RBF_INTERP_ND
Multidimensional Interpolation with Radial Basis Functions

RBF_INTERP_ND is a MATLAB library 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 GNU LGPL license.

Languages:

RBF_INTERP_ND 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:

LAGRANGE_INTERP_ND, a MATLAB library 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, 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.

RBF_INTERP_2D, a MATLAB library which defines and evaluates radial basis function (RBF) interpolants to 2D data.

SHEPARD_INTERP_ND, a MATLAB library which defines and evaluates Shepard interpolants to multidimensional data, based on inverse distance weighting.

SPARSE_INTERP_ND a MATLAB library which can be used to define a sparse interpolant to a function f(x) of a multidimensional argument.

SPINTERP, a MATLAB library 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, a MATLAB library 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.
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:

• 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.

Examples and Tests:

• rbf_interp_nd_test.m, calls all the tests;
• rbf_interp_nd_test_output.txt, the output file.
• rbf_interp_nd_test01.m, tests RBF_WEIGHT and RBF_INTERP_ND with PHI1.
• rbf_interp_nd_test01_exact.png, a plot of the exact function f(x,y).
• rbf_interp_nd_test01_interp.png, a plot of the interpolant function using phi1(r).
• rbf_interp_nd_test02.m, tests RBF_WEIGHT and RBF_INTERP_ND with PHI2.
• rbf_interp_nd_test02_exact.png, a plot of the exact function f(x,y).
• rbf_interp_nd_test02_interp.png, a plot of the interpolant function using phi2(r).
• rbf_interp_nd_test03.m, tests RBF_WEIGHT and RBF_INTERP_ND with PHI3.
• rbf_interp_nd_test03_exact.png, a plot of the exact function f(x,y).
• rbf_interp_nd_test03_interp.png, a plot of the interpolant function using phi3(r).
• rbf_interp_nd_test04.m, tests RBF_WEIGHT and RBF_INTERP_ND with PHI4.
• rbf_interp_nd_test04_exact.png, a plot of the exact function f(x,y).
• rbf_interp_nd_test04_interp.png, a plot of the interpolant function using phi4(r).

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