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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 C++ 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 = "../../c_src/legendre_polynomial/legendre_polynomial.html">a C version</a> and
<a href = "../../cpp_src/legendre_polynomial/legendre_polynomial.html">a C++ version</a> and
<a href = "../../f77_src/legendre_polynomial/legendre_polynomial.html">a FORTRAN77 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 = "../../cpp_src/bernstein_polynomial/bernstein_polynomial.html">
BERNSTEIN_POLYNOMIAL</a>,
a C++ library which
evaluates the Bernstein polynomials,
useful for uniform approximation of functions;
</p>
<p>
<a href = "../../cpp_src/chebyshev_polynomial/chebyshev_polynomial.html">
CHEBYSHEV_POLYNOMIAL</a>,
a C++ library which
considers the Chebyshev polynomials T(i,x), U(i,x), V(i,x) and W(i,x).
Functions are provided to evaluate the polynomials, determine their zeros,
produce their polynomial coefficients, produce related quadrature rules,
project other functions onto these polynomial bases, and integrate
double and triple products of the polynomials.
</p>
<p>
<a href = "../../cpp_src/hermite_polynomial/hermite_polynomial.html">
HERMITE_POLYNOMIAL</a>,
a C++ library which
evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial,
the Hermite function, and related functions.
</p>
<p>
<a href = "../../cpp_src/int_exactness_legendre/int_exactness_legendre.html">
INT_EXACTNESS_LEGENDRE</a>,
a C++ program which
tests the polynomial exactness of Gauss-Legendre quadrature rules.
</p>
<p>
<a href = "../../cpp_src/jacobi_polynomial/jacobi_polynomial.html">
JACOBI_POLYNOMIAL</a>,
a C++ library which
evaluates the Jacobi polynomial and associated functions.
</p>
<p>
<a href = "../../cpp_src/laguerre_polynomial/laguerre_polynomial.html">
LAGUERRE_POLYNOMIAL</a>,
a C++ library which
evaluates the Laguerre polynomial, the generalized Laguerre polynomial,
and the Laguerre function.
</p>
<p>
<a href = "../../cpp_src/legendre_product_polynomial/legendre_product_polynomial.html">
LEGENDRE_PRODUCT_POLYNOMIAL</a>,
a C++ library which
defines Legendre product polynomials, creating a multivariate
polynomial as the product of univariate Legendre polynomials.
</p>
<p>
<a href = "../../cpp_src/legendre_rule/legendre_rule.html">
LEGENDRE_RULE</a>,
a C++ program which
computes a 1D Gauss-Legendre quadrature rule.
</p>
<p>
<a href = "../../cpp_src/lobatto_polynomial/lobatto_polynomial.html">
LOBATTO_POLYNOMIAL</a>,
a C++ library which
evaluates Lobatto polynomials, similar to Legendre polynomials
except that they are zero at both endpoints.
</p>
<p>
<a href = "../../cpp_src/polpak/polpak.html">
POLPAK</a>,
a C++ library which
evaluates a variety of mathematical functions.
</p>
<p>
<a href = "../../cpp_src/test_values/test_values.html">
TEST_VALUES</a>,
a C++ 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.cpp">legendre_polynomial.cpp</a>, the source code.
</li>
<li>
<a href = "legendre_polynomial.hpp">legendre_polynomial.hpp</a>, the include file.
</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.cpp">legendre_polynomial_prb.cpp</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 = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 14 March 2012.
</i>
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