jacobi_polynomial/jacobi_polynomial.html

<html>

  <head>
    <title>
      JACOBI_POLYNOMIAL - Jacobi Polynomials
    </title>
  </head>

  <body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">

    <h1 align = "center">
      JACOBI_POLYNOMIAL <br> Jacobi Polynomials
    </h1>

    <hr>

    <p>
      <b>JACOBI_POLYNOMIAL</b>
      is a FORTRAN90 library which
      evaluates the Jacobi polynomial.
    </p>

    <p>
      For a given choice of the parameters a and b, both greater than -1,
      the Jacobi polynomials are a set of polynomials which are
      pairwise orthogonal with respect to the integral:
      <pre>
        integral (-1<=x<=+1) J(i,a,b,x) J(j,a,b,x) (1-x)^a (1+x)^b dx
      </pre>
      That is, this integral is 0 unless i = j.  J(i,a,b,x) indicates the 
      Jacobi polynomial of degree i.
    </p>

    <p>
      The standard Jacobi polynomials can be defined by a three term 
      recurrence formula that is a bit too ugly to quote here.
    </p>

    <p>
      It is worth noting that the definition of the Jacobi polynomials
      is general enough that it includes some familiar families as
      special cases:
      <ul>
        <li>
          if a = b = 0, we have the Legendre polynomials, P(n,x);
        </li>
        <li>
          if a = b = -1/2, we have the Chebyshev polynomials of the first kind, T(n,x);
        </li>
        <li>
          if a = b = 1/2, we have the Chebyshev polynomials of the second kind, U(n,x);
        </li>
        <li>
          if a = b, we have the Gegenbauer polynomials;
        </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>JACOBI_POLYNOMIAL</b> is available in
      <a href = "../../cpp_src/jacobi_polynomial/jacobi_polynomial.html">a C++ version</a> and
      <a href = "../../f_src/jacobi_polynomial/jacobi_polynomial.html">a FORTRAN90 version</a> and
      <a href = "../../m_src/jacobi_polynomial/jacobi_polynomial.html">a MATLAB version</a>.
    </p>

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

    <p>
      <a href = "../../f_src/chebyshev_polynomial/chebyshev_polynomial.html">
      CHEBYSHEV_POLYNOMIAL</a>,
      a FORTRAN90 library which
      evaluates the Chebyshev polynomial and associated functions.
    </p>

    <p>
      <a href = "../../f_src/hermite_polynomial/hermite_polynomial.html">
      HERMITE_POLYNOMIAL</a>,
      a FORTRAN90 library which
      evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial,
      the Hermite function, and related functions.
    </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_polynomial/laguerre_polynomial.html">
      LAGUERRE_POLYNOMIAL</a>,
      a FORTRAN90 library which
      evaluates the Laguerre polynomial, the generalized Laguerre polynomials,
      and the Laguerre function.
    </p>

    <p>
      <a href = "../../f_src/legendre_polynomial/legendre_polynomial.html">
      LEGENDRE_POLYNOMIAL</a>,
      a FORTRAN90 library which
      evaluates the Legendre polynomial and associated functions.
    </p>

    <p>
      <a href = "../../f_src/polpak/polpak.html">
      POLPAK</a>,
      a FORTRAN90 library which
      evaluates a variety of mathematical functions.
    </p>

    <p>
      <a href = "../../f_src/test_values/test_values.html">
      TEST_VALUES</a>,
      a FORTRAN90 library which
      supplies test values of various mathematical functions.
    </p>

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

    <p>
      <ol>
        <li>
          Theodore Chihara,<br>
          An Introduction to Orthogonal Polynomials,<br>
          Gordon and Breach, 1978,<br>
          ISBN: 0677041500,<br>
          LC: QA404.5 C44.
        </li>
        <li>
          Walter Gautschi,<br>
          Orthogonal Polynomials: Computation and Approximation,<br>
          Oxford, 2004,<br>
          ISBN: 0-19-850672-4,<br>
          LC: QA404.5 G3555.
        </li>
        <li>
          Frank Olver, Daniel Lozier, Ronald Boisvert, Charles Clark,<br>
          NIST Handbook of Mathematical Functions,<br>
          Cambridge University Press, 2010,<br>
          ISBN: 978-0521192255,<br>
          LC: QA331.N57.
        </li>
        <li>
          Gabor Szego,<br>
          Orthogonal Polynomials,<br>
          American Mathematical Society, 1992,<br>
          ISBN: 0821810235,<br>
          LC: QA3.A5.v23.
        </li>
      </ol>
    </p>

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

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

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

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

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

    <p>
      <ul>
        <li>
          <b>IMTQLX</b> diagonalizes a symmetric tridiagonal matrix.
        </li>
        <li>
          <b>J_DOUBLE_PRODUCT_INTEGRAL:</b> integral of J(i,x)*J(j,x)*(1-x)^a*(1+x)^b.
        </li>
        <li>
          <b>J_INTEGRAL</b> evaluates a monomial integral associated with J(n,a,b,x).
        </li>
        <li>
          <b>J_POLYNOMIAL</b> evaluates the Jacobi polynomials J(n,a,b,x).
        </li>
        <li>
          <b>J_POLYNOMIAL_VALUES</b> returns some values of the Jacobi polynomial.
        </li>
        <li>
          <b>J_POLYNOMIAL_ZEROS:</b> zeros of Jacobi polynomial J(n,a,b,x).
        </li>
        <li>
          <b>J_QUADRATURE_RULE:</b> Gauss-Jacobi quadrature based on J(n,a,b,x).
        </li>
        <li>
          <b>R8_FACTORIAL</b> computes the factorial of N.
        </li>
        <li>
          <b>R8_SIGN</b> returns the sign of an R8.
        </li>
        <li>
          <b>R8MAT_PRINT</b> prints an R8MAT.
        </li>
        <li>
          <b>R8MAT_PRINT_SOME</b> prints some of an R8MAT.
        </li>
        <li>
          <b>R8VEC_PRINT</b> prints an R8VEC.
        </li>
        <li>
          <b>R8VEC2_PRINT</b> prints an R8VEC2.
        </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 modified on 19 April 2012.
    </i>

    <!-- John Burkardt -->

  </body>

</html>