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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 FORTRAN90 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 = "../../cpp_src/jacobi_polynomial/jacobi_polynomial.html">a C++ 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 = "../../f_src/chebyshev_polynomial/chebyshev_polynomial.html">
CHEBYSHEV_POLYNOMIAL</a>,
a FORTRAN90 library which
evaluates the Chebyshev polynomial and associated functions.
</p>
<p>
<a href = "../../f_src/hermite_polynomial/hermite_polynomial.html">
HERMITE_POLYNOMIAL</a>,
a FORTRAN90 library which
evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial,
the Hermite function, and related functions.
</p>
<p>
<a href = "../../f_src/jacobi_rule/jacobi_rule.html">
JACOBI_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Jacobi quadrature rule.
</p>
<p>
<a href = "../../f_src/laguerre_polynomial/laguerre_polynomial.html">
LAGUERRE_POLYNOMIAL</a>,
a FORTRAN90 library which
evaluates the Laguerre polynomial, the generalized Laguerre polynomials,
and the Laguerre function.
</p>
<p>
<a href = "../../f_src/legendre_polynomial/legendre_polynomial.html">
LEGENDRE_POLYNOMIAL</a>,
a FORTRAN90 library which
evaluates the Legendre polynomial and associated functions.
</p>
<p>
<a href = "../../f_src/polpak/polpak.html">
POLPAK</a>,
a FORTRAN90 library which
evaluates a variety of mathematical functions.
</p>
<p>
<a href = "../../f_src/test_values/test_values.html">
TEST_VALUES</a>,
a FORTRAN90 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.f90">jacobi_polynomial.f90</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.f90">jacobi_polynomial_prb.f90</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 = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last modified on 19 April 2012.
</i>
<!-- John Burkardt -->
</body>
</html>