rbf_interp_nd

rbf_interp_nd, an Octave code which defines and evaluates radial basis function (RBF) interpolants to multidimensional 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)
      

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)

RBF_INTERP_ND needs access to the R8LIB library.

Licensing:

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

Languages:

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

Related Data and Programs:

rbf_interp_nd_test

lagrange_interp_nd, an Octave code which defines and evaluates the Lagrange polynomial p(x) which interpolates a set of data depending on a multidimensional argument x that was evaluated on a product grid, so that p(x(i)) = z(i).

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

rbf_interp_1d, an Octave code which defines and evaluates radial basis function (RBF) interpolants to 1d data.

rbf_interp_2d, an Octave code which defines and evaluates radial basis function (RBF) interpolants to 2d data.

shepard_interp_nd, an Octave code which defines and evaluates Shepard interpolants to multidimensional data, based on inverse distance weighting.

sparse_interp_nd an Octave code which can be used to define a sparse interpolant to a function f(x) of a multidimensional argument.

spinterp, an Octave code which carries out piecewise multilinear hierarchical sparse grid interpolation; an earlier version of this software is ACM TOMS algorithm 847, by Andreas Klimke;

test_interp_nd, an Octave code which defines test problems for interpolation of data z(x), depending on an m-dimensional argument.

Reference:

1. Richard Franke,
   Scattered Data Interpolation: Tests of Some Methods,
   Mathematics of Computation,
   Volume 38, Number 157, January 1982, pages 181-200.

Source Code:

• phi1.m, evaluates the multiquadric radial basis function.
• phi2.m, evaluates the inverse multiquadric radial basis function.
• phi3.m, evaluates the thin-plate spline radial basis function.
• phi4.m, evaluates the gaussian radial basis function.
• rbf_interp_nd.m, evaluates a radial basis function interpolant.
• rbf_weight.m, computes weights for radial basis function interpolation.