rbf_interp_nd/rbf_interp_nd.html

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      RBF_INTERP_ND - Multidimensional Interpolation with Radial Basis Functions
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      RBF_INTERP_ND <br> Multidimensional Interpolation with Radial Basis Functions
    </h1>

    <hr>

    <p>
      <b>RBF_INTERP_ND</b>
      is a FORTRAN90 library which
      defines and evaluates radial basis function (RBF) interpolants to multidimensional data.
    </p>

    <p>
      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.
    </p>

    <p>
      Given nd multidimensional points xd with function values fd, and a 
      basis function phi(r), the form of the interpolant is
      <pre>
       f(x) = sum ( 1 <= i <= nd ) w(i) * phi(||x-xd(i)||)
      </pre>
      where the weights w have been precomputed by solving
      <pre>
        sum ( 1 <= i <= nd ) w(i) * phi(||xd(j)-xd(i)||) = fd(j)
      </pre>
    </p>

    <p>
      Four families of radial basis functions are provided. 
      <ul>
        <li>
          phi1(r) = sqrt ( r^2 + r0^2 ) (multiquadric)
        </li>
        <li>
          phi2(r) = 1 / sqrt ( r^2 + r0^2 ) (inverse multiquadric)
        </li>
        <li>
          phi3(r) = r^2 * log ( r / r0 )   (thin plate spline)
        </li>
        <li>
          phi4(r) = exp ( -0.5 r^2 / r0^2 )   (gaussian)
        </li>
      </ul>
      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.
    <p>

    <p>
      <b>RBF_INTERP_ND</b> needs access to the R8LIB library.
    </p>

    <h3 align = "center">
      Licensing:
    </h3>

    <p>
      The computer code and data files described and made available on this web page
      are distributed under
      <a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
    </p>

    <h3 align = "center">
      Languages:
    </h3>

    <p>
      <b>RBF_INTERP_ND</b> is available in
      <a href = "../../c_src/rbf_interp_nd/rbf_interp_nd.html">a C version</a> and
      <a href = "../../cpp_src/rbf_interp_nd/rbf_interp_nd.html">a C++ version</a> and
      <a href = "../../f77_src/rbf_interp_nd/rbf_interp_nd.html">a FORTRAN77 version</a> and
      <a href = "../../f_src/rbf_interp_nd/rbf_interp_nd.html">a FORTRAN90 version</a> and
      <a href = "../../m_src/rbf_interp_nd/rbf_interp_nd.html">a MATLAB version</a>.
    </p>

    <h3 align = "center">
      Related Data and Programs:
    </h3>

    <p>
      <a href = "../../f_src/lagrange_interp_nd/lagrange_interp_nd.html">
      LAGRANGE_INTERP_ND</a>,
      a FORTRAN90 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).
    </p>

    <p>
      <a href = "../../f_src/r8lib/r8lib.html">
      R8LIB</a>,
      a FORTRAN90 library which
      contains many utility routines using double precision real (R8) arithmetic.
    </p>

    <p>
      <a href = "../../f_src/rbf_interp_1d/rbf_interp_1d.html">
      RBF_INTERP_1D</a>,
      a FORTRAN90 library which
      defines and evaluates radial basis function (RBF) interpolants to 1D data.
    </p>

    <p>
      <a href = "../../f_src/rbf_interp_2d/rbf_interp_2d.html">
      RBF_INTERP_2D</a>,
      a FORTRAN90 library which
      defines and evaluates radial basis function (RBF) interpolants to 2D data.
    </p>

    <p>
      <a href = "../../f_src/shepard_interp_nd/shepard_interp_nd.html">
      SHEPARD_INTERP_ND</a>,
      a FORTRAN90 library which
      defines and evaluates Shepard interpolants to multidimensional data,
      based on inverse distance weighting.
    </p>

    <p>
      <a href = "../../f_src/sparse_interp_nd/sparse_interp_nd.html">
      SPARSE_INTERP_ND</a>
      a FORTRAN90 library which
      can be used to define a sparse interpolant to a function f(x) of a 
      multidimensional argument.
    </p>

    <p>
      <a href = "../../m_src/spinterp/spinterp.html">
      SPINTERP</a>,
      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;
    </p>

    <p>
      <a href = "../../f_src/test_interp_nd/test_interp_nd.html">
      TEST_INTERP_ND</a>,
      a FORTRAN90 library which
      defines test problems for interpolation of data z(x),
      depending on an M-dimensional argument.
    </p>

    <h3 align = "center">
      Reference:
    </h3>

    <p>
      <ol>
        <li>
          Richard Franke,<br>
          Scattered Data Interpolation: Tests of Some Methods,<br>
          Mathematics of Computation,<br>
          Volume 38, Number 157, January 1982, pages 181-200.
        </li>
        <li>
          William Press, Brian Flannery, Saul Teukolsky, William Vetterling,<br>
          Numerical Recipes in FORTRAN: The Art of Scientific Computing,<br>
          Third Edition,<br>
          Cambridge University Press, 2007,<br>
          ISBN13: 978-0-521-88068-8,<br>
          LC: QA297.N866.
        </li>
      </ol>
    </p>


    <h3 align = "center">
      Source Code:
    </h3>

    <p>
      <ul>
        <li>
          <a href = "rbf_interp_nd.f90">rbf_interp_nd.f90</a>, the source code.
        </li>
        <li>
          <a href = "rbf_interp_nd.sh">rbf_interp_nd.sh</a>,
          BASH commands to compile the source code.
        </li>
      </ul>
    </p>

    <h3 align = "center">
      Examples and Tests:
    </h3>

    <p>
      <ul>
        <li>
          <a href = "rbf_interp_nd_prb.f90">rbf_interp_nd_prb.f90</a>,
          a sample calling program.
        </li>
        <li>
          <a href = "rbf_interp_nd_prb.sh">rbf_interp_nd_prb.sh</a>,
          BASH commands to compile and run the sample program.
        </li>
        <li>
          <a href = "rbf_interp_nd_prb_output.txt">rbf_interp_nd_prb_output.txt</a>,
          the output file.
        </li>
      </ul>
    </p>

    <h3 align = "center">
      List of Routines:
    </h3>

    <p>
      <ul>
        <li>
          <b>DAXPY</b> computes constant times a vector plus a vector.
        </li>
        <li>
          <b>DDOT</b> forms the dot product of two vectors.
        </li>
        <li>
          <b>DNRM2</b> returns the euclidean norm of a vector.
        </li>
        <li>
          <b>DROT</b> applies a plane rotation.
        </li>
        <li>
          <b>DROTG</b> constructs a Givens plane rotation.
        </li>
        <li>
          <b>DSCAL</b> scales a vector by a constant.
        </li>
        <li>
          <b>DSVDC</b> computes the singular value decomposition of a real rectangular matrix.
        </li>
        <li>
          <b>DSWAP</b> interchanges two vectors.
        </li>
        <li>
          <b>PHI1</b> evaluates the multiquadric radial basis function.
        </li>
        <li>
          <b>PHI2</b> evaluates the inverse multiquadric radial basis function.
        </li>
        <li>
          <b>PHI3</b> evaluates the thin-plate spline radial basis function.
        </li>
        <li>
          <b>PHI4</b> evaluates the gaussian radial basis function.
        </li>
        <li>
          <b>R8MAT_SOLVE_SVD</b> solves a linear system A*x=b using the SVD.
        </li>
        <li>
          <b>RBF_INTERP_ND</b> evaluates a radial basis function interpolant.
        </li>
        <li>
          <b>RBF_WEIGHT</b> computes weights for radial basis function interpolation.
        </li>
      </ul>
    </p>

    <p>
      You can go up one level to <a href = "../f_src.html">
      the FORTRAN90 source codes</a>.
    </p>

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    <i>
      Last revised on 27 September 2012.
    </i>

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