patterson_rule/patterson_rule.html

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      PATTERSON_RULE - Gauss-Patterson Quadrature Rules
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      PATTERSON_RULE <br> Gauss-Patterson Quadrature Rules
    </h1>

    <hr>

    <p>
      <b>PATTERSON_RULE</b>
      is a FORTRAN90 program which
      generates a specific Gauss-Patterson quadrature rule,
      based on user input.
    </p>

    <p>
      The rule is written to three files for easy use as input
      to other programs.
    </p>

    <p>
      The Gauss-Patterson quadrature is a nested family which begins with
      the Gauss-Legendre rules of orders 1 and 3, and then succesively inserts
      one new abscissa in each subinterval.  Thus, after the second rule, the
      Gauss-Patterson rules do not have the super-high precision of the
      Gauss-Legendre rules.  They trade this precision in exchange for the
      advantages of nestedness.  This means that Gauss-Patterson rules are
      only available for orders of 1, 3, 7, 15, 31, 63, 127 or 255.
    </p>

    <p>
      The <i>standard Gauss-Patterson quadrature rule </i> is used as follows:
      <pre>
        Integral ( A &lt;= x &lt;= B ) f(x) dx
      </pre>
      is to be approximated by
      <pre>
        Sum ( 1 &lt;= i &lt;= order ) w(i) * f(x(i))
      </pre>
    </p>

    <p>
      The polynomial precision of a Gauss-Patterson rule can be checked
      numerically by the <b>INT_EXACTNESS_LEGENDRE</b> program.  We should expect
      <table border=1>
        <tr>
          <th>Index</th><th>Order</th><th>Free+Fixed</th><th>Expected Precision</th><th>Actual Precision</th>
        </tr>
        <tr>
          <td>0</td><td>1</td><td>1 +  0</td><td>2*1+0-1=1</td><td>1</td>
        </tr>
        <tr>
          <td>1</td><td>3</td><td>3 +  0</td><td>2*3+0-1=5</td><td>5</td>
        </tr>
        <tr>
          <td>2</td><td>7</td><td>4 +  3</td><td>2*4+3-1=10</td><td>10 + 1 = 11</td>
        </tr>
        <tr>
          <td>3</td><td>15</td><td>8 +  7</td><td>2*8+7-1=22</td><td>22 + 1 = 23</td>
        </tr>
        <tr>
          <td>4</td><td>31</td><td>16 + 15</td><td>2*16+15-1=46</td><td>46 + 1 = 47</td>
        </tr>
        <tr>
          <td>5</td><td>63</td><td>32 + 31</td><td>2*32+31-1=94</td><td>94 + 1 = 95</td>
        </tr>
        <tr>
          <td>6</td><td>127</td><td>64 + 63</td><td>2*64+63-1=190</td><td>190 + 1 = 191</td>
        </tr>
        <tr>
          <td>7</td><td>255</td><td>128 + 127</td><td>2*128+127-1=382</td><td>382 + 1 = 383</td>
        </tr>
      </table>
      where the extra 1 degree of precision comes about because the rules are symmetric,
      and can integrate any odd monomial exactly.  Thus, after the first rule, the
      precision is 3*2^index - 1.
    </p>

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

    <p>
      <blockquote>
        <b>patterson_rule</b> <i>order</i> <i>a</i> <i>b</i> <i>filename</i>
      </blockquote>
      where
      <ul>
        <li>
          <i>order</i> is the number of points in the quadrature rule.  Acceptable values are
          1, 3, 7, 15, 31, 63, 127 or 255.
        </li>
        <li>
          <i>a</i> is the left endpoint;
        </li>
        <li>
          <i>b</i> is the right endpoint;
        </li>
        <li>
          <i>filename</i> specifies the output filenames:
          <i>filename</i><b>_w.txt</b>,
          <i>filename</i><b>_x.txt</b>, and <i>filename</i><b>_r.txt</b>,
          containing the weights, abscissas, and interval limits.
        </li>
      </ul>
    </p>

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

    <p>
      <b>PATTERSON_RULE</b> is available in
      <a href = "../../cpp_src/patterson_rule/patterson_rule.html">a C++ version</a> and
      <a href = "../../f_src/patterson_rule/patterson_rule.html">a FORTRAN90 version</a> and
      <a href = "../../m_src/patterson_rule/patterson_rule.html">a MATLAB version</a>.
    </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">
      Related Data and Programs:
    </h3>

    <p>
      <a href = "../../f_src/ccn_rule/ccn_rule.html">
      CCN_RULE</a>,
      a FORTRAN90 program which
      defines a nested Clenshaw Curtis quadrature rule.
    </p>

    <p>
      <a href = "../../f_src/chebyshev1_rule/chebyshev1_rule.html">
      CHEBYSHEV1_RULE</a>,
      a FORTRAN90 program which
      can compute and print a Gauss-Chebyshev type 1 quadrature rule.
    </p>

    <p>
      <a href = "../../f_src/chebyshev2_rule/chebyshev2_rule.html">
      CHEBYSHEV2_RULE</a>,
      a FORTRAN90 program which
      can compute and print a Gauss-Chebyshev type 2 quadrature rule.
    </p>

    <p>
      <a href = "../../f_src/clenshaw_curtis_rule/clenshaw_curtis_rule.html">
      CLENSHAW_CURTIS_RULE</a>,
      a FORTRAN90 program which
      defines a Clenshaw Curtis quadrature rule.
    </p>

    <p>
      <a href = "../../f_src/gegenbauer_rule/gegenbauer_rule.html">
      GEGENBAUER_RULE</a>,
      a FORTRAN90 program which
      can compute and print a Gauss-Gegenbauer quadrature rule.
    </p>

    <p>
      <a href = "../../f_src/gen_hermite_rule/gen_hermite_rule.html">
      GEN_HERMITE_RULE</a>,
      a FORTRAN90 program which
      can compute and print a generalized Gauss-Hermite quadrature rule.
    </p>

    <p>
      <a href = "../../f_src/gen_laguerre_rule/gen_laguerre_rule.html">
      GEN_LAGUERRE_RULE</a>,
      a FORTRAN90 program which
      can compute and print a generalized Gauss-Laguerre quadrature rule.
    </p>

    <p>
      <a href = "../../f_src/hermite_rule/hermite_rule.html">
      HERMITE_RULE</a>,
      a FORTRAN90 program which
      can compute and print a Gauss-Hermite quadrature rule.
    </p>

    <p>
      <a href = "../../f_src/int_exactness_legendre/int_exactness_legendre.html">
      INT_EXACTNESS_LEGENDRE</a>,
      a FORTRAN90 program which
      checks the polynomial exactness of a Gauss-Legendre quadrature rule.
    </p>

    <p>
      <a href = "../../f_src/jacobi_rule/jacobi_rule.html">
      JACOBI_RULE</a>,
      a FORTRAN90 program which
      can compute and print a Gauss-Jacobi quadrature rule.
    </p>

    <p>
      <a href = "../../f_src/laguerre_rule/laguerre_rule.html">
      LAGUERRE_RULE</a>,
      a FORTRAN90 program which
      can compute and print a Gauss-Laguerre quadrature rule.
    </p>

    <p>
      <a href = "../../f_src/legendre_rule/legendre_rule.html">
      LEGENDRE_RULE</a>,
      a FORTRAN90 program which
      can compute and print a Gauss-Legendre quadrature rule.
    </p>

    <p>
      <a href = "../../f_src/legendre_rule_fast/legendre_rule_fast.html">
      LEGENDRE_RULE_FAST</a>,
      a FORTRAN90 program which
      uses a fast (order N) algorithm to compute a Gauss-Legendre quadrature rule of given order.
    </p>

    <p>
      <a href = "../../f_src/product_rule/product_rule.html">
      PRODUCT_RULE</a>,
      a FORTRAN90 program which
      constructs a product rule
      from 1D factor rules.
    </p>

    <p>
      <a href = "../../f_src/quadpack/quadpack.html">
      QUADPACK</a>,
      a FORTRAN90 library which
      contains routines for numerical estimation of integrals in 1D.
    </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 = "../../datasets/quadrature_rules_patterson/quadrature_rules_patterson.html">
      QUADRATURE_RULES_PATTERSON</a>,
      a dataset directory which
      contains triples of files defining standard Gauss-Patterson
      quadrature rules.
    </p>

    <p>
      <a href = "../../f_src/quadrule/quadrule.html">
      QUADRULE</a>,
      a FORTRAN90 library which
      defines 1-dimensional quadrature rules.
    </p>

    <p>
      <a href = "../../f_src/tanh_sinh_rule/tanh_sinh_rule.html">
      TANH_SINH_RULE</a>,
      a FORTRAN90 program which
      computes and writes out a tanh-sinh quadrature rule of given order.
    </p>

    <p>
      <a href = "../../f_src/test_int/test_int.html">
      TEST_INT</a>,
      a FORTRAN90 library which
      contains functions that may be used as test integrands for
      quadrature rules in 1D.
    </p>

    <p>
      <a href = "../../f77_src/toms699/toms699.html">
      TOMS699</a>,
      a FORTRAN77 library which
      implements a new representation of Patterson's quadrature formula;<br>
      this is ACM TOMS algorithm 699.
    </p>

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

    <p>
      <ol>
        <li>
          Milton Abramowitz, Irene Stegun,<br>
          Handbook of Mathematical Functions,<br>
          National Bureau of Standards, 1964,<br>
          ISBN: 0-486-61272-4,<br>
          LC: QA47.A34.
        </li>
        <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>
        <li>
          Arthur Stroud, Don Secrest,<br>
          Gaussian Quadrature Formulas,<br>
          Prentice Hall, 1966,<br>
          LC: QA299.4G3S7.
        </li>
      </ol>
    </p>

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

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

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

    <p>
      <ul>
        <li>
          <a href = "gp_o15_r.txt">gp_o15_r.txt</a>,
          the region file created by the command
          <pre><b>
            patterson_rule 15 gp_o15
          </b></pre>
        </li>
        <li>
          <a href = "gp_o15_w.txt">gp_o15_w.txt</a>,
          the weight file created by the command
          <pre><b>
            patterson_rule 15 gp_o15
          </b></pre>
        </li>
        <li>
          <a href = "gp_o15_x.txt">gp_o15_x.txt</a>,
          the abscissa file created by the command
          <pre><b>
            patterson_rule 15 gp_o15
          </b></pre>
        </li>
      </ul>
    </p>

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

    <p>
      <ul>
        <li>
          <b>MAIN</b> is the main program for PATTERSON_RULE.
        </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>GET_UNIT</b> returns a free FORTRAN unit number.
        </li>
        <li>
          <b>ORDER_CHECK</b> checks the value of ORDER.
        </li>
        <li>
          <b>PATTERSON_HANDLE</b> looks up the requested Gauss-Patterson rule and outputs it.
        </li>
        <li>
          <b>PATTERSON_SET</b> sets abscissas and weights for Gauss-Patterson quadrature.
        </li>
        <li>
          <b>R8MAT_WRITE</b> writes an R8MAT file.
        </li>
        <li>
          <b>RESCALE</b> rescales a Legendre quadrature rule from [-1,+1] to [A,B].
        </li>
        <li>
          <b>S_TO_I4</b> reads an I4 from a string.
        </li>
        <li>
          <b>S_TO_R8</b> reads an R8 value from 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 16 February 2010.
    </i>

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