test_int_hermite/test_int_hermite.html

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    <title>
      TEST_INT_HERMITE - Quadrature Tests for Infinite Intervals
    </title>
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    <h1 align = "center">
      TEST_INT_HERMITE <br> Quadrature Tests for Infinite Intervals
    </h1>

    <hr>

    <p>
      <b>TEST_INT_HERMITE</b>
      is a FORTRAN90 library which
      defines integration problems over infinite intervals of the form (-oo,+oo).
    </p>

    <p>
      The test integrands would normally be used to testing one
      dimensional quadrature software.  It is possible to invoke a
      particular function by index, or to try out all available functions,
      as demonstrated in the sample calling program.
    </p>

    <p>
      For a given integrand function f(x), the problem is to estimate
      <pre>
        I(f) = integral ( -oo < x < +oo ) w(x) * f(x) dx
      </pre>
    </p>

    <p>
      We consider three variations of the problem, depending on the
      form of the weight factor w(x):
      <ul>
        <li>
          <b>option</b> = 0, the unweighted integral:
          <pre>
            Integral ( -oo &lt; x &lt; +oo ) f(x) dx
          </pre>
        </li>
        <li>
          <b>option</b> = 1, the physicist weighted integral:
          <pre>
            Integral ( -oo &lt; x &lt; +oo ) exp(-x*x) f(x) dx
          </pre>
        <li>
          <b>option</b> = 2, the probabilist weighted integral:
          <pre>
            Integral ( -oo &lt; x &lt; +oo ) exp(-x*x/2) f(x) dx
          </pre>
        </li>
      </ul>
    </p>

    <p>
      For option 0, the test integrands have the form:
      <ol>
        <li>
          f1(x) = exp(-x*x) * cos(2*omega*x);
        </li>
        <li>
          f2(x) = exp(-x*x);
        </li>
        <li>
          f3(x) = exp(-px)/(1+exp(-qx));
        </li>
        <li>
          f4(x) = sin ( x^2 );
        </li>
        <li>
          f5(x) = 1 / (1+x^2) sqrt (4+3x^2) );
        </li>
        <li>
          f6(x) = exp(-x*x) * x^m;
        </li>
        <li>
          f7(x) = x^2 cos(x) exp(-x*x);
        </li>
        <li>
          f8(x) = sqrt ( 1 + x * x / 2 ) * exp(-x*x/2);
        </li>
      </ol>
    </p>

    <p>
      For option 1, the test integrands have the form:
      <ol>
        <li>
          f1(x) = cos(2*omega*x);
        </li>
        <li>
          f2(x) = 1
        </li>
        <li>
          f3(x) = exp(x*x) * exp(-px)/(1+exp(-qx));
        </li>
        <li>
          f4(x) = exp(x*x) * sin ( x^2 );
        </li>
        <li>
          f5(x) = exp(x*x) * 1 / (1+x^2) sqrt (4+3x^2) );
        </li>
        <li>
          f6(x) = x^m;
        </li>
        <li>
          f7(x) = x^2 cos(x);
        </li>
        <li>
          f8(x) = sqrt ( 1 + x * x / 2 ) * exp(+x*x/2);
        </li>
      </ol>
    </p>

    <p>
      For option 2, the test integrands have the form:
      <ol>
        <li>
          f1(x) = exp(-x*x/2) * cos(2*omega*x);
        </li>
        <li>
          f2(x) = exp(-x*x/2);
        </li>
        <li>
          f3(x) = exp(+x*x/2) * exp(-px)/(1+exp(-qx));
        </li>
        <li>
          f4(x) = exp(+x*x/2) * sin ( x^2 );
        </li>
        <li>
          f5(x) = exp(+x*x/2) * 1 / (1+x^2) sqrt (4+3x^2) );
        </li>
        <li>
          f6(x) = exp(-x*x/2) * x^m;
        </li>
        <li>
          f7(x) = x^2 cos(x) exp(-x*x/2);
        </li>
        <li>
          f8(x) = sqrt ( 1 + x * x / 2 );
        </li>
      </ol>
    </p>

    <p>
      The library includes not just the integrand, but also the exact value
      of the integral (or, typically, an estimate of this value), and
      a title for the problem.
      Thus, for each integrand function, several routines are supplied.  For
      instance, for function #1, we have the routines:
      <ul>
        <li>
          <b>P01_FUN</b> evaluates the integrand for problem 1.
        </li>
        <li>
          <b>P01_EXACT</b> returns the estimated integral for problem 1.
        </li>
        <li>
          <b>P01_TITLE</b> returns a title for problem 1.
        </li>
      </ul>
      So once you have the calling sequences for these routines, you
      can easily evaluate the function, or integrate it on the
      appropriate interval, or compare your estimate of the integral
      to the exact value.
    </p>

    <p>
      Moreover, since the same interface is used for each function,
      if you wish to work with problem 5 instead, you simply change
      the "01" to "05" in your routine calls.
    </p>

    <p>
      If you wish to call <i>all</i> of the functions, then you
      simply use the generic interface, which requires you to specify
      the problem number as an extra input argument:
      <ul>
        <li>
          <b>P00_FUN</b> evaluates the integrand for any problem.
        </li>
        <li>
          <b>P00_EXACT</b> returns the exact integral for any problem.
        </li>
        <li>
          <b>P00_TITLE</b> returns a title for any problem.
        </li>
      </ul>
    </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>TEST_INT_HERMITE</b> is available in
      <a href = "../../c_src/test_int_hermite/test_int_hermite.html">a C version</a> and
      <a href = "../../cpp_src/test_int_hermite/test_int_hermite.html">a C++ version</a> and
      <a href = "../../f77_src/test_int_hermite/test_int_hermite.html">a FORTRAN77 version</a> and
      <a href = "../../f_src/test_int_hermite/test_int_hermite.html">a FORTRAN90 version</a> and
      <a href = "../../m_src/test_int_hermite/test_int_hermite.html">a MATLAB version</a>.
    </p>

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

    <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_hermite/int_exactness_hermite.html">
      INT_EXACTNESS_HERMITE</a>,
      a FORTRAN90 program which
      tests the polynomial exactness of Gauss-Hermite quadrature rules.
    </p>

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

    <p>
      <a href = "../../datasets/quadrature_rules_hermite_physicist/quadrature_rules_hermite_physicist.html">
      QUADRATURE_RULES_HERMITE_PHYSICIST</a>,
      a dataset directory which
      contains Gauss-Hermite quadrature rules, for integration
      on the interval (-oo,+oo), with weight function exp(-x^2).
    </p>

    <p>
      <a href = "../../datasets/quadrature_rules_hermite_probabilist/quadrature_rules_hermite_probabilist.html">
      QUADRATURE_RULES_HERMITE_PROBABILIST</a>,
      a dataset directory which
      contains Gauss-Hermite quadrature rules, for integration
      on the interval (-oo,+oo), with weight function exp(-x^2/2).
    </p>

    <p>
      <a href = "../../datasets/quadrature_rules_hermite_unweighted/quadrature_rules_hermite_unweighted.html">
      QUADRATURE_RULES_HERMITE_UNWEIGHTED</a>,
      a dataset directory which
      contains Gauss-Hermite quadrature rules, for integration
      on the interval (-oo,+oo), with weight function 1.
    </p>

    <p>
      <a href = "../../f_src/r16_hermite_rule/r16_hermite_rule.html">
      R16_HERMITE_RULE</a>,
      a FORTRAN90 program which
      can compute and print a Gauss-Hermite quadrature rule, using
      "quadruple precision real" arithmetic.
    </p>

    <p>
      <a href = "../../f_src/test_int/test_int.html">
      TEST_INT</a>,
      a FORTRAN90 library which
      defines some test integration problems over finite intervals.
    </p>

    <p>
      <a href = "../../f_src/test_int_2d/test_int_2d.html">
      TEST_INT_2D</a>,
      a FORTRAN90 library which
      defines test integrands for 2D quadrature rules.
    </p>

    <p>
      <a href = "../../f_src/test_int_laguerre/test_int_laguerre.html">
      TEST_INT_LAGUERRE</a>,
      a FORTRAN90 library which
      defines test integrands for integration over [-ALPHA,+oo).
    </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>
        <li>
          Prem Kythe, Michael Schaeferkotter,<br>
          Handbook of Computational Methods for Integration,<br>
          Chapman and Hall, 2004,<br>
          ISBN: 1-58488-428-2,<br>
          LC: QA299.3.K98.
        </li>
        <li>
          Robert Piessens, Elise deDoncker-Kapenga,
          Christian Ueberhuber, David Kahaner,<br>
          QUADPACK: A Subroutine Package for Automatic Integration,<br>
          Springer, 1983,<br>
          ISBN: 3540125531,<br>
          LC: QA299.3.Q36.
        </li>
        <li>
          William Squire,<br>
          Comparison of Gauss-Hermite and Midpoint Quadrature with Application
          to the Voigt Function,<br>
          in Numerical Integration:
          Recent Developments, Software and Applications,<br>
          edited by Patrick Keast, Graeme Fairweather,<br>
          Reidel, 1987, pages 337-340,<br>
          ISBN: 9027725144,<br>
          LC: QA299.3.N38.
        </li>
        <li>
          Arthur Stroud, Don Secrest,<br>
          Gaussian Quadrature Formulas,<br>
          Prentice Hall, 1966,<br>
          LC: QA299.4G3S7.
        </li>
        <li>
          Alan Turing,<br>
          A Method for the Calculation of the Zeta Function,<br>
          Proceedings of the London Mathematical Society,<br>
          Volume 48, 1943, pages 180-197.
        </li>
      </ol>
    </p>

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

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

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

    <p>
      <ul>
        <li>
          <a href = "test_int_hermite_prb.f90">test_int_hermite_prb.f90</a>,
          the calling program;
        </li>
        <li>
          <a href = "test_int_hermite_prb.sh">test_int_hermite_prb.sh</a>,
          commands to compile, link and run the calling program;
        </li>
        <li>
          <a href = "test_int_hermite_prb_output.txt">test_int_hermite_prb_output.txt</a>,
          the output file.
        </li>
      </ul>
    </p>

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

    <p>
      <ul>
        <li>
          <b>HERMITE_COMPUTE</b> computes a Gauss-Hermite quadrature rule.
        </li>
        <li>
          <b>HERMITE_INTEGRAL</b> returns the value of a Hermite polynomial integral.
        </li>
        <li>
          <b>HERMITE_RECUR</b> finds the value and derivative of a Hermite polynomial.
        </li>
        <li>
          <b>HERMITE_ROOT</b> improves an approximate root of a Hermite polynomial.
        </li>
        <li>
          <b>I4_FACTORIAL2</b> computes the double factorial function.
        </li>
        <li>
          <b>P00_EXACT</b> returns the exact integral for any problem.
        </li>
        <li>
          <b>P00_FUN</b> evaluates the integrand for any problem.
        </li>
        <li>
          <b>P00_GAUSS_HERMITE</b> applies a Gauss-Hermite quadrature rule.
        </li>
        <li>
          <b>P00_MONTE_CARLO</b> applies a Monte Carlo procedure to Hermite integrals.
        </li>
        <li>
          <b>P00_PROBLEM_NUM</b> returns the number of test integration problems.
        </li>
        <li>
          <b>P00_TITLE</b> returns the title for any problem.
        </li>
        <li>
          <b>P00_TURING</b> applies the Turing quadrature rule.
        </li>
        <li>
          <b>P01_EXACT</b> returns the exact integral for problem 1.
        </li>
        <li>
          <b>P01_FUN</b> evaluates the integrand for problem 1.
        </li>
        <li>
          <b>P01_TITLE</b> returns the title for problem 1.
        </li>
        <li>
          <b>P02_EXACT</b> returns the exact integral for problem 2.
        </li>
        <li>
          <b>P02_FUN</b> evaluates the integrand for problem 2.
        </li>
        <li>
          <b>P02_TITLE</b> returns the title for problem 2.
        </li>
        <li>
          <b>P03_EXACT</b> returns the exact integral for problem 3.
        </li>
        <li>
          <b>P03_FUN</b> evaluates the integrand for problem 3.
        </li>
        <li>
          <b>P03_TITLE</b> returns the title for problem 3.
        </li>
        <li>
          <b>P04_EXACT</b> returns the exact integral for problem 4.
        </li>
        <li>
          <b>P04_FUN</b> evaluates the integrand for problem 4.
        </li>
        <li>
          <b>P04_TITLE</b> returns the title for problem 4.
        </li>
        <li>
          <b>P05_EXACT</b> returns the exact integral for problem 5.
        </li>
        <li>
          <b>P05_FUN</b> evaluates the integrand for problem 5.
        </li>
        <li>
          <b>P05_TITLE</b> returns the title for problem 5.
        </li>
        <li>
          <b>P06_EXACT</b> returns the exact integral for problem 6.
        </li>
        <li>
          <b>P06_FUN</b> evaluates the integrand for problem 6.
        </li>
        <li>
          <b>P06_PARAM</b> gets or sets parameters for problem 6.
        </li>
        <li>
          <b>P06_TITLE</b> returns the title for problem 6.
        </li>
        <li>
          <b>P07_EXACT</b> returns the exact integral for problem 7.
        </li>
        <li>
          <b>P07_FUN</b> evaluates the integrand for problem 7.
        </li>
        <li>
          <b>P07_TITLE</b> returns the title for problem 7.
        </li>
        <li>
          <b>P08_EXACT</b> returns the exact integral for problem 8.
        </li>
        <li>
          <b>P08_FUN</b> evaluates the integrand for problem 8.
        </li>
        <li>
          <b>P08_TITLE</b> returns the title for problem 8.
        </li>
        <li>
          <b>R8_GAMMA</b> evaluates Gamma(X) for a real argument.
        </li>
        <li>
          <b>R8_UNIFORM_01</b> returns a unit pseudorandom R8.
        </li>
        <li>
          <b>R8VEC_NORMAL_01</b> returns a unit pseudonormal R8VEC.
        </li>
        <li>
          <b>R8VEC_UNIFORM_01</b> returns a unit pseudorandom R8VEC.
        </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 30 July 2010.
    </i>

    <!-- John Burkardt -->

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