nint_exactness/nint_exactness.html

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    <title>
      NINT_EXACTNESS - Exactness of Multidimensional Quadrature
    </title>
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    <h1 align = "center">
      NINT_EXACTNESS <br> Exactness of Multidimensional Quadrature
    </h1>

    <hr>

    <p>
      <b>NINT_EXACTNESS</b>
      is a FORTRAN90 program which
      investigates the polynomial exactness of a multidimensional
      quadrature rule which is defined over a finite rectangular product region.
    </p>

    <p>
      The polynomial exactness of a quadrature rule is defined as the
      highest total degree <b>D</b> such that the quadrature rule is
      guaranteed to integrate exactly all polynomials of total degree
      <b>DEGREE_MAX</b> or less, ignoring roundoff.  The total degree of a polynomial
      is the maximum of the degrees of all its monomial terms.  The degree
      of a monomial term is the sum of the exponents.  Thus, for instance,
      the <b>DEGREE</b> of
      <blockquote><b>
        x<sup>2</sup>y z<sup>5</sup>
      </b></blockquote>
      is 2+1+5=8.
    </p>

    <p>
      To be thorough, the program starts at <b>DEGREE</b> = 0, and then
      proceeds to <b>DEGREE</b> = 1, 2, and so on up to a maximum degree
      <b>DEGREE_MAX</b> specified by the user.  At each value of <b>DEGREE</b>,
      the program generates every possible monomial term, applies the
      quadrature rule to it, and determines the quadrature error.  The program
      uses a scaling factor on each monomial so that the exact integral
      should always be 1; therefore, each reported error can be compared
      on a fixed scale.
    </p>

    <p>
      The program is very flexible and interactive.  The quadrature rule
      is defined by three files, to be read at input, and the
      maximum degree is specified by the user as well.
    </p>

    <p>
      Note that the three files that define the quadrature rule
      are assumed to have related names, of the form
      <ul>
        <li>
          <i>prefix</i>_<b>x.txt</b>
        </li>
        <li>
          <i>prefix</i>_<b>w.txt</b>
        </li>
        <li>
          <i>prefix</i>_<b>r.txt</b>
        </li>
      </ul>
      When running the program, the user only enters the common <i>prefix</i>
      part of the file names, which is enough information for the program
      to find all three files.
    </p>

    <p>
      For information on the form of these files, see the
      <b>QUADRATURE_RULES</b> directory listed below.
    </p>

    <p>
      The exactness results are written to an output file with the
      corresponding name:
      <ul>
        <li>
          <i>prefix</i>_<b>exact.txt</b>
        </li>
      </ul>
    </p>

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

    <p>
      <blockquote>
        <b>nint_exactness</b> <i>prefix</i> <i>degree_max</i>
      </blockquote>
      where
      <ul>
        <li>
          <i>prefix</i> is the common prefix for the files containing the abscissa, weight
          and region information of the quadrature rule;
        </li>
        <li>
          <i>degree_max</i> is the maximum total monomial degree to check.  This should be
          a relatively small nonnegative number, particularly if the
          spatial dimension is high.  A value of 5 or 10 might be
          reasonable, but a value of 50 or 100 is probably never a
          good input!
        </li>
      </ul>
    </p>

    <p>
      If the arguments are not supplied on the command line, the
      program will prompt for them.
    </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>NINT_EXACTNESS</b> is available in
      <a href = "../../cpp_src/nint_exactness/nint_exactness.html">a C++ version</a> and
      <a href = "../../f_src/nint_exactness/nint_exactness.html">a FORTRAN90 version</a> and
      <a href = "../../m_src/nint_exactness/nint_exactness.html">a MATLAB version</a>.
    </p>

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

    <p>
      <a href = "../../f_src/int_exactness/int_exactness.html">
      INT_EXACTNESS</a>,
      a FORTRAN90 program which
      tests the polynomial exactness of one dimensional quadrature rules.
    </p>

    <p>
      <a href = "../../f_src/integral_test/integral_test.html">
      INTEGRAL_TEST</a>,
      a FORTRAN90 program which
      uses test integrals to measure the effectiveness of
      certain sets of quadrature rules.
    </p>

    <p>
      <a href = "../../f_src/nint_exactness_mixed/nint_exactness_mixed.html">
      NINT_EXACTNESS_MIXED</a>,
      a FORTRAN90 program which
      measures the polynomial exactness of a multidimensional quadrature rule
      based on a mixture of 1D quadrature rule factors.
    </p>

    <p>
      <a href = "../../f_src/pyramid_exactness/pyramid_exactness.html">
      PYRAMID_EXACTNESS</a>,
      a FORTRAN90 program which
      investigates the polynomial exactness of a quadrature rule for the pyramid.
    </p>

    <p>
      <a href = "../../datasets/quadrature_rules/quadrature_rules.html">
      QUADRATURE_RULES</a>,
      a dataset directory which
      contains sets of files that define quadrature
      rules over various 1D intervals or multidimensional hypercubes.
    </p>

    <p>
      <a href = "../../f_src/quadrule/quadrule.html">
      QUADRULE</a>,
      a FORTRAN90 library which
      defines quadrature rules on a
      variety of intervals with different weight functions.
    </p>

    <p>
      <a href = "../../f_src/sphere_exactness/sphere_exactness.html">
      SPHERE_EXACTNESS</a>,
      a FORTRAN90 program which
      tests the polynomial exactness of a quadrature rule for the unit sphere;
    </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 multiple dimensions.
    </p>

    <p>
      <a href = "../../f_src/test_nint/test_nint.html">
      TEST_NINT</a>,
      a FORTRAN90 library which
      defines integrand functions for testing
      M-dimensional quadrature routines.
    </p>

    <p>
      <a href = "../../f_src/testpack/testpack.html">
      TESTPACK</a>,
      a FORTRAN90 library which
      defines a set of integrands used to test M-dimensional quadrature.
    </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 = "../../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>

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

    <p>
      <ol>
        <li>
          Philip Davis, Philip Rabinowitz,<br>
          Methods of Numerical Integration,<br>
          Second Edition,<br>
          Dover, 2007,<br>
          ISBN: 0486453391,<br>
          LC: QA299.3.D28.
        </li>
      </ol>
    </p>

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

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

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

    <p>
      <b>CC_D1_O2</b> is a Clenshaw-Curtis order 2 rule for 1D.
      <ul>
        <li>
          <a href = "../../datasets/quadrature_rules/cc_d1_o2_x.txt">cc_d1_o2_x.txt</a>,
          the abscissas of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/cc_d1_o2_w.txt">cc_d1_o2_w.txt</a>,
          the weights of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/cc_d1_o2_r.txt">cc_d1_o2_r.txt</a>,
          defines the region for the rule.
        </li>
        <li>
          <a href = "cc_d1_o2_exact.txt">cc_d1_o2_exact.txt</a>,
          the results of the exactness test, up to degree 5.
        </li>
      </ul>
    </p>

    <p>
      <b>CC_D1_O3</b> is a Clenshaw-Curtis order 3 rule for 1D.
      If you are paying attention, you may be surprised to see that
      a Clenshaw Curtis rule of odd order has one more degree of
      accuracy than you'd expect!
      <ul>
        <li>
          <a href = "../../datasets/quadrature_rules/cc_d1_o3_x.txt">cc_d1_o3_x.txt</a>,
          the abscissas of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/cc_d1_o3_w.txt">cc_d1_o3_w.txt</a>,
          the weights of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/cc_d1_o3_r.txt">cc_d1_o3_r.txt</a>,
          defines the region for the rule.
        </li>
        <li>
          <a href = "cc_d1_o3_exact.txt">cc_d1_o3_exact.txt</a>,
          the results of the exactness test, up to degree 5.
        </li>
      </ul>
    </p>

    <p>
      <b>CC_D2_O3x3</b> is a Clenshaw-Curtis 3x3 product rule for 2D.
      <ul>
        <li>
          <a href = "../../datasets/quadrature_rules/cc_d2_o3x3_x.txt">cc_d2_o3x3_x.txt</a>,
          the abscissas of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/cc_d2_o3x3_w.txt">cc_d2_o3x3_w.txt</a>,
          the weights of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/cc_d2_o3x3_r.txt">cc_d2_o3x3_r.txt</a>,
          defines the region for the rule.
        </li>
        <li>
          <a href = "cc_d2_o3x3_exact.txt">cc_d2_o3x3_exact.txt</a>,
          the results of the exactness test, up to degree 8.
        </li>
      </ul>
    </p>

    <p>
      <b>CC_D3_O3x3x3</b> is a Clenshaw-Curtis 3x3x3 product rule for 3D.
      <ul>
        <li>
          <a href = "../../datasets/quadrature_rules/cc_d3_o3x3x3_x.txt">cc_d3_o3x3x3_x.txt</a>,
          the abscissas of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/cc_d3_o3x3x3_w.txt">cc_d3_o3x3x3_w.txt</a>,
          the weights of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/cc_d3_o3x3x3_r.txt">cc_d3_o3x3x3_r.txt</a>,
          defines the region for the rule.
        </li>
        <li>
          <a href = "cc_d3_o3x3x3_exact.txt">cc_d3_o3x3x3_exact.txt</a>,
          the results of the exactness test, up to degree 5.
        </li>
      </ul>
    </p>

    <p>
      <b>CCGL_D2_O3x2</b> is a product rule for 2D whose factors are
      a Clenshaw-Curtis of order 3 and a Gauss-Legendre rule of order 2.
      <ul>
        <li>
          <a href = "../../datasets/quadrature_rules/ccgl_d2_o3x2_x.txt">ccgl_d2_o3x2_x.txt</a>,
          the abscissas of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/ccgl_d2_o3x2_w.txt">ccgl_d2_o3x2_w.txt</a>,
          the weights of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/ccgl_d2_o3x2_r.txt">ccgl_d2_o3x2_r.txt</a>,
          defines the region for the rule.
        </li>
        <li>
          <a href = "ccgl_d2_o3x2_exact.txt">ccgl_d2_o3x2_exact.txt</a>,
          the results of the exactness test, up to degree 5.
        </li>
      </ul>
    </p>

    <p>
      <b>CC_D2_LEVEL0</b> is a Clenshaw Curtis sparse grid rule for 2D
      of level 0 and order 1.
      <ul>
        <li>
          <a href = "../../datasets/sparse_grid_cc/cc_d2_level0_x.txt">cc_d2_level0_x.txt</a>,
          the abscissas of the rule.
        </li>
        <li>
          <a href = "../../datasets/sparse_grid_cc/cc_d2_level0_w.txt">cc_d2_level0_w.txt</a>,
          the weights of the rule.
        </li>
        <li>
          <a href = "../../datasets/sparse_grid_cc/cc_d2_level0_r.txt">cc_d2_level0_r.txt</a>,
          defines the region for the rule.
        </li>
        <li>
          <a href = "cc_d2_level0_exact.txt">cc_d2_level0_exact.txt</a>,
          the results of the exactness test, up to degree 4.
        </li>
      </ul>
    </p>

    <p>
      <b>CC_D2_LEVEL1</b> is a Clenshaw Curtis sparse grid rule for 2D
      of level 1 and order 5.
      <ul>
        <li>
          <a href = "../../datasets/sparse_grid_cc/cc_d2_level1_x.txt">cc_d2_level1_x.txt</a>,
          the abscissas of the rule.
        </li>
        <li>
          <a href = "../../datasets/sparse_grid_cc/cc_d2_level1_w.txt">cc_d2_level1_w.txt</a>,
          the weights of the rule.
        </li>
        <li>
          <a href = "../../datasets/sparse_grid_cc/cc_d2_level1_r.txt">cc_d2_level1_r.txt</a>,
          defines the region for the rule.
        </li>
        <li>
          <a href = "cc_d2_level1_exact.txt">cc_d2_level1_exact.txt</a>,
          the results of the exactness test, up to degree 4.
        </li>
      </ul>
    </p>

    <p>
      <b>CC_D2_LEVEL2</b> is a Clenshaw Curtis sparse grid rule for 2D
      of level 2 and order 13.
      <ul>
        <li>
          <a href = "../../datasets/sparse_grid_cc/cc_d2_level2_x.txt">cc_d2_level2_x.txt</a>,
          the abscissas of the rule.
        </li>
        <li>
          <a href = "../../datasets/sparse_grid_cc/cc_d2_level2_w.txt">cc_d2_level2_w.txt</a>,
          the weights of the rule.
        </li>
        <li>
          <a href = "../../datasets/sparse_grid_cc/cc_d2_level2_r.txt">cc_d2_level2_r.txt</a>,
          defines the region for the rule.
        </li>
        <li>
          <a href = "cc_d2_level2_exact.txt">cc_d2_level2_exact.txt</a>,
          the results of the exactness test, up to degree 6.
        </li>
      </ul>
    </p>

    <p>
      <b>CC_D2_LEVEL3</b> is a Clenshaw Curtis sparse grid rule for 2D
      of level 3 and order 25.
      <ul>
        <li>
          <a href = "../../datasets/sparse_grid_cc/cc_d2_level3_x.txt">cc_d2_level3_x.txt</a>,
          the abscissas of the rule.
        </li>
        <li>
          <a href = "../../datasets/sparse_grid_cc/cc_d2_level3_w.txt">cc_d2_level3_w.txt</a>,
          the weights of the rule.
        </li>
        <li>
          <a href = "../../datasets/sparse_grid_cc/cc_d2_level3_r.txt">cc_d3_level3_r.txt</a>,
          defines the region for the rule.
        </li>
        <li>
          <a href = "cc_d2_level3_exact.txt">cc_d2_level3_exact.txt</a>,
          the results of the exactness test, up to degree 9.
        </li>
      </ul>
    </p>

    <p>
      <b>CC_D2_LEVEL4</b> is a Clenshaw Curtis sparse grid rule for 2D
      of level 4 and order 65.
      <ul>
        <li>
          <a href = "../../datasets/sparse_grid_cc/cc_d2_level4_x.txt">cc_d2_level4_x.txt</a>,
          the abscissas of the rule.
        </li>
        <li>
          <a href = "../../datasets/sparse_grid_cc/cc_d2_level4_w.txt">cc_d2_level4_w.txt</a>,
          the weights of the rule.
        </li>
        <li>
          <a href = "../../datasets/sparse_grid_cc/cc_d2_level4_r.txt">cc_d3_level4_r.txt</a>,
          defines the region for the rule.
        </li>
        <li>
          <a href = "cc_d2_level4_exact.txt">cc_d2_level4_exact.txt</a>,
          the results of the exactness test, up to degree 17.
        </li>
      </ul>
    </p>

    <p>
      <b>CC_D100_LEVEL1</b> is a Clenshaw Curtis sparse grid rule for 100D
      of level 1 and order 201.
      <ul>
        <li>
          <a href = "../../datasets/sparse_grid_cc/cc_d100_level1_x.txt">cc_d100_level1_x.txt</a>,
          the abscissas of the rule.
        </li>
        <li>
          <a href = "../../datasets/sparse_grid_cc/cc_d100_level1_w.txt">cc_d100_level1_w.txt</a>,
          the weights of the rule.
        </li>
        <li>
          <a href = "../../datasets/sparse_grid_cc/cc_d100_level1_r.txt">cc_d100_level1_r.txt</a>,
          defines the region for the rule.
        </li>
        <li>
          <a href = "cc_d100_level1_exact.txt">cc_d100_level1_exact.txt</a>,
          the results of the exactness test, up to degree 2, demonstrating
          the accurate integration of 10,101 separate monomial terms in 100D,
          using a 201 point rule.
        </li>
      </ul>
    </p>

    <p>
      <b>CCS_D2_LEVEL4</b> is a Clenshaw Curtis "Slow-Exponential-Growth" sparse grid rule for 2D
      of level 4 and order 49.
      <ul>
        <li>
          <a href = "../../datasets/sparse_grid_ccs/ccs_d2_level4_x.txt">ccs_d2_level4_x.txt</a>,
          the abscissas of the rule.
        </li>
        <li>
          <a href = "../../datasets/sparse_grid_ccs/ccs_d2_level4_w.txt">ccs_d2_level4_w.txt</a>,
          the weights of the rule.
        </li>
        <li>
          <a href = "../../datasets/sparse_grid_ccs/ccs_d2_level4_r.txt">ccs_d3_level4_r.txt</a>,
          defines the region for the rule.
        </li>
        <li>
          <a href = "ccs_d2_level4_exact.txt">ccs_d2_level4_exact.txt</a>,
          the results of the exactness test, up to degree 17.
        </li>
      </ul>
    </p>

    <p>
      <b>GL_D1_O3</b> is a Gauss-Legendre order 3 rule for 1D.
      <ul>
        <li>
          <a href = "../../datasets/quadrature_rules/gl_d1_o3_x.txt">gl_d1_o3_x.txt</a>,
          the abscissas of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/gl_d1_o3_w.txt">gl_d1_o3_w.txt</a>,
          the weights of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/gl_d1_o3_r.txt">gl_d1_o3_r.txt</a>,
          defines the region for the rule.
        </li>
        <li>
          <a href = "gl_d1_o3_exact.txt">gl_d1_o3_exact.txt</a>,
          the results of the exactness test, up to degree 5.
        </li>
      </ul>
    </p>

    <p>
      <b>GL_D2_O3x3</b> is a Gauss-Legendre 3x3 product rule for 2D.
      <ul>
        <li>
          <a href = "../../datasets/quadrature_rules/gl_d2_o3x3_x.txt">gl_d2_o3x3_x.txt</a>,
          the abscissas of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/gl_d2_o3x3_w.txt">gl_d2_o3x3_w.txt</a>,
          the weights of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/gl_d2_o3x3_r.txt">gl_d2_o3x3_r.txt</a>,
          defines the region for the rule.
        </li>
        <li>
          <a href = "gl_d2_o3x3_exact.txt">gl_d2_o3x3_exact.txt</a>,
          the results of the exactness test, up to degree 5.
        </li>
      </ul>
    </p>

    <p>
      <b>GL_D3_O3x3x3</b> is a Gauss-Legendre 3x3x3 product rule for 3D.
      <ul>
        <li>
          <a href = "../../datasets/quadrature_rules/gl_d3_o3x3x3_x.txt">gl_d3_o3x3x3_x.txt</a>,
          the abscissas of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/gl_d3_o3x3x3_w.txt">gl_d3_o3x3x3_w.txt</a>,
          the weights of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/gl_d3_o3x3x3_r.txt">gl_d3_o3x3x3_r.txt</a>,
          defines the region for the rule.
        </li>
        <li>
          <a href = "gl_d3_o3x3x3_exact.txt">gl_d3_o3x3x3_exact.txt</a>,
          the results of the exactness test, up to degree 5.
        </li>
      </ul>
    </p>

    <p>
      <b>NCC_D1_O5</b> is a Newton-Cotes Closed order 5 rule for 1D.
      <ul>
        <li>
          <a href = "../../datasets/quadrature_rules/ncc_d1_o5_x.txt">ncc_d1_o5_x.txt</a>,
          the abscissas of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/ncc_d1_o5_w.txt">ncc_d1_o5_w.txt</a>,
          the weights of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/ncc_d1_o5_r.txt">ncc_d1_o5_r.txt</a>,
          defines the region for the rule.
        </li>
        <li>
          <a href = "ncc_d1_o5_exact.txt">ncc_d1_o5_exact.txt</a>,
          the results of the exactness test, up to degree 7.
        </li>
      </ul>
    </p>

    <p>
      <b>NCC_D2_O5x5</b> is a Newton-Cotes Closed 5x5 product rule for 2D.
      <ul>
        <li>
          <a href = "../../datasets/quadrature_rules/ncc_d2_o5x5_x.txt">ncc_d2_o5x5_x.txt</a>,
          the abscissas of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/ncc_d2_o5x5_w.txt">ncc_d2_o5x5_w.txt</a>,
          the weights of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/ncc_d2_o5x5_r.txt">ncc_d2_o5x5_r.txt</a>,
          defines the region for the rule.
        </li>
        <li>
          <a href = "ncc_d2_o5x5_exact.txt">ncc_d2_o5x5_exact.txt</a>,
          the results of the exactness test, up to degree 7.
        </li>
      </ul>
    </p>

    <p>
      <b>NCC_D3_O5x5x5</b> is a Newton-Cotes Closed 5x5x5 product rule for 3D.
      <ul>
        <li>
          <a href = "../../datasets/quadrature_rules/ncc_d3_o5x5x5_x.txt">ncc_d3_o5x5x5_x.txt</a>,
          the abscissas of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/ncc_d3_o5x5x5_w.txt">ncc_d3_o5x5x5_w.txt</a>,
          the weights of the rule.
        </li>
        <li>
          <a href = "../../datasets/quadrature_rules/ncc_d3_o5x5x5_r.txt">ncc_d3_o5x5x5_r.txt</a>,
          defines the region for the rule.
        </li>
        <li>
          <a href = "ncc_d3_o5x5x5_exact.txt">ncc_d3_o5x5x5_exact.txt</a>,
          the results of the exactness test, up to degree 7.
        </li>
      </ul>
    </p>

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

    <p>
      <ul>
        <li>
          <b>MAIN</b> is the main program for NINT_EXACTNESS.
        </li>
        <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>COMP_NEXT</b> computes the compositions of the integer N into K parts.
        </li>
        <li>
          <b>FILE_COLUMN_COUNT</b> counts the number of columns in the first line of a file.
        </li>
        <li>
          <b>FILE_ROW_COUNT</b> counts the number of row records in a file.
        </li>
        <li>
          <b>GET_UNIT</b> returns a free FORTRAN unit number.
        </li>
        <li>
          <b>MONOMIAL_INT01</b> returns the integral of a monomial over the [0,1] hypercube.
        </li>
        <li>
          <b>MONOMIAL_QUADRATURE</b> applies a quadrature rule to a monomial.
        </li>
        <li>
          <b>MONOMIAL_VALUE</b> evaluates a monomial.
        </li>
        <li>
          <b>R8MAT_DATA_READ</b> reads data from an R8MAT file.
        </li>
        <li>
          <b>R8MAT_HEADER_READ</b> reads the header from an R8MAT file.
        </li>
        <li>
          <b>S_TO_I4</b> reads an I4 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>S_WORD_COUNT</b> counts the number of "words" in a string.
        </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>

    <hr>

    <i>
      Last revised on 27 January 2010.
    </i>

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