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<html>
<head>
<title>
LOBATTO_POLYNOMIAL - Lobatto Polynomials
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
LOBATTO_POLYNOMIAL <br> Lobatto Polynomials
</h1>
<hr>
<p>
<b>LOBATTO_POLYNOMIAL</b>
is a MATLAB library which
evaluates the completed Lobatto polynomial and associated functions.
</p>
<p>
The completed Lobatto polynomial Lo(n,x) can be defined by:
<pre>
Lo(n,x) = n * ( P(n-1,x) - x * P(n,x) )
</pre>
where n is a positive integer called the order, x is a real value
between -1 and +1, and P(n,x) is the Legendre polynomial.
</p>
<p>
The completed Lobatto polynomial Lo(n,x) has degree n+1, and is
zero at x = -1 and x = +1.
</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>LOBATTO_POLYNOMIAL</b> is available in
<a href = "../../c_src/lobatto_polynomial/lobatto_polynomial.html">a C version</a> and
<a href = "../../cpp_src/lobatto_polynomial/lobatto_polynomial.html">a C++ version</a> and
<a href = "../../f77_src/lobatto_polynomial/lobatto_polynomial.html">a FORTRAN77 version</a> and
<a href = "../../f_src/lobatto_polynomial/lobatto_polynomial.html">a FORTRAN90 version</a> and
<a href = "../../m_src/lobatto_polynomial/lobatto_polynomial.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/bernstein_polynomial/bernstein_polynomial.html">
BERNSTEIN_POLYNOMIAL</a>,
a MATLAB library which
evaluates the Bernstein polynomials,
useful for uniform approximation of functions;
</p>
<p>
<a href = "../../m_src/chebyshev_polynomial/chebyshev_polynomial.html">
CHEBYSHEV_POLYNOMIAL</a>,
a MATLAB 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 = "../../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/jacobi_polynomial/jacobi_polynomial.html">
JACOBI_POLYNOMIAL</a>,
a MATLAB library which
evaluates the Jacobi polynomial and associated functions.
</p>
<p>
<a href = "../../m_src/laguerre_polynomial/laguerre_polynomial.html">
LAGUERRE_POLYNOMIAL</a>,
a MATLAB library which
evaluates the Laguerre polynomial, the generalized Laguerre polynomial,
and the Laguerre function.
</p>
<p>
<a href = "../../m_src/legendre_polynomial/legendre_polynomial.html">
LEGENDRE_POLYNOMIAL</a>,
a MATLAB library which
evaluates the Legendre polynomial and associated functions.
</p>
<p>
<a href = "../../m_src/polpak/polpak.html">
POLPAK</a>,
a MATLAB library which
evaluates a variety of mathematical functions.
</p>
<p>
<a href = "../../m_src/test_values/test_values.html">
TEST_VALUES</a>,
a MATLAB library which
supplies test values of various mathematical functions.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Milton Abramowitz, Irene Stegun,<br>
Handbook of Mathematical Functions,<br>
National Bureau of Standards, 1964,<br>
ISBN: 0-486-61272-4,<br>
LC: QA47.A34.
</li>
<li>
Larry Andrews,<br>
Special Functions of Mathematics for Engineers,<br>
Second Edition,<br>
Oxford University Press, 1998,<br>
ISBN: 0819426164,<br>
LC: QA351.A75.
</li>
<li>
Daniel Zwillinger, editor,<br>
CRC Standard Mathematical Tables and Formulae,<br>
30th Edition,<br>
CRC Press, 1996.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "lobatto_polynomial_derivative.m">lobatto_polynomial_derivative.m</a>,
returns the derivative of the completed Lobatto polynomials Lo'(n,x).
</li>
<li>
<a href = "lobatto_polynomial_plot.m">lobatto_polynomial_plot.m</a>,
plots completed Lobatto polynomials Lo(n,x).
</li>
<li>
<a href = "lobatto_polynomial_value.m">lobatto_polynomial_value.m</a>,
returns the value of the completed Lobatto polynomials Lo(n,x).
</li>
<li>
<a href = "lobatto_polynomial_values.m">lobatto_polynomial_values.m</a>,
returns some tabulated values of the completed Lobatto polynomials Lo(n,x).
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>,
prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "lobatto_polynomial_test.m">lobatto_polynomial_test.m</a>,
calls all the tests.
</li>
<li>
<a href = "lobatto_polynomial_test_output.txt">lobatto_polynomial_test_output.txt</a>,
the output file.
</li>
<li>
<a href = "lobatto_polynomial_derivative_test.m">lobatto_polynomial_derivative_test.m</a>
</li>
<li>
<a href = "lobatto_polynomial_plot_test.m">lobatto_polynomial_plot_test.m</a>
</li>
<li>
<a href = "lobatto_polynomial_value.png">lobatto_polynomial_value.png</a>
</li>
<li>
<a href = "lobatto_polynomial_derivative.png">lobatto_polynomial_derivative.png</a>
</li>
<li>
<a href = "lobatto_polynomial_value_test.m">lobatto_polynomial_value_test.m</a>
</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 revised on 02 May 2013.
</i>
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