RBF interpolant.
• p03_phi3_poly.png, the data for problem p03 with a PHI3 RBF interpolant.
• p03_phi4_poly.png, the data for problem p03 with a PHI4 RBF interpolant.
• p04_data.png, the data for problem p04 with a linear interpolant.
• p04_phi1_poly.png, the data for problem p04 with a PHI1 RBF interpolant.
• p04_phi2_poly.png, the data for problem p04 with a PHI2 RBF interpolant.
• p04_phi3_poly.png, the data for problem p04 with a PHI3 RBF interpolant.
• p04_phi4_poly.png, the data for problem p04 with a PHI4 RBF interpolant.
• p05_data.png, the data for problem p05 with a linear interpolant.
• p05_phi1_poly.png, the data for problem p05 with a PHI1 RBF interpolant.
• p05_phi2_poly.png, the data for problem p05 with a PHI2 RBF interpolant.
• p05_phi3_poly.png, the data for problem p05 with a PHI3 RBF interpolant.
• p05_phi4_poly.png, the data for problem p05 with a PHI4 RBF interpolant.
• p06_data.png, the data for problem p06 with a linear interpolant.
• p06_phi1_poly.png, the data for problem p06 with a PHI1 RBF interpolant.
• p06_phi2_poly.png, the data for problem p06 with a PHI2 RBF interpolant.
• p06_phi3_poly.png, the data for problem p06 with a PHI3 RBF interpolant.
• p06_phi4_poly.png, the data for problem p06 with a PHI4 RBF interpolant.
• p07_data.png, the data for problem p07 with a linear interpolant.
• p07_phi1_poly.png, the data for problem p07 with a PHI1 RBF interpolant.
• p07_phi2_poly.png, the data for problem p07 with a PHI2 RBF interpolant.
• p07_phi3_poly.png, the data for problem p07 with a PHI3 RBF interpolant.
• p07_phi4_poly.png, the data for problem p07 with a PHI4 RBF interpolant.
• p08_data.png, the data for problem p08 with a linear interpolant.
• p08_phi1_poly.png, the data for problem p08 with a PHI1 RBF interpolant.
• p08_phi2_poly.png, the data for problem p08 with a PHI2 RBF interpolant.
• p08_phi3_poly.png, the data for problem p08 with a PHI3 RBF interpolant.
• p08_phi4_poly.png, the data for problem p08 with a PHI4 RBF interpolant.