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<html>
<head>
<title>
HERMITE_PRODUCT_DISPLAY - Display 2D Cartesian Products of Hermite Polynomials
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
HERMITE_PRODUCT_DISPLAY <br> Display 2D Cartesian Products of Hermite Polynomials
</h1>
<hr>
<p>
<b>HERMITE_PRODUCT_DISPLAY</b>
is a MATLAB program which
displays an image of a function created by the Cartesian product of two Hermite polynomials,
such as f(x,y) = h(3,x) * h(1,y).
</p>
<p>
There are five types of Hermite polynomial available. Perhaps the best behaved
are "Hen(n,x)" and "Hf(n,x)", which don't blow up within the plotting interval as fast as
the other functions do.
</p>
<p>
The physicist's Hermite polynomial H(n,x) can be defined by:
<pre>
H(n,x) = (-1)^n exp(x^2/2) * d^n/dx^n ( exp(-x^2/2) )
</pre>
</p>
<p>
The normalized physicist's Hermite polynomial Hn(n,x) is scaled so that
<pre>
Integral ( -oo < X < +oo ) exp ( - X^2 ) * Hn(M,X) Hn(N,X) dX = delta ( N, M )
</pre>
</p>
<p>
The probabilist's Hermite polynomial He(n,x) is related to H(n,x) by:
<pre>
He(n,x) = H(n,x/sqrt(2)) / sqrt ( 2^n )
</pre>
</p>
<p>
The normalized probabilist's Hermite polynomial Hen(n,x) is scaled so that
<pre>
Integral ( -oo < X < +oo ) exp ( - 0.5*X^2 ) * Hen(M,X) Hen(N,X) dX = delta ( N, M )
</pre>
</p>
<p>
The Hermite function Hf(n,x) is related to H(n,x) by:
<pre>
Hf(n,x) = H(n,x) * exp(-x^2/2) / sqrt ( 2^n * n! * sqrt ( pi ) )
</pre>
and is scaled so that:
<pre>
Integral ( -oo < X < +oo ) Hf(M,X) Hf(N,X) dX = delta ( N, M )
</pre>
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>hermite_product_display</b> ( <i>'name'</i>, <i>i</i>, <i>j</i> )
</blockquote>
where
<ul>
<li>
<i>'name'</i> is 'h', 'hn', 'he', 'hen', or 'hf';
</li>
<li>
<i>i</i> is the index of the polynomial in the X direction;
</li>
<li>
<i>j</i> is the index of the polynomial in the Y direction.
</li>
</ul>
</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_PRODUCT_DISPLAY</b> is available in
<a href = "../../m_src/hermite_product_display/hermite_product_display.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/fem_basis_q4_display/fem_basis_q4_display.html">
FEM_BASIS_Q4_DISPLAY</a>,
a MATLAB program which
displays a basis function associated with a linear quadrilateral ("Q4") mesh.
</p>
<p>
<a href = "../../m_src/fem_basis_t3_display/fem_basis_t3_display.html">
FEM_BASIS_T3_DISPLAY</a>,
a MATLAB program which
displays a basis function associated with a 3-node triangle "T3" mesh.
</p>
<p>
<a href = "../../m_src/fem_basis_t4_display/fem_basis_t4_display.html">
FEM_BASIS_T4_DISPLAY</a>,
a MATLAB program which
displays a basis function associated with a 4-node triangle "T4" mesh.
</p>
<p>
<a href = "../../m_src/fem_basis_t6_display/fem_basis_t6_display.html">
FEM_BASIS_T6_DISPLAY</a>,
a MATLAB program which
displays a basis function associated with a 6-node triangle "T6" mesh.
</p>
<p>
<a href = "../../m_src/hermite_polynomial/hermite_polynomial.html">
HERMITE_POLYNOMIAL</a>,
a MATLAB library which
evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial,
the Hermite function, and related functions.
</p>
<p>
<a href = "../../m_src/hermite_product_polynomial/hermite_product_polynomial.html">
HERMITE_PRODUCT_POLYNOMIAL</a>,
a MATLAB library which
defines Hermite product polynomials, creating a multivariate
polynomial as the product of univariate Hermite polynomials.
</p>
<p>
<a href = "../../m_src/polygonal_surface_display/polygonal_surface_display.html">
POLYGONAL_SURFACE_DISPLAY</a>,
a MATLAB program which
displays a surface in 3D described as a set of polygons;
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "hermite_product_display.m">hermite_product_display.m</a>,
displays a 2D Hermite product such as H(i,x)*H(j,y).
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
Here we plot all the possible products of orders 0 through 5, using the
"hen" polynomial:
<ul>
<li>
<a href = "hen_00.png">hen_00.png</a>, Hen(0,x)*Hen(0,y);
</li>
<li>
<a href = "hen_01.png">hen_01.png</a>, Hen(0,x)*Hen(1,y);
</li>
<li>
<a href = "hen_02.png">hen_02.png</a>, Hen(0,x)*Hen(2,y);
</li>
<li>
<a href = "hen_03.png">hen_03.png</a>, Hen(0,x)*Hen(3,y);
</li>
<li>
<a href = "hen_04.png">hen_04.png</a>, Hen(0,x)*Hen(4,y);
</li>
<li>
<a href = "hen_05.png">hen_05.png</a>, Hen(0,x)*Hen(5,y);
</li>
<li>
<a href = "hen_11.png">hen_11.png</a>, Hen(1,x)*Hen(1,y);
</li>
<li>
<a href = "hen_12.png">hen_12.png</a>, Hen(1,x)*Hen(2,y);
</li>
<li>
<a href = "hen_13.png">hen_13.png</a>, Hen(1,x)*Hen(3,y);
</li>
<li>
<a href = "hen_14.png">hen_14.png</a>, Hen(1,x)*Hen(4,y);
</li>
<li>
<a href = "hen_15.png">hen_15.png</a>, Hen(1,x)*Hen(5,y);
</li>
<li>
<a href = "hen_22.png">hen_22.png</a>, Hen(2,x)*Hen(2,y);
</li>
<li>
<a href = "hen_23.png">hen_23.png</a>, Hen(2,x)*Hen(3,y);
</li>
<li>
<a href = "hen_24.png">hen_24.png</a>, Hen(2,x)*Hen(4,y);
</li>
<li>
<a href = "hen_25.png">hen_25.png</a>, Hen(2,x)*Hen(5,y);
</li>
<li>
<a href = "hen_33.png">hen_33.png</a>, Hen(3,x)*Hen(3,y);
</li>
<li>
<a href = "hen_34.png">hen_34.png</a>, Hen(3,x)*Hen(4,y);
</li>
<li>
<a href = "hen_35.png">hen_35.png</a>, Hen(3,x)*Hen(5,y);
</li>
<li>
<a href = "hen_44.png">hen_44.png</a>, Hen(4,x)*Hen(4,y);
</li>
<li>
<a href = "hen_45.png">hen_45.png</a>, Hen(4,x)*Hen(5,y);
</li>
<li>
<a href = "hen_55.png">hen_55.png</a>, Hen(5,x)*Hen(5,y);
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last modified on 19 February 2012.
</i>
<!-- John Burkardt -->
</body>
</html>