# RBF_INTERP_1D Radial Basis Function Interpolation in 1D

RBF_INTERP_1D is a FORTRAN77 library 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)||)
```
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 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)
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.

RBF_INTERP_1D needs the R8LIB library. The test code also needs the TEST_INTERP library.

### Languages:

RBF_INTERP_1D 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:

BARYCENTRIC_INTERP_1D, a FORTRAN77 library 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 FORTRAN77 library which determines the combination of Chebyshev polynomials which interpolates a set of data, so that p(x(i)) = y(i).

DIVDIF, a FORTRAN77 library which uses divided differences to compute the polynomial interpolant to a given set of data.

HERMITE, a FORTRAN77 library which computes the Hermite interpolant, a polynomial that matches function values and derivatives.

LAGRANGE_INTERP_1D, a FORTRAN77 library 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 FORTRAN77 library which interpolates a set of data using a piecewise constant interpolant defined by the nearest neighbor criterion.

PWL_INTERP_1D, a FORTRAN77 library which interpolates a set of data using a piecewise linear interpolant.

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

RBF_INTERP_2D, a FORTRAN77 library which defines and evaluates radial basis function (RBF) interpolants to 2D data.

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

SHEPARD_INTERP_1D, a FORTRAN77 library which defines and evaluates Shepard interpolants to 1D data, based on inverse distance weighting.

TEST_INTERP, a FORTRAN77 library which defines a number of test problems for interpolation, provided as a set of (x,y) data.

TEST_INTERP_1D, a FORTRAN77 library which defines test problems for interpolation of data y(x), depending on a 2D argument.

VANDERMONDE_INTERP_1D, a FORTRAN77 library which finds a polynomial interpolant to data y(x) of a 1D 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.