legendre_polynomial/legendre_polynomial.html

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      LEGENDRE_POLYNOMIAL - Legendre Polynomials
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      LEGENDRE_POLYNOMIAL <br> Legendre Polynomials
    </h1>

    <hr>

    <p>
      <b>LEGENDRE_POLYNOMIAL</b>
      is a FORTRAN90 library which
      evaluates the Legendre polynomial and associated functions.
    </p>

    <p>
      The Legendre polynomial P(n,x) can be defined by:
      <pre>
        P(0,x) = 1
        P(1,x) = x
        P(n,x) = (2*n-1)/n * x * P(n-1,x) - (n-1)/n * P(n-2,x)
      </pre>
      where n is a nonnegative integer.
    </p>

    <p>
      The N zeroes of P(n,x) are the abscissas used for Gauss-Legendre
      quadrature of the integral of a function F(X) with weight function 1
      over the interval [-1,1].
    </p>

    <p>
      The Legendre polynomials are orthogonal under the inner product defined
      as integration from -1 to 1:
      <pre>
        Integral ( -1 <= x <= 1 ) P(i,x) * P(j,x) dx 
          = 0 if i =/= j
          = 2 / ( 2*i+1 ) if i = j.
      </pre>
    </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>LEGENDRE_POLYNOMIAL</b> is available in
      <a href = "../../cpp_src/legendre_polynomial/legendre_polynomial.html">a C++ version</a> and
      <a href = "../../f_src/legendre_polynomial/legendre_polynomial.html">a FORTRAN90 version</a> and
      <a href = "../../m_src/legendre_polynomial/legendre_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/int_exactness_legendre/int_exactness_legendre.html">
      INT_EXACTNESS_LEGENDRE</a>,
      a FORTRAN90 program which
      tests the polynomial exactness of Gauss-Legendre quadrature rules.
    </p>

    <p>
      <a href = "../../f_src/jacobi_polynomial/jacobi_polynomial.html">
      JACOBI_POLYNOMIAL</a>,
      a FORTRAN90 library which
      evaluates the Jacobi polynomial and associated functions.
    </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 polynomial,
      and the Laguerre function.
    </p>

    <p>
      <a href = "../../f_src/legendre_rule/legendre_rule.html">
      LEGENDRE_RULE</a>,
      a FORTRAN90 program which
      computes a 1D Gauss-Legendre quadrature rule.
    </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>

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      Source Code:
    </h3>

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

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      Examples and Tests:
    </h3>

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

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      List of Routines:
    </h3>

    <p>
      <ul>
        <li>
          <b>IMTQLX</b> diagonalizes a symmetric tridiagonal matrix.
        </li>
        <li>
          <b>P_EXPONENTIAL_PRODUCT:</b> exponential products for P(n,x).
        </li>
        <li>
          <b>P_INTEGRAL</b> evaluates a monomial integral associated with P(n,x).
        </li>
        <li>
          <b>P_POLYNOMIAL</b> evaluates the Legendre polynomials P(n,x).
        </li>
        <li>
          <b>P_POLYNOMIAL_COEFFICIENTS:</b> coefficients of Legendre polynomials P(n,x).
        </li>
        <li>
          <b>P_POLYNOMIAL_PRIME</b> evaluates the derivative of Legendre polynomials P'(n,x).
        </li>
        <li>
          <b>P_POLYNOMIAL_VALUES</b> returns values of the Legendre polynomials P(n,x).
        </li>
        <li>
          <b>P_POLYNOMIAL_ZEROS:</b> zeros of Legendre function P(n,x).
        </li>
        <li>
          <b>P_POWER_PRODUCT:</b> power products for Legendre polynomial P(n,x).
        </li>
        <li>
          <b>P_QUADRATURE_RULE:</b> quadrature for Legendre function P(n,x).
        </li>
        <li>
          <b>PM_POLYNOMIAL</b> evaluates the Legendre polynomials Pm(n,m,x).
        </li>
        <li>
          <b>PM_POLYNOMIAL_VALUES</b> returns values of Legendre polynomials Pm(n,m,x).
        </li>
        <li>
          <b>LEGENDRE_ASSOCIATED_NORMALIZED_VALUES:</b> normalized associated Legendre.
        </li>
        <li>
          <b>LEGENDRE_FUNCTION_Q_VALUES</b> returns values of the Legendre Q function.
        </li>
        <li>
          <b>LEGENDRE_ASSOCIATED_NORMALIZED:</b> normalized associated Legendre functions.
        </li>
        <li>
          <b>LEGENDRE_FUNCTION_Q</b> evaluates the Legendre Q functions.
        </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>

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    <i>
      Last revised on 12 March 2012.
    </i>

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