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<html>
<head>
<title>
HERMITE_POLYNOMIAL - Hermite Polynomials
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
HERMITE_POLYNOMIAL <br> Hermite Polynomials
</h1>
<hr>
<p>
<b>HERMITE_POLYNOMIAL</b>
is a C++ library which
evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial,
the Hermite function, and related functions.
</p>
<p>
The physicist's Hermite polynomial H(i,x) can be defined by:
<pre>
H(i,x) = (-1)^i exp(x^2/2) * d^i/dx^i ( exp(-x^2/2) )
</pre>
</p>
<p>
The normalized physicist's Hermite polynomial Hn(i,x) is scaled so that
<pre>
Integral ( -oo < x < +oo ) exp ( - x^2 ) * Hn(i,x) Hn(j,x) dx = delta ( i, j )
</pre>
</p>
<p>
The probabilist's Hermite polynomial He(i,x) is related to H(i,x) by:
<pre>
He(i,x) = H(i,x/sqrt(2)) / sqrt ( 2^in )
</pre>
</p>
<p>
The normalized probabilist's Hermite polynomial Hen(i,x) is scaled so that
<pre>
Integral ( -oo < x < +oo ) exp ( - 0.5*x^2 ) * Hen(i,x) Hen(j,x) dx = delta ( i, j )
</pre>
</p>
<p>
The Hermite function Hf(i,x) is related to H(i,x) by:
<pre>
Hf(i,x) = H(i,x) * exp(-x^2/2) / sqrt ( 2^i * i! * sqrt ( pi ) )
</pre>
</p>
<p>
The Hermite function Hf(i,x) is scaled so that:
<pre>
Integral ( -oo < x < +oo ) Hf(i,x) Hf(j,x) dx = delta ( 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>HERMITE_POLYNOMIAL</b> is available in
<a href = "../../c_src/hermite_polynomial/hermite_polynomial.html">a C version</a> and
<a href = "../../cpp_src/hermite_polynomial/hermite_polynomial.html">a C++ version</a> and
<a href = "../../f77_src/hermite_polynomial/hermite_polynomial.html">a FORTRAN77 version</a> and
<a href = "../../f_src/hermite_polynomial/hermite_polynomial.html">a FORTRAN90 version</a> and
<a href = "../../m_src/hermite_polynomial/hermite_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_POLYOMIAL</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/gen_hermite_rule/gen_hermite_rule.html">
GEN_HERMITE_RULE</a>,
a C++ program which
can compute and print a generalized Gauss-Hermite quadrature rule.
</p>
<p>
<a href = "../../cpp_src/hermite_product_polynomial/hermite_product_polynomial.html">
HERMITE_PRODUCT_POLYNOMIAL</a>,
a C++ library which
defines Hermite product polynomials, creating a multivariate
polynomial as the product of univariate Hermite polynomials.
</p>
<p>
<a href = "../../cpp_src/hermite_rule/hermite_rule.html">
HERMITE_RULE</a>,
a C++ program which
can compute and print a Gauss-Hermite quadrature rule.
</p>
<p>
<a href = "../../cpp_src/hermite_test_int/hermite_test_int.html">
HERMITE_TEST_INT</a>,
a C++ library which
defines test integrands for Hermite integrals with
interval (-oo,+oo) and density exp(-x^2).
</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 polynomials,
and the Laguerre function.
</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/pce_burgers/pce_burgers.html">
PCE_BURGERS</a>,
a C++ program which
defines and solves a version of the time-dependent viscous Burgers equation,
with uncertain viscosity, using a polynomial chaos expansion in terms
of Hermite polynomials,
by Gianluca Iaccarino.
</p>
<p>
<a href = "../../cpp_src/pce_ode_hermite/pce_ode_hermite.html">
PCE_ODE_HERMITE</a>,
a C++ program which
sets up a simple scalar ODE for exponential decay with an uncertain
decay rate, using a polynomial chaos expansion in terms of Hermite polynomials.
</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 = "hermite_polynomial.cpp">hermite_polynomial.cpp</a>, the source code.
</li>
<li>
<a href = "hermite_polynomial.hpp">hermite_polynomial.hpp</a>, the include file.
</li>
<li>
<a href = "hermite_polynomial.sh">hermite_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 = "hermite_polynomial_prb.cpp">hermite_polynomial_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "hermite_polynomial_prb.sh">hermite_polynomial_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "hermite_polynomial_prb_output.txt">hermite_polynomial_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>H_INTEGRAL</b> evaluates the integral of H(i,x).
</li>
<li>
<b>H_POLYNOMIAL_COEFFICIENTS:</b> coefficients of H(i,x).
</li>
<li>
<b>H_POLYNOMIAL_VALUE</b> evaluates H(i,x).
</li>
<li>
<b>H_POLYNOMIAL_VALUES:</b> tabulated values of H(i,x).
</li>
<li>
<b>H_POLYNOMIAL_ZEROS:</b> zeros of H(i,x).
</li>
<li>
<b>H_QUADRATURE_RULE:</b> quadrature for H(i,x).
</li>
<li>
<b>HE_DOUBLE_PRODUCT_INTEGRAL:</b> integral of He(i,x)*He(j,x)*e^(-x^2/2).
</li>
<li>
<b>HE_INTEGRAL</b> evaluates the integral of He(i,x).
</li>
<li>
<b>HE_POLYNOMIAL_COEFFICIENTS:</b> coefficients of He(i,x).
</li>
<li>
<b>HE_POLYNOMIAL_VALUE</b> evaluates He(i,x).
</li>
<li>
<b>HE_POLYNOMIAL_VALUES:</b> tabulated values of He(i,x).
</li>
<li>
<b>HE_POLYNOMIAL_ZEROS:</b> zeros of He(i,x).
</li>
<li>
<b>HE_QUADRATURE_RULE:</b> quadrature for He(i,x).
</li>
<li>
<b>HE_TRIPLE_PRODUCT_INTEGRAL:</b> integral of He(i,x)*He(j,x)*He(k,x)*e^(-x^2/2).
</li>
<li>
<b>HEN_EXPONENTIAL_PRODUCT:</b> exponential product exp(b*x)*Hen(i,x)*Hen(j,x).
</li>
<li>
<b>HEN_POLYNOMIAL_VALUE:</b> evaluates Hen(i,x).
</li>
<li>
<b>HEN_POWER_PRODUCT:</b> power products, x^e*Hen(i,x)*Hen(j,x).
</li>
<li>
<b>HF_EXPONENTIAL_PRODUCT:</b> exponential products, exp(b*x)*Hf(i,x)*Hf(j,x).
</li>
<li>
<b>HF_FUNCTION_VALUE</b> evaluates Hf(i,x).
</li>
<li>
<b>HF_FUNCTION_VALUES:</b> tabulated values of Hf(i,x).
</li>
<li>
<b>HF_POWER_PRODUCT:</b> power products x^e*Hf(i,x)*Hf(j,x).
</li>
<li>
<b>HF_QUADRATURE_RULE:</b> quadrature for Hf(i,x).
</li>
<li>
<b>HN_EXPONENTIAL_PRODUCT:</b> exponential products exp(b*x)*Hn(i,x)*Hn(j,x).
</li>
<li>
<b>HN_POLYNOMIAL_VALUE</b> evaluates Hn(i,x).
</li>
<li>
<b>HN_POWER_PRODUCT:</b> power products x^e*Hn(i,x)*Hn(j,x).
</li>
<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>R8_EPSILON</b> returns the R8 roundoff unit.
</li>
<li>
<b>R8_FACTORIAL</b> computes the factorial of N.
</li>
<li>
<b>R8_FACTORIAL2</b> computes the double factorial function.
</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 19 October 2014.
</i>
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