RBF_INTERP_1D
Radial Basis Function Interpolation in 1D
RBF_INTERP_1D
is a MATLAB 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(xxd(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.
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_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 MATLAB 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 MATLAB library which
determines the combination of Chebyshev polynomials which
interpolates a set of data, so that p(x(i)) = y(i).
DIVDIF,
a MATLAB library which
uses divided differences to compute the polynomial interpolant
to a given set of data.
HERMITE,
a MATLAB library which
computes the Hermite interpolant, a polynomial that matches function values
and derivatives.
LAGRANGE_INTERP_1D,
a MATLAB 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 MATLAB library which
interpolates a set of data using a piecewise constant interpolant
defined by the nearest neighbor criterion.
NEWTON_INTERP_1D,
a MATLAB library which
finds a polynomial interpolant to data using Newton divided differences.
PWL_INTERP_1D,
a MATLAB library which
interpolates a set of data using a piecewise linear interpolant.
R8LIB,
a MATLAB library which
contains many utility routines using double precision real (R8) arithmetic.
RBF_INTERP_2D,
a MATLAB library which
defines and evaluates radial basis function (RBF) interpolants to 2D data.
RBF_INTERP_ND,
a MATLAB library which
defines and evaluates radial basis function (RBF) interpolants to multidimensional data.
SHEPARD_INTERP_1D,
a MATLAB library which
defines and evaluates Shepard interpolants to 1D data,
which are based on inverse distance weighting.
TEST_INTERP,
a MATLAB library which
defines a number of test problems for interpolation,
provided as a set of (x,y) data.
TEST_INTERP_1D,
a MATLAB library which
defines test problems for interpolation of data y(x),
depending on a 2D argument.
VANDERMONDE_INTERP_1D,
a MATLAB library which
finds a polynomial interpolant to a function of 1D data
by setting up and solving a linear system for the polynomial coefficients,
involving the Vandermonde matrix.
Reference:

Richard Franke,
Scattered Data Interpolation: Tests of Some Methods,
Mathematics of Computation,
Volume 38, Number 157, January 1982, pages 181200.

William Press, Brian Flannery, Saul Teukolsky, William Vetterling,
Numerical Recipes in FORTRAN: The Art of Scientific Computing,
Third Edition,
Cambridge University Press, 2007,
ISBN13: 9780521880688,
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 thinplate spline radial basis function.

phi4.m,
evaluates the gaussian radial basis function.

rbf_interp_1d.m,
evaluates a radial basis function interpolant.

rbf_weight.m,
computes weights for radial basis function interpolation.
Examples and Tests:
Running these tests requires access to the test_interp library.
Should that library be available in a directory at the same level, this
can be accomplished with the command "addpath ( '../test_interp' )".
The test program makes a number of plots.

p01_data.png,
the data for problem p01 with a linear interpolant.

p01_phi1_poly.png,
the data for problem p01 with a PHI1 RBF interpolant.

p01_phi2_poly.png,
the data for problem p01 with a PHI2 RBF interpolant.

p01_phi3_poly.png,
the data for problem p01 with a PHI3 RBF interpolant.

p01_phi4_poly.png,
the data for problem p01 with a PHI4 RBF interpolant.

p02_data.png,
the data for problem p02 with a linear interpolant.

p02_phi1_poly.png,
the data for problem p02 with a PHI1 RBF interpolant.

p02_phi2_poly.png,
the data for problem p02 with a PHI2 RBF interpolant.

p02_phi3_poly.png,
the data for problem p02 with a PHI3 RBF interpolant.

p02_phi4_poly.png,
the data for problem p02 with a PHI4 RBF interpolant.

p03_data.png,
the data for problem p03 with a linear interpolant.

p03_phi1_poly.png,
the data for problem p03 with a PHI1 RBF interpolant.

p03_phi2_poly.png,
the data for problem p03 with a PHI2 RBF interpolant.

p03_phi3_poly.png,
the data for problem p03 with a PHI3 RBF interpolant.

p03_phi4_poly.png,
the data for problem p03 with a PHI4 RBF interpolant.

p04_data.png,
the data for problem p04 with a linear interpolant.

p04_phi1_poly.png,
the data for problem p04 with a PHI1 RBF interpolant.

p04_phi2_poly.png,
the data for problem p04 with a PHI2 RBF interpolant.

p04_phi3_poly.png,
the data for problem p04 with a PHI3 RBF interpolant.

p04_phi4_poly.png,
the data for problem p04 with a PHI4 RBF interpolant.

p05_data.png,
the data for problem p05 with a linear interpolant.

p05_phi1_poly.png,
the data for problem p05 with a PHI1 RBF interpolant.

p05_phi2_poly.png,
the data for problem p05 with a PHI2 RBF interpolant.

p05_phi3_poly.png,
the data for problem p05 with a PHI3 RBF interpolant.

p05_phi4_poly.png,
the data for problem p05 with a PHI4 RBF interpolant.

p06_data.png,
the data for problem p06 with a linear interpolant.

p06_phi1_poly.png,
the data for problem p06 with a PHI1 RBF interpolant.

p06_phi2_poly.png,
the data for problem p06 with a PHI2 RBF interpolant.

p06_phi3_poly.png,
the data for problem p06 with a PHI3 RBF interpolant.

p06_phi4_poly.png,
the data for problem p06 with a PHI4 RBF interpolant.

p07_data.png,
the data for problem p07 with a linear interpolant.

p07_phi1_poly.png,
the data for problem p07 with a PHI1 RBF interpolant.

p07_phi2_poly.png,
the data for problem p07 with a PHI2 RBF interpolant.

p07_phi3_poly.png,
the data for problem p07 with a PHI3 RBF interpolant.

p07_phi4_poly.png,
the data for problem p07 with a PHI4 RBF interpolant.

p08_data.png,
the data for problem p08 with a linear interpolant.

p08_phi1_poly.png,
the data for problem p08 with a PHI1 RBF interpolant.

p08_phi2_poly.png,
the data for problem p08 with a PHI2 RBF interpolant.

p08_phi3_poly.png,
the data for problem p08 with a PHI3 RBF interpolant.

p08_phi4_poly.png,
the data for problem p08 with a PHI4 RBF interpolant.
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the MATLAB source codes.
Last modified on 24 June 2013.