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<html>
<head>
<title>
JACOBI_POLYNOMIAL - Jacobi Polynomials
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
JACOBI_POLYNOMIAL <br> Jacobi Polynomials
</h1>
<hr>
<p>
<b>JACOBI_POLYNOMIAL</b>
is a C++ library which
evaluates the Jacobi polynomial.
</p>
<p>
For a given choice of the parameters a and b, both greater than -1,
the Jacobi polynomials are a set of polynomials which are
pairwise orthogonal with respect to the integral:
<pre>
integral (-1<=x<=+1) J(i,a,b,x) J(j,a,b,x) (1-x)^a (1+x)^b dx
</pre>
That is, this integral is 0 unless i = j. J(i,a,b,x) indicates the
Jacobi polynomial of degree i.
</p>
<p>
The standard Jacobi polynomials can be defined by a three term
recurrence formula that is a bit too ugly to quote here.
</p>
<p>
It is worth noting that the definition of the Jacobi polynomials
is general enough that it includes some familiar families as
special cases:
<ul>
<li>
if a = b = 0, we have the Legendre polynomials, P(n,x);
</li>
<li>
if a = b = -1/2, we have the Chebyshev polynomials of the first kind, T(n,x);
</li>
<li>
if a = b = 1/2, we have the Chebyshev polynomials of the second kind, U(n,x);
</li>
<li>
if a = b, we have the Gegenbauer polynomials;
</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>JACOBI_POLYNOMIAL</b> is available in
<a href = "../../c_src/jacobi_polynomial/jacobi_polynomial.html">a C version</a> and
<a href = "../../cpp_src/jacobi_polynomial/jacobi_polynomial.html">a C++ version</a> and
<a href = "../../f77_src/jacobi_polynomial/jacobi_polynomial.html">a FORTRAN77 version</a> and
<a href = "../../f_src/jacobi_polynomial/jacobi_polynomial.html">a FORTRAN90 version</a> and
<a href = "../../m_src/jacobi_polynomial/jacobi_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/jacobi_rule/jacobi_rule.html">
JACOBI_RULE</a>,
a C++ program which
can compute and print a Gauss-Jacobi quadrature rule.
</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/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 = "jacobi_polynomial.cpp">jacobi_polynomial.cpp</a>, the source code.
</li>
<li>
<a href = "jacobi_polynomial.hpp">jacobi_polynomial.hpp</a>, the source code.
</li>
<li>
<a href = "jacobi_polynomial.sh">jacobi_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 = "jacobi_polynomial_prb.cpp">jacobi_polynomial_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "jacobi_polynomial_prb.sh">jacobi_polynomial_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "jacobi_polynomial_prb_output.txt">jacobi_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>J_DOUBLE_PRODUCT_INTEGRAL:</b> integral of J(i,x)*J(j,x)*(1-x)^a*(1+x)^b.
</li>
<li>
<b>J_INTEGRAL</b> evaluates a monomial integral associated with J(n,a,b,x).
</li>
<li>
<b>J_POLYNOMIAL</b> evaluates the Jacobi polynomials J(n,a,b,x).
</li>
<li>
<b>J_POLYNOMIAL_VALUES</b> returns some values of the Jacobi polynomial.
</li>
<li>
<b>J_POLYNOMIAL_ZEROS:</b> zeros of Jacobi polynomial J(n,a,b,x).
</li>
<li>
<b>J_QUADRATURE_RULE:</b> Gauss-Jacobi quadrature based on J(n,a,b,x).
</li>
<li>
<b>R8_FACTORIAL</b> computes the factorial of N.
</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_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 modified on 19 April 2012.
</i>
<!-- John Burkardt -->
</body>
</html>