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<html>
<head>
<title>
LAGUERRE_POLYNOMIAL - Laguerre Polynomials
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
LAGUERRE_POLYNOMIAL <br> Laguerre Polynomials
</h1>
<hr>
<p>
<b>LAGUERRE_POLYNOMIAL</b>
is a C++ library which
evaluates the Laguerre polynomial, the generalized Laguerre polynomials,
and the Laguerre function.
</p>
<p>
The Laguerre polynomial L(n,x) can be defined by:
<pre>
L(n,x) = exp(x)/n! * d^n/dx^n ( exp(-x) * x^n )
</pre>
where n is a nonnegative integer.
</p>
<p>
The generalized Laguerre polynomial Lm(n,m,x) can be defined by:
<pre>
Lm(n,m,x) = exp(x)/(x^m*n!) * d^n/dx^n ( exp(-x) * x^(m+n) )
</pre>
where n and m are nonnegative integers.
</p>
<p>
The Laguerre function can be defined by:
<pre>
Lf(n,alpha,x) = exp(x)/(x^alpha*n!) * d^n/dx^n ( exp(-x) * x^(alpha+n) )
</pre>
where n is a nonnegative integer and -1.0 < alpha is a real number.
</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>LAGUERRE_POLYNOMIAL</b> is available in
<a href = "../../c_src/laguerre_polynomial/laguerre_polynomial.html">a C version</a> and
<a href = "../../cpp_src/laguerre_polynomial/laguerre_polynomial.html">a C++ version</a> and
<a href = "../../f77_src/laguerre_polynomial/laguerre_polynomial.html">a FORTRAN77 version</a> and
<a href = "../../f_src/laguerre_polynomial/laguerre_polynomial.html">a FORTRAN90 version</a> and
<a href = "../../m_src/laguerre_polynomial/laguerre_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
evaluates the Chebyshev polynomial and associated functions.
</p>
<p>
<a href = "../../cpp_src/gen_laguerre_rule/gen_laguerre_rule.html">
GEN_LAGUERRE_RULE</a>,
a C++ program which
can compute and print a generalized Gauss-Laguerre quadrature rule.
</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/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_rule/laguerre_rule.html">
LAGUERRE_RULE</a>,
a C++ program which
can compute and print a Gauss-Laguerre quadrature rule.
</p>
<p>
<a href = "../../cpp_src/laguerre_test_int/laguerre_test_int.html">
LAGUERRE_TEST_INT</a>,
a C++ library which
defines test integrands for integration over [A,+oo).
</p>
<p>
<a href = "../../cpp_src/legendre_polynomial/legendre_polynomial.html">
LEGENDRE_POLYNOMIAL</a>,
a C++ library which
evaluates the Legendre polynomial and associated functions.
</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 = "laguerre_polynomial.cpp">laguerre_polynomial.cpp</a>, the source code.
</li>
<li>
<a href = "laguerre_polynomial.hpp">laguerre_polynomial.hpp</a>, the include file.
</li>
<li>
<a href = "laguerre_polynomial.sh">laguerre_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 = "laguerre_polynomial_prb.cpp">laguerre_polynomial_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "laguerre_polynomial_prb.sh">laguerre_polynomial_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "laguerre_polynomial_prb_output.txt">laguerre_polynomial_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>I4_MAX</b> returns the maximum of two I4's.
</li>
<li>
<b>I4_MIN</b> returns the minimum of two I4's.
</li>
<li>
<b>IMTQLX</b> diagonalizes a symmetric tridiagonal matrix.
</li>
<li>
<b>L_EXPONENTIAL_PRODUCT:</b> exponential product table for L(n,x).
</li>
<li>
<b>L_INTEGRAL</b> evaluates a monomial integral associated with L(n,x).
</li>
<li>
<b>L_POLYNOMIAL</b> evaluates the Laguerre polynomials L(n,x).
</li>
<li>
<b>L_POLYNOMIAL_COEFFICIENTS:</b> coeffs for Laguerre polynomial L(n,x).
</li>
<li>
<b>L_POLYNOMIAL_VALUES</b> returns some values of the Laguerre polynomial L(n,x).
</li>
<li>
<b>L_POLYNOMIAL_ZEROS:</b> zeros of the Laguerre polynomial L(n,x).
</li>
<li>
<b>L_POWER_PRODUCT:</b> power product table for L(n,x).
</li>
<li>
<b>L_QUADRATURE_RULE:</b> Gauss-Laguerre quadrature based on L(n,x).
</li>
<li>
<b>LF_FUNCTION</b> evaluates the Laguerre function Lf(n,alpha,x).
</li>
<li>
<b>LF_FUNCTION_VALUES:</b> some values of the Laguerre function Lf(n,alpha,x).
</li>
<li>
<b>LF_FUNCTION_ZEROS</b> returns the zeros of Lf(n,alpha,x).
</li>
<li>
<b>LF_INTEGRAL</b> evaluates a monomial integral associated with Lf(n,alpha,x).
</li>
<li>
<b>LF_QUADRATURE_RULE:</b> Gauss-Laguerre quadrature rule for Lf(n,alpha,x);
</li>
<li>
<b>LM_INTEGRAL</b> evaluates a monomial integral associated with Lm(n,m,x).
</li>
<li>
<b>LM_POLYNOMIAL</b> evaluates Laguerre polynomials Lm(n,m,x).
</li>
<li>
<b>LM_POLYNOMIAL_COEFFICIENTS:</b> coefficients of Laguerre polynomial Lm(n,m,x).
</li>
<li>
<b>LM_POLYNOMIAL_VALUES:</b> some values of the Laguerre polynomial Lm(n,m,x).
</li>
<li>
<b>LM_POLYNOMIAL_ZEROS</b> returns the zeros for Lm(n,m,x).
</li>
<li>
<b>LM_QUADRATURE_RULE:</b> Gauss-Laguerre quadrature rule for Lm(n,m,x);
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_ADD</b> adds two R8's.
</li>
<li>
<b>R8_EPSILON</b> returns the R8 roundoff unit.
</li>
<li>
<b>R8_FACTORIAL</b> computes the factorial of N.
</li>
<li>
<b>R8_GAMMA</b> evaluates Gamma(X) for an R8.
</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_DOT_PRODUCT</b> computes the dot product of a pair of R8VEC's.
</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 11 March 2012.
</i>
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