# RBF_INTERP_2D Radial Basis Function Interpolation in 2D

RBF_INTERP_2D is a FORTRAN77 library which defines and evaluates radial basis function (RBF) interpolants to 2D 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)
```

Although the technique is generally applied in a multidimensional setting, in this directory we look specifically at the case involving 2D 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)
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_2D 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_2D, a FORTRAN77 library which defines and evaluates the Lagrange polynomial p(x,y) which interpolates a set of data depending on a 2D argument that was evaluated on a product grid, so that p(x(i),y(j)) = z(i,j).

PADUA, a FORTRAN77 library which returns the points and weights for Padu sets, useful for interpolation in 2D. GNUPLOT is used to plot the points.

PWL_INTERP_2D, a FORTRAN77 library which evaluates a piecewise linear interpolant to data defined on a regular 2D grid.

R8LIB, a FORTRAN77 library which contains many utility routines, using double precision real (R8) arithmetic.

RBF_INTERP_1D, a FORTRAN77 library which defines and evaluates radial basis function (RBF) interpolants to 1D data.

RBF_INTERP_ND, a FORTRAN77 library which defines and evaluates radial basis function (RBF) interpolants to multidimensional data.

SHEPARD_INTERP_2D, a FORTRAN77 library which defines and evaluates Shepard interpolants to 2D data, based on inverse distance weighting.

TEST_INTERP_2D, a FORTRAN77 library which defines test problems for interpolation of data z(x,y) of a 2D argument.

TOMS886, a FORTRAN77 library which defines the Padua points for interpolation in a 2D region, including the rectangle, triangle, and ellipse, by Marco Caliari, Stefano de Marchi, Marco Vianello. This is ACM TOMS algorithm 886.

VANDERMONDE_INTERP_2D, a FORTRAN77 library which finds a polynomial interpolant to data z(x,y) of a 2D argument 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.

### List of Routines:

• DAXPY computes constant times a vector plus a vector.
• DDOT forms the dot product of two vectors.
• DNRM2 returns the euclidean norm of a vector.
• DROT applies a plane rotation.
• DROTG constructs a Givens plane rotation.
• DSCAL scales a vector by a constant.
• DSVDC computes the singular value decomposition of a real rectangular matrix.
• DSWAP interchanges two vectors.