gm_rule/gm_rule.html

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      GM_RULE - Grundmann-Moeller Quadrature Rules for M-dimensional Simplexes.
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      GM_RULE <br> Grundmann-Moeller Quadrature Rules for M-dimensional Simplexes.
    </h1>

    <hr>

    <p>
      <b>GM_RULE</b>
      is a FORTRAN90 library which
      defines Grundmann-Moeller quadrature rules for a simplex
      (generalized triangle) in M dimensions.
    </p>

    <p>
      The user can choose the spatial dimension M, thus defining the region
      to be a triangle (M = 2), tetrahedron (M = 3) or a general M-dimensional
      simplex.
    </p>

    <p>
      The user chooses the index <b>S</b> of the rule.  Rules are available
      with index <b>S</b> = 0 on up.  A rule of index <b>S</b> will exactly
      integrate any polynomial of total degree <b>2*S+1</b> or less.
    </p>

    <p>
      The rules are defined on the unit M-dimensional simplex.  A simple
      linear transformation can be used to map the vertices and weights
      to an arbitrary simplex, while preserving the accuracy of the rule.
    </p>

    <p>
      The formula for the weight involves the ratio of factorials and a
      high power.  Direct evaluation of this formula results in disastrous
      inaccuracies, even for relatively low order rules.  The original
      code has been revised to take a more cautious and accurate approach
      to this computation.
    </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>GM_RULE</b> is available in
      <a href = "../../c_src/gm_rule/gm_rule.html">a C version</a> and
      <a href = "../../cpp_src/gm_rule/gm_rule.html">a C++ version</a> and
      <a href = "../../f77_src/gm_rule/gm_rule.html">a FORTRAN77 version</a> and
      <a href = "../../f_src/gm_rule/gm_rule.html">a FORTRAN90 version</a> and
      <a href = "../../m_src/gm_rule/gm_rule.html">a MATLAB version</a>.
    </p>

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

    <p>
      <a href = "../../f_src/cubpack/cubpack.html">
      CUBPACK</a>,
      a FORTRAN90 library which
      estimates the integral of a function
      (or vector of functions) over a collection of M-dimensional
      hyperrectangles and simplices.
    </p>

    <p>
      <a href = "../../f_src/dunavant/dunavant.html">
      DUNAVANT</a>,
      a FORTRAN90 library which
      defines Dunavant rules for quadrature on a triangle.
    </p>

    <p>
      <a href = "../../f_src/fekete/fekete.html">
      FEKETE</a>,
      a FORTRAN90 library which
      defines Fekete rules for interpolation or quadrature on a triangle.
    </p>

    <p>
      <a href = "../../f_src/felippa/felippa.html">
      FELIPPA</a>,
      a FORTRAN90 library which
      defines quadrature rules for lines, triangles, quadrilaterals,
      pyramids, wedges, tetrahedrons and hexahedrons.
    </p>

    <p>
      <a href = "../../f_src/keast/keast.html">
      KEAST</a>,
      a FORTRAN90 library which
      defines quadrature rules for a tetrahedron.
    </p>

    <p>
      <a href = "../../f_src/lyness_rule/lyness_rule.html">
      LYNESS_RULE</a>,
      a FORTRAN90 library which
      returns Lyness-Jespersen quadrature rules for the triangle.
    </p>

    <p>
      <a href = "../../f_src/ncc_tetrahedron/ncc_tetrahedron.html">
      NCC_TETRAHEDRON</a>,
      a FORTRAN90 library which
      defines Newton-Cotes Closed quadrature
      rules on a tetrahedron.
    </p>

    <p>
      <a href = "../../f_src/ncc_triangle/ncc_triangle.html">
      NCC_TRIANGLE</a>,
      a FORTRAN90 library which
      defines Newton-Cotes Closed quadrature rules on a triangle.
    </p>

    <p>
      <a href = "../../f_src/nco_tetrahedron/nco_tetrahedron.html">
      NCO_TETRAHEDRON</a>,
      a FORTRAN90 library which
      defines Newton-Cotes Open quadrature rules on a tetrahedron.
    </p>

    <p>
      <a href = "../../f_src/nco_triangle/nco_triangle.html">
      NCO_TRIANGLE</a>,
      a FORTRAN90 library which
      defines Newton-Cotes Open quadrature rules on a triangle.
    </p>

    <p>
      <a href = "../../datasets/quadrature_rules_tet/quadrature_rules_tet.html">
      QUADRATURE_RULES_TET</a>,
      a dataset directory which
      contains triples of files defining various quadrature rules on tetrahedrons.
    </p>

    <p>
      <a href = "../../datasets/quadrature_rules_tri/quadrature_rules_tri.html">
      QUADRATURE_RULES_TRI</a>,
      a dataset directory which
      contains triples of files defining various quadrature rules on triangles.
    </p>

    <p>
      <a href = "../../f77_src/simpack/simpack.html">
      SIMPACK</a>,
      a FORTRAN77 library which
      approximate the integral of a function or vector of functions
      over a M-dimensional simplex, or a region which is the
      sum of M-dimensional simplexes.
    </p>

    <p>
      <a href = "../../f_src/simplex_coordinates/simplex_coordinates.html">
      SIMPLEX_COORDINATES</a>,
      a FORTRAN90 library which
      computes the Cartesian coordinates of the vertices of a regular
      simplex in M dimensions.
    </p>

    <p>
      <a href = "../../f_src/stroud/stroud.html">
      STROUD</a>,
      a FORTRAN90 library which
      defines quadrature rules for a variety of unusual areas, surfaces and volumes
      in 2D, 3D and M-dimensions.
    </p>

    <p>
      <a href = "../../f_src/test_tri_int/test_tri_int.html">
      TEST_TRI_INT</a>,
      a FORTRAN90 library which
      can be used to test algorithms for quadrature over a triangle.
    </p>

    <p>
      <a href = "../../f_src/tetrahedron_exactness/tetrahedron_exactness.html">
      TETRAHEDRON_EXACTNESS</a>,
      a FORTRAN90 program which
      investigates the polynomial exactness of a quadrature rule for the tetrahedron.
    </p>

    <p>
      <a href = "../../m_src/tetrahedron_monte_carlo/tetrahedron_monte_carlo.html">
      TETRAHEDRON_MONTE_CARLO</a>,
      a MATLAB program which
      uses the Monte Carlo method to estimate integrals over a tetrahedron.
    </p>

    <p>
      <a href = "../../f_src/triangle_exactness/triangle_exactness.html">
      TRIANGLE_EXACTNESS</a>,
      a FORTRAN90 program which
      investigates the polynomial exactness of a quadrature rule for the triangle.
    </p>

    <p>
      <a href = "../../f_src/triangle_monte_carlo/triangle_monte_carlo.html">
      TRIANGLE_MONTE_CARLO</a>,
      a FORTRAN90 program which
      uses the Monte Carlo method to estimate integrals over a triangle.
    </p>

    <p>
      <a href = "../../f_src/wandzura/wandzura.html">
      WANDZURA</a>,
      a FORTRAN90 library which
      definine Wandzura rules for quadrature on a triangle.
    </p>

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

    <p>
      <ol>
        <li>
          Paul Bratley, Bennett Fox, Linus Schrage,<br>
          A Guide to Simulation,<br>
          Second Edition,<br>
          Springer, 1987,<br>
          ISBN: 0387964673,<br>
          LC: QA76.9.C65.B73.
        </li>
        <li>
          Bennett Fox,<br>
          Algorithm 647:
          Implementation and Relative Efficiency of Quasirandom
          Sequence Generators,<br>
          ACM Transactions on Mathematical Software,<br>
          Volume 12, Number 4, December 1986, pages 362-376.
        </li>
        <li>
          Axel Grundmann, Michael Moeller,<br>
          Invariant Integration Formulas for the N-Simplex
          by Combinatorial Methods,<br>
          SIAM Journal on Numerical Analysis,<br>
          Volume 15, Number 2, April 1978, pages 282-290.
        </li>
        <li>
          Pierre LEcuyer,<br>
          Random Number Generation,<br>
          in Handbook of Simulation,<br>
          edited by Jerry Banks,<br>
          Wiley, 1998,<br>
          ISBN: 0471134031,<br>
          LC: T57.62.H37.
        </li>
        <li>
          Peter Lewis, Allen Goodman, James Miller,<br>
          A Pseudo-Random Number Generator for the System/360,<br>
          IBM Systems Journal,<br>
          Volume 8, 1969, pages 136-143.
        </li>
        <li>
          Albert Nijenhuis, Herbert Wilf,<br>
          Combinatorial Algorithms for Computers and Calculators,<br>
          Second Edition,<br>
          Academic Press, 1978,<br>
          ISBN: 0-12-519260-6,<br>
          LC: QA164.N54.
        </li>
        <li>
          ML Wolfson, HV Wright,<br>
          Algorithm 160:
          Combinatorial of M Things Taken N at a Time,<br>
          Communications of the ACM,<br>
          Volume 6, Number 4, April 1963, page 161.
        </li>
      </ol>
    </p>

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

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

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

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

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

    <p>
      <ul>
        <li>
          <b>COMP_NEXT</b> computes the compositions of the integer N into K parts.
        </li>
        <li>
          <b>GET_UNIT</b> returns a free FORTRAN unit number.
        </li>
        <li>
          <b>GM_RULE_SET</b> sets a Grundmann-Moeller rule.
        </li>
        <li>
          <b>GM_RULE_SET_OLD</b> sets a Grundmann-Moeller rule.  (OBSOLETE VERSION)
        </li>
        <li>
          <b>GM_RULE_SIZE</b> determines the size of a Grundmann-Moeller rule.
        </li>
        <li>
          <b>I4_CHOOSE</b> computes the binomial coefficient C(N,K).
        </li>
        <li>
          <b>I4_HUGE</b> returns a "huge" I4.
        </li>
        <li>
          <b>MONOMIAL_VALUE</b> evaluates a monomial.
        </li>
        <li>
          <b>R8_FACTORIAL</b> computes the factorial.
        </li>
        <li>
          <b>R8VEC_UNIFORM_01</b> returns a unit pseudorandom R8VEC.
        </li>
        <li>
          <b>SIMPLEX_UNIT_MONOMIAL_INT</b> integrates a monomial over a simplex.
        </li>
        <li>
          <b>SIMPLEX_UNIT_MONOMIAL_QUADRATURE:</b> quadrature of monomials in a unit simplex.
        </li>
        <li>
          <b>SIMPLEX_UNIT_SAMPLE</b> returns uniformly random points from a general simplex.
        </li>
        <li>
          <b>SIMPLEX_UNIT_TO_GENERAL</b> maps the unit simplex to a general simplex.
        </li>
        <li>
          <b>SIMPLEX_UNIT_VOLUME</b> computes the volume of the unit simplex.
        </li>
        <li>
          <b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
        </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 06 February 2011.
    </i>

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