cvt/cvt.html

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      CVT - Centroidal Voronoi Tessellations
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      CVT <br> Centroidal Voronoi Tessellations
    </h1>

    <hr>

    <p>
      <b>CVT</b> 
      is a FORTRAN90 library which 
      computes Centroidal Voronoi Tessellation (CVT) datasets.
    </p>

    <p>
      The generation of a CVT dataset is of necessity more complicated than
      that of a quasirandom dataset.  A quasirandom dataset is generally
      simply the first N values of a deterministic sequence.  But to compute
      a CVT requires an iteration, so there must be an initial assignment 
      for the generators, and then a number of iterations.  Moreover, in 
      each iteration, estimates must be made of the volume and location
      of the Voronoi cells.  This is typically done by Monte Carlo 
      sampling.  The accuracy of the resulting CVT depends in part on 
      the number of sampling points and the number of iterations taken.  
    </p>

    <p>
      The library is mostly used to generate a dataset of points 
      uniformly distributed in the unit hypersquare.  However, a user 
      may be interested in computations with other geometries or 
      point densities.  To do this, the user needs to replace the
      <b>USER</b> routine in the <b>CVT</b> library, and then specify
      the appropriate values <b>init=3</b> and <b>sample=3</b>.      
    </p>

    <p>
      The <b>USER</b> routine returns a set of sample points from the 
      region of interest.  The default <b>USER</b> routine samples points
      uniformly from the unit circle.  But other geometries are 
      easy to set up.  Changing the point density simply requires
      weighting the sampling in the region.
    </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>CVT</b> is available in
      <a href = "../../cpp_src/cvt/cvt.html">a C++ version</a> and
      <a href = "../../f_src/cvt/cvt.html">a FORTRAN90 version</a> and
      <a href = "../../m_src/cvt/cvt.html">a MATLAB version</a>.
    </p>

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

    <p>
      <a href = "../../datasets/cvt/cvt.html">
      CVT</a>,
      a dataset directory which
      contains a variety of CVT datasets.
    </p>

    <p>
      <a href = "../../f_src/cvt_basis/cvt_basis.html">
      CVT_BASIS</a>,
      a FORTRAN90 program which 
      seeks to produce a small set of basis vectors from a large set of data vectors.
    </p>

    <p>
      <a href = "../../f_src/cvt_dataset/cvt_dataset.html">
      CVT_DATASET</a>,
      a FORTRAN90 program which
      computes a CVT dataset.
    </p>

    <p>
      <a href = "../../m_src/cvt_demo/cvt_demo.html">
      CVT_DEMO</a>,
      a MATLAB library which
      demonstrates the computation of a CVT.
    </p>

    <p>
      <a href = "../../f_src/cvt_fixed/cvt_fixed.html">
      CVT_FIXED</a>,
      a FORTRAN90 library which
      computes CVT's in which some values
      are to be fixed.
    </p>

    <p>
      <a href = "../../f_src/cvt_mod/cvt_mod.html">
      CVT_MOD</a>,
      a FORTRAN90 library which
      computes CVT's on a unit
      square with "mod" geometry, which essentially makes the unit
      square a tile in a periodic pattern.
    </p>

    <p>
      <a href = "../../f_src/cvt_triangulation/cvt_triangulation.html">
      CVT_TRIANGULATION</a>,
      a FORTRAN90 program which
      constructs a CVT for a region
      specified by the <b>TEST_TRIANGULATION</b> library.
    </p>

    <p>
      <a href = "../../f_src/faure/faure.html">
      FAURE</a>,
      a FORTRAN90 library which
      computes elements of a Faure quasirandom sequence.
    </p>

    <p>
      <a href = "../../f_src/grid/grid.html">
      GRID</a>, 
      a FORTRAN90 library which
      computes elements of a grid dataset.
    </p>

    <p>
      <a href = "../../f_src/halton/halton.html">
      HALTON</a>,
      a FORTRAN90 library which
      computes elements of a Halton quasirandom sequence.
    </p>

    <p>
      <a href = "../../f_src/hammersley/hammersley.html">
      HAMMERSLEY</a>, 
      a FORTRAN90 library which
      computes elements of a Hammersley quasirandom sequence.
    </p>

    <p>
      <a href = "../../f_src/hex_grid/hex_grid.html">
      HEX_GRID</a>,
      a FORTRAN90 library which 
      computes elements of a hexagonal grid dataset.
    </p>

    <p>
      <a href = "../../f_src/hex_grid_angle/hex_grid_angle.html">
      HEX_GRID_ANGLE</a>, 
      a FORTRAN90 library which
      computes elements of an angled hexagonal grid dataset.
    </p>

    <p>
      <a href = "../../f_src/ihs/ihs.html">
      IHS</a>,
      a FORTRAN90 library which
      computes elements of an improved distributed Latin hypercube dataset.
    </p>

    <p>
      <a href = "../../f_src/latin_center/latin_center.html">
      LATIN_CENTER</a>,
      a FORTRAN90 library which 
      computes elements of a Latin Hypercube dataset, choosing center points.
    </p>

    <p>
      <a href = "../../f_src/latin_edge/latin_edge.html">
      LATIN_EDGE</a>, 
      a FORTRAN90 library which
      computes elements of a Latin Hypercube dataset, choosing edge points.
    </p>

    <p>
      <a href = "../../f_src/latin_random/latin_random.html">
      LATIN_RANDOM</a>, 
      a FORTRAN90 library which
      computes elements of a Latin Hypercube dataset, choosing 
      points at random.
    </p>

    <p>
      <a href = "../../f_src/lattice_rule/lattice_rule.html">
      LATTICE_RULE</a>, 
      a FORTRAN90 library which 
      approximates multidimensional integrals using lattice rules.
    </p>

    <p>
      <a href = "../../f_src/lcvt/lcvt.html">
      LCVT</a>, 
      a FORTRAN90 library which
      computes a latinized 
      Centroidal Voronoi Tessellation.
    </p>

    <p>
      <a href = "../../f_src/niederreiter2/niederreiter2.html">
      NIEDERREITER2</a>, 
      a FORTRAN90 library which
      computes elements of a Niederreiter quasirandom sequence with base 2.
    </p>

    <p>
      <a href = "../../f_src/normal/normal.html">
      NORMAL</a>, 
      a FORTRAN90 library which
      computes elements of a 
      sequence of pseudorandom normally distributed values.
    </p>

    <p>
      <a href = "../../f_src/sandia_cvt/sandia_cvt.html">
      SANDIA_CVT</a>,
      a FORTRAN90 library which
      computes CVT's.
    </p>

    <p>
      <a href = "../../f_src/sobol/sobol.html">
      SOBOL</a>, 
      a FORTRAN90 library which
      computes elements of a Sobol quasirandom sequence.
    </p>

    <p>
      <a href = "../../f_src/uniform/uniform.html">
      UNIFORM</a>,
      a FORTRAN90 library which
      computes elements of a uniform pseudorandom sequence.
    </p>

    <p>
      <a href = "../../f_src/van_der_corput/van_der_corput.html">
      VAN_DER_CORPUT</a>, 
      a FORTRAN90 library which
      computes elements of a van der Corput quasirandom sequence.
    </p>

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

    <p>
      <ol>
        <li>
          Franz Aurenhammer,<br>
          Voronoi diagrams - 
          a study of a fundamental geometric data structure,<br>
          ACM Computing Surveys,<br>
          Volume 23, Number 3, pages 345-405, September 1991.
        </li>
        <li>
          Paul Bratley, Bennett Fox, Linus Schrage,<br>
          A Guide to Simulation,<br>
          Springer Verlag, pages 201-202, 1983.
        </li>
        <li>
          John Burkardt, Max Gunzburger, Janet Peterson, Rebecca Brannon,<br>
          User Manual and Supporting Information for Library of Codes
          for Centroidal Voronoi Placement and Associated Zeroth,
          First, and Second Moment Determination,<br>
          Sandia National Laboratories Technical Report SAND2002-0099,<br>
          February 2002.
        </li>
        <li>
          Qiang Du, Vance Faber, Max Gunzburger,<br>
          Centroidal Voronoi Tessellations: Applications and Algorithms,<br>
          SIAM Review,<br>
          Volume 41, 1999, pages 637-676.
        </li>
        <li>
          Bennett Fox,<br>
          Algorithm 647:<br>
          Implementation and Relative Efficiency of Quasirandom
          Sequence Generators,<br>
          ACM Transactions on Mathematical Software,<br>
          Volume 12, Number 4, pages 362-376, 1986.
        </li>
        <li>
          John Halton,<br>
          On the efficiency of certain quasi-random sequences of points
          in evaluating multi-dimensional integrals,<br>
          Numerische Mathematik,<br>
          Volume 2, pages 84-90, 1960.
        </li>
        <li>
          Lili Ju, Qiang Du, Max Gunzburger,<br>
          Probabilistic methods for centroidal Voronoi tessellations
          and their parallel implementations,<br>
          Parallel Computing,<br>
          Volume 28, 2002, pages 1477-1500.
        </li>
      </ol>
    </p>

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

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

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

    <p>
      <ul>
        <li>
          <a href = "cvt_prb.f90">cvt_prb.f90</a>,
          a sample calling program.
        </li>
        <li>
          <a href = "cvt_prb.sh">cvt_prb.sh</a>,
          commands to compile and run the sample program.
        </li>
        <li>
          <a href = "cvt_prb_output.txt">cvt_prb_output.txt</a>,
          the output file.
        </li>
        <li>
          <a href = "cvt_circle.txt">cvt_circle.txt</a>,
          a set of 100 CVT points in a circle, generated using
          the built in <b>USER</b> routine.
        </li>
        <li>
          <a href = "cvt_circle.png">cvt_circle.png</a>,
          a <a href = "../../data/png/png.html">PNG</a> 
          image of the 100 CVT points in a circle.
        </li>
      </ul>
    </p>

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

    <p>
      <ul>
        <li>
          <b>CH_CAP</b> capitalizes a single character.
        </li>
        <li>
          <b>CH_EQI</b> is a case insensitive comparison of two characters for equality.
        </li>
        <li>
          <b>CH_TO_DIGIT</b> returns the integer value of a base 10 digit.
        </li>
        <li>
          <b>CVT</b> computes a Centroidal Voronoi Tessellation.
        </li>
        <li>
          <b>CVT_ENERGY</b> computes the CVT energy of a dataset.
        </li>
        <li>
          <b>CVT_ITERATE</b> takes one step of the CVT iteration.
        </li>
        <li>
          <b>CVT_SAMPLE</b> returns sample points.
        </li>
        <li>
          <b>CVT_WRITE</b> writes a CVT dataset to a file.
        </li>
        <li>
          <b>DATA_READ</b> reads generator coordinate data from a file.
        </li>
        <li>
          <b>FIND_CLOSEST</b> finds the nearest R point to each S point.
        </li>
        <li>
          <b>GET_SEED</b> returns a seed for the random number generator.
        </li>
        <li>
          <b>GET_UNIT</b> returns a free FORTRAN unit number.
        </li>
        <li>
          <b>HALHAM_LEAP_CHECK</b> checks LEAP for a Halton or Hammersley sequence.
        </li>
        <li>
          <b>HALHAM_N_CHECK</b> checks N for a Halton or Hammersley sequence.
        </li>
        <li>
          <b>HALHAM_DIM_NUM_CHECK</b> checks DIM_NUM for a Halton or Hammersley sequence.
        </li>
        <li>
          <b>HALHAM_SEED_CHECK</b> checks SEED for a Halton or Hammersley sequence.
        </li>
        <li>
          <b>HALHAM_STEP_CHECK</b> checks STEP for a Halton or Hammersley sequence.
        </li>
        <li>
          <b>HALTON_BASE_CHECK</b> checks BASE for a Halton sequence.
        </li>
        <li>
          <b>I4_TO_HALTON_SEQUENCE</b> computes N elements of a leaped Halton subsequence.
        </li>
        <li>
          <b>I4VEC_TRANSPOSE_PRINT</b> prints an integer vector "transposed".
        </li>
        <li>
          <b>PRIME</b> returns any of the first PRIME_MAX prime numbers.
        </li>
        <li>
          <b>R8MAT_TRANSPOSE_PRINT</b> prints a real matrix, transposed.
        </li>
        <li>
          <b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of a real matrix, transposed.
        </li>
        <li>
          <b>R8MAT_UNIFORM_01</b> fills an array with pseudorandom numbers.
        </li>
        <li>
          <b>RANDOM_INITIALIZE</b> initializes the FORTRAN90 random number seed.
        </li>
        <li>
          <b>S_CAP</b> replaces any lowercase letters by uppercase ones in a string.
        </li>
        <li>
          <b>S_EQI</b> is a case insensitive comparison of two strings for equality.
        </li>
        <li>
          <b>S_TO_I4</b> reads an integer value from a string.
        </li>
        <li>
          <b>S_TO_R8</b> reads an R8 from a string.
        </li>
        <li>
          <b>S_TO_R8VEC</b> reads an R8VEC from a string.
        </li>
        <li>
          <b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
        </li>
        <li>
          <b>TIMESTRING</b> writes the current YMDHMS date into a string.
        </li>
        <li>
          <b>TUPLE_NEXT_FAST</b> computes the next element of a tuple space, "fast".
        </li>
        <li>
          <b>USER</b> samples points in a user-specified region with given density.
        </li>
      </ul>
    </p>

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

    <hr>

    <i>
      Last revised on 12 November 2006.
    </i>

    <!-- John Burkardt -->

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