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<html>
<head>
<title>
LEGENDRE_POLYNOMIAL - Legendre Polynomials
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
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>
<h3 align = "center">
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>
<h3 align = "center">
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>
<h3 align = "center">
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>
<hr>
<i>
Last revised on 12 March 2012.
</i>
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