# rbf_interp_nd

rbf_interp_nd, a FORTRAN90 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)||)
```
where the weights w have been precomputed by solving
```        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)
Each uses a "scale factor" r0, whose value is recommended to be greater than the minimal distance between points, and rather less than the maximal distance. Changing the value of r0 changes the shape of the interpolant function.

### Languages:

rbf_interp_nd is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

LAGRANGE_INTERP_ND, a FORTRAN90 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, a FORTRAN90 code which contains many utility routines using double precision real (R8) arithmetic.

RBF_INTERP_1D, a FORTRAN90 code which defines and evaluates radial basis function (RBF) interpolants to 1D data.

RBF_INTERP_2D, a FORTRAN90 code which defines and evaluates radial basis function (RBF) interpolants to 2D data.

SHEPARD_INTERP_ND, a FORTRAN90 code which defines and evaluates Shepard interpolants to multidimensional data, based on inverse distance weighting.

SPARSE_INTERP_ND a FORTRAN90 code 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 FORTRAN90 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.
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:

Last revised on 24 August 2020.