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<html>
<head>
<title>
SANDIA_RULES - Quadrature Rules of Gaussian Type
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SANDIA_RULES <br> Quadrature Rules of Gaussian Type
</h1>
<hr>
<p>
<b>SANDIA_RULES</b>
is a C++ library which
generates a variety of quadrature rules of various orders.
</p>
<p>
This library is used, in turn, by several other libraries, including
<b>SPARSE_GRID_MIXED</b>, <b>SPARSE_GRID_MIXED_GROWTH</b>, and <b>SGMGA</b>.
This means that a program that calls any one of those libraries must have
access to a compiled copy of <b>SANDIA_RULES</b> as well.
</p>
<p>
<table border=1>
<tr>
<th>Name</th>
<th>Usual domain</th>
<th>Weight function</th>
</tr>
<tr>
<td>Gauss-Legendre</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>Clenshaw-Curtis</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>Fejer Type 2</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>Gauss-Chebyshev 1</td>
<td>[-1,+1]</td>
<td>1/sqrt(1-x<sup>2</sup>)</td>
</tr>
<tr>
<td>Gauss-Chebyshev 2</td>
<td>[-1,+1]</td>
<td>sqrt(1-x<sup>2</sup>)</td>
</tr>
<tr>
<td>Gauss-Gegenbauer</td>
<td>[-1,+1]</td>
<td>(1-x<sup>2</sup>)<sup>alpha</sup></td>
</tr>
<tr>
<td>Gauss-Jacobi</td>
<td>[-1,+1]</td>
<td>(1-x)<sup>alpha</sup> (1+x)<sup>beta</sup></td>
</tr>
<tr>
<td>Gauss-Laguerre</td>
<td>[0,+oo)</td>
<td>e<sup>-x</sup></td>
</tr>
<tr>
<td>Generalized Gauss-Laguerre</td>
<td>[0,+oo)</td>
<td>x<sup>alpha</sup> e<sup>-x</sup></td>
</tr>
<tr>
<td>Gauss-Hermite</td>
<td>(-oo,+oo)</td>
<td>e<sup>-x*x</sup></td>
</tr>
<tr>
<td>Generalized Gauss-Hermite</td>
<td>(-oo,+oo)</td>
<td>|x|<sup>alpha</sup> e<sup>-x*x</sup></td>
</tr>
<tr>
<td>Hermite Genz-Keister</td>
<td>(-oo,+oo)</td>
<td>e<sup>-x*x</sup></td>
</tr>
<tr>
<td>Newton-Cotes-Closed</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>Newton-Cotes-Open</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>Newton-Cotes-Open-Half</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
</table>
</p>
<p>
For example, a Gauss-Gegenbauer quadrature rule is used to approximate:
<pre>
Integral ( -1 <= x <= +1 ) f(x) (1-x^2)^alpha dx
</pre>
where <b>alpha</b> is a real parameter chosen by the user.
</p>
<p>
The approximation to the integral is formed by computing a weighted sum
of function values at specific points:
<pre>
Sum ( 1 <= i <= n ) w(i) * f(x(i))
</pre>
The quantities <b>x</b> are the <i>abscissas</i> of the quadrature rule,
the values <b>w</b> are the <i>weights</i> of the quadrature rule, and the
number of terms <b>n</b> in the sum is the <i>order</i> of the quadrature rule.
</p>
<p>
As a matter of convenience, most of the quadrature rules are available
through three related functions:
<ul>
<li>
<b>name_COMPUTE</b> returns points X and weights W;
</li>
<li>
<b>name_COMPUTE_POINTS</b> returns points X;
</li>
<li>
<b>name_COMPUTE_WEIGHTS</b> returns weights W;
</li>
</ul>
In some cases, it is possible to compute points or weights separately;
in other cases, the point and weight functions actually call the
underlying function for the entire rule, and then discard the unrequested
information.
</p>
<p>
Some of these quadrature rules expect a parameter ALPHA, and perhaps also
a parameter BETA, in order to fully define the rule. Therefore, the
argument lists of these functions vary. They always include the input
quantity ORDER, but may have one or two additional inputs. In order to offer
a uniform interface, there is also a family of functions with a standard
set of input arguments, ORDER, NP, and P. Here NP is parameter counter,
and P is the parameter value vector P. Using this interface, it is possible
to call all the quadrature functions with the same argument list.
The uniform interface functions can be identified by the
suffix <b>_NP</b> that appears in their names. Generally, these functions
"unpack" the parameter vector where needed, and then call the corresponding
basic function. Of course, for many rules NP is zero and P may be a null
pointer.
<ul>
<li>
<b>name_COMPUTE_NP ( ORDER, NP, P, X, W )</b>
unpacks parameters, calls name_COMPUTE, returns points X and weights W;
</li>
<li>
<b>name_COMPUTE_POINTS_NP ( ORDER, NP, P, X )</b>
unpacks parameters, calls name_COMPUTE_POINTS, returns points X;
</li>
<li>
<b>name_COMPUTE_WEIGHTS_NP ( ORDER, NP, P, W )</b>
unpacks parameters, calls name_COMPUTE_WEIGHTS, returns weights W;
</li>
</ul>
</p>
<p>
There is yet a third possible interface, in which no ALPHA or BETA parameters
appear in the function call; this interface is primarily intended for a particular
software environment. The interfaces are made available in a separate library
called <b>SANDIA_RULES2</b>.
</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>SANDIA_RULES</b> is available in
<a href = "../../c_src/sandia_rules/sandia_rules.html">a C version</a> and
<a href = "../../cpp_src/sandia_rules/sandia_rules.html">a C++ version</a> and
<a href = "../../f_src/sandia_rules/sandia_rules.html">a FORTRAN90 version</a> and
<a href = "../../m_src/sandia_rules/sandia_rules.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/chebyshev1_rule/chebyshev1_rule.html">
CHEBYSHEV1_RULE</a>,
a C++ program which
can compute and print a Gauss-Chebyshev type 1 quadrature rule.
</p>
<p>
<a href = "../../cpp_src/chebyshev2_rule/chebyshev2_rule.html">
CHEBYSHEV2_RULE</a>,
a C++ program which
can compute and print a Gauss-Chebyshev type 2 quadrature rule.
</p>
<p>
<a href = "../../cpp_src/gegenbauer_rule/gegenbauer_rule.html">
GEGENBAUER_RULE</a>,
a C++ program which
can compute and print a Gauss-Gegenbauer quadrature rule.
</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/gen_laguerre_rule/gen_laguerre_rule.html">
GEN_LAGUERRE_RULE</a>,
a C++ program which
can compute and print a generalized Gauss-Laguerre quadrature rule.
</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/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_rule/laguerre_rule.html">
LAGUERRE_RULE</a>,
a C++ program which
can compute and print a Gauss-Laguerre quadrature rule.
</p>
<p>
<a href = "../../cpp_src/legendre_rule/legendre_rule.html">
LEGENDRE_RULE</a>,
a C++ program which
can compute and print a Gauss-Legendre quadrature rule.
</p>
<p>
<a href = "../../datasets/quadrature_rules/quadrature_rules.html">
QUADRATURE_RULES</a>,
a dataset directory which
contains sets of files that define quadrature
rules over various 1D intervals or multidimensional hypercubes.
</p>
<p>
<a href = "../../datasets/quadrature_rules_legendre/quadrature_rules_legendre.html">
QUADRATURE_RULES_LEGENDRE</a>,
a dataset directory which
contains triples of files defining standard Gauss-Legendre quadrature rules.
</p>
<p>
<a href = "../../cpp_src/sandia_rules2/sandia_rules2.html">
SANDIA_RULES2</a>,
a C++ library which
contains a very small selection of functions which serve as an interface
between SANDIA_SGMG or SANDIA_SGMGA and SANDIA_RULES.
</p>
<p>
<a href = "../../cpp_src/sgmga/sgmga.html">
SGMGA</a>,
a C++ library which
creates sparse grids based on a mixture of 1D quadrature rules,
allowing anisotropic weights for each dimension.
</p>
<p>
<a href = "../../cpp_src/sparse_grid_mixed/sparse_grid_mixed.html">
SPARSE_GRID_MIXED</a>,
a C++ library which
creates a sparse grid dataset based on a mixed set of 1D factor rules.
</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>
William Cody,<br>
An Overview of Software Development for Special Functions,<br>
in Numerical Analysis Dundee, 1975,<br>
edited by GA Watson,<br>
Lecture Notes in Mathematics 506,<br>
Springer, 1976.
</li>
<li>
Philip Davis, Philip Rabinowitz,<br>
Methods of Numerical Integration,<br>
Second Edition,<br>
Dover, 2007,<br>
ISBN: 0486453391,<br>
LC: QA299.3.D28.
</li>
<li>
Sylvan Elhay, Jaroslav Kautsky,<br>
Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of
Interpolatory Quadrature,<br>
ACM Transactions on Mathematical Software,<br>
Volume 13, Number 4, December 1987, pages 399-415.
</li>
<li>
Alan Genz, Bradley Keister,<br>
Fully symmetric interpolatory rules for multiple integrals
over infinite regions with Gaussian weight,<br>
Journal of Computational and Applied Mathematics,<br>
Volume 71, 1996, pages 299-309.
</li>
<li>
John Hart, Ward Cheney, Charles Lawson, Hans Maehly,
Charles Mesztenyi, John Rice, Henry Thatcher,
Christoph Witzgall,<br>
Computer Approximations,<br>
Wiley, 1968,<br>
LC: QA297.C64.
</li>
<li>
Knut Petras,<br>
Smolyak Cubature of Given Polynomial Degree with Few Nodes
for Increasing Dimension,<br>
Numerische Mathematik,<br>
Volume 93, Number 4, February 2003, pages 729-753.
</li>
<li>
Arthur Stroud, Don Secrest,<br>
Gaussian Quadrature Formulas,<br>
Prentice Hall, 1966,<br>
LC: QA299.4G3S7.
</li>
<li>
Shanjie Zhang, Jianming Jin,<br>
Computation of Special Functions,<br>
Wiley, 1996,<br>
ISBN: 0-471-11963-6,<br>
LC: QA351.C45
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "sandia_rules.cpp">sandia_rules.cpp</a>, the source code.
</li>
<li>
<a href = "sandia_rules.hpp">sandia_rules.hpp</a>, the include file.
</li>
<li>
<a href = "sandia_rules.sh">sandia_rules.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "sandia_rules_prb.cpp">sandia_rules_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "sandia_rules_prb.sh">sandia_rules_prb.sh</a>,
commands to compile, link and run the sample calling program.
</li>
<li>
<a href = "sandia_rules_prb_output.txt">sandia_rules_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>BINARY_VECTOR_NEXT</b> generates the next binary vector.
</li>
<li>
<b>CCN_COMPUTE</b> computes a nested Clenshaw Curtis quadrature rule.
</li>
<li>
<b>CCN_COMPUTE_NP</b> computes a nested Clenshaw Curtis quadrature rule.
</li>
<li>
<b>CCN_COMPUTE_POINTS:</b> compute nested Clenshaw Curtis points.
</li>
<li>
<b>CCN_COMPUTE_POINTS_NP:</b> nested Clenshaw Curtis quadrature points.
</li>
<li>
<b>CCN_COMPUTE_WEIGHTS:</b> weights for nested Clenshaw Curtis rule.
</li>
<li>
<b>CCN_COMPUTE_WEIGHTS_NP:</b> nested Clenshaw Curtis quadrature weights.
</li>
<li>
<b>CHEBYSHEV1_COMPUTE</b> computes a Chebyshev type 1 quadrature rule.
</li>
<li>
<b>CHEBYSHEV1_COMPUTE_NP</b> computes a Chebyshev type 1 quadrature rule.
</li>
<li>
<b>CHEBYSHEV1_COMPUTE_POINTS</b> computes Chebyshev type 1 quadrature points.
</li>
<li>
<b>CHEBYSHEV1_COMPUTE_POINTS_NP</b> computes Chebyshev type 1 quadrature points.
</li>
<li>
<b>CHEBYSHEV1_COMPUTE_WEIGHTS</b> computes Chebyshev type 1 quadrature weights.
</li>
<li>
<b>CHEBYSHEV1_COMPUTE_WEIGHTS_NP:</b> Chebyshev type 1 quadrature weights.
</li>
<li>
<b>CHEBYSHEV1_INTEGRAL</b> evaluates a monomial Chebyshev type 1 integral.
</li>
<li>
<b>CHEBYSHEV2_COMPUTE</b> computes a Chebyshev type 2 quadrature rule.
</li>
<li>
<b>CHEBYSHEV2_COMPUTE_NP</b> computes a Chebyshev type 2 quadrature rule.
</li>
<li>
<b>CHEBYSHEV2_COMPUTE_POINTS</b> computes Chebyshev type 2 quadrature points.
</li>
<li>
<b>CHEBYSHEV2_COMPUTE_POINTS_NP</b> computes Chebyshev type 2 quadrature points.
</li>
<li>
<b>CHEBYSHEV2_COMPUTE_WEIGHTS</b> computes Chebyshev type 2 quadrature weights.
</li>
<li>
<b>CHEBYSHEV2_COMPUTE_WEIGHTS_NP:</b> Chebyshev type 2 quadrature weights.
</li>
<li>
<b>CHEBYSHEV2_INTEGRAL</b> evaluates a monomial Chebyshev type 2 integral.
</li>
<li>
<b>CLENSHAW_CURTIS_COMPUTE</b> computes a Clenshaw Curtis quadrature rule.
</li>
<li>
<b>CLENSHAW_CURTIS_COMPUTE_NP</b> computes a Clenshaw Curtis quadrature rule.
</li>
<li>
<b>CLENSHAW_CURTIS_COMPUTE_POINTS</b> computes Clenshaw Curtis quadrature points.
</li>
<li>
<b>CLENSHAW_CURTIS_COMPUTE_POINTS_NP:</b> Clenshaw Curtis quadrature points.
</li>
<li>
<b>CLENSHAW_CURTIS_COMPUTE_WEIGHTS</b> computes Clenshaw Curtis quadrature weights.
</li>
<li>
<b>CLENSHAW_CURTIS_COMPUTE_WEIGHTS_NP:</b> Clenshaw Curtis quadrature weights.
</li>
<li>
<b>COMP_NEXT</b> computes the compositions of the integer N into K parts.
</li>
<li>
<b>CPU_TIME</b> reports the elapsed CPU time.
</li>
<li>
<b>DIF_DERIV</b> computes the derivative of a polynomial in divided difference form.
</li>
<li>
<b>DIF_SHIFT_X</b> replaces one abscissa of a divided difference table with a new one.
</li>
<li>
<b>DIF_SHIFT_ZERO</b> shifts a divided difference table so that all abscissas are zero.
</li>
<li>
<b>DIF_TO_R8POLY</b> converts a divided difference table to a standard polynomial.
</li>
<li>
<b>FEJER2_COMPUTE</b> computes a Fejer type 2 rule.
</li>
<li>
<b>FEJER2_COMPUTE_NP</b> computes a Fejer type 2 rule.
</li>
<li>
<b>FEJER2_COMPUTE_POINTS</b> computes Fejer type 2 quadrature points.
</li>
<li>
<b>FEJER2_COMPUTE_POINTS_NP</b> computes Fejer type 2 quadrature points.
</li>
<li>
<b>FEJER2_COMPUTE_WEIGHTS</b> computes Fejer type 2 quadrature weights.
</li>
<li>
<b>FEJER2_COMPUTE_WEIGHTS_NP</b> computes Fejer type 2 quadrature weights.
</li>
<li>
<b>GEGENBAUER_COMPUTE</b> computes a Gegenbauer quadrature rule.
</li>
<li>
<b>GEGENBAUER_COMPUTE_NP</b> computes a Gegenbauer quadrature rule.
</li>
<li>
<b>GEGENBAUER_COMPUTE_POINTS</b> computes Gegenbauer quadrature points.
</li>
<li>
<b>GEGENBAUER_COMPUTE_POINTS_NP</b> computes Gegenbauer quadrature points.
</li>
<li>
<b>GEGENBAUER_COMPUTE_WEIGHTS</b> computes Gegenbauer quadrature weights.
</li>
<li>
<b>GEGENBAUER_COMPUTE_WEIGHTS_NP</b> computes Gegenbauer quadrature weights.
</li>
<li>
<b>GEGENBAUER_INTEGRAL</b> integrates a monomial with Gegenbauer weight.
</li>
<li>
<b>GEGENBAUER_RECUR</b> evaluates a Gegenbauer polynomial.
</li>
<li>
<b>GEGENBAUER_ROOT</b> improves an approximate root of a Gegenbauer polynomial.
</li>
<li>
<b>GEN_HERMITE_COMPUTE</b> computes a generalized Gauss-Hermite quadrature rule.
</li>
<li>
<b>GEN_HERMITE_COMPUTE_NP</b> computes a Generalized Hermite quadrature rule.
</li>
<li>
<b>GEN_HERMITE_COMPUTE_POINTS</b> computes Generalized Hermite quadrature points.
</li>
<li>
<b>GEN_HERMITE_COMPUTE_POINTS_NP:</b> Generalized Hermite quadrature points.
</li>
<li>
<b>GEN_HERMITE_COMPUTE_WEIGHTS</b> computes Generalized Hermite quadrature weights.
</li>
<li>
<b>GEN_HERMITE_COMPUTE_WEIGHTS_NP:</b> Generalized Hermite quadrature weights.
</li>
<li>
<b>GEN_HERMITE_DR_COMPUTE</b> computes a Generalized Hermite quadrature rule.
</li>
<li>
<b>GEN_HERMITE_INTEGRAL</b> evaluates a monomial Generalized Hermite integral.
</li>
<li>
<b>GEN_LAGUERRE_COMPUTE:</b> generalized Gauss-Laguerre quadrature rule.
</li>
<li>
<b>GEN_LAGUERRE_COMPUTE_NP</b> computes a Generalized Laguerre quadrature rule.
</li>
<li>
<b>GEN_LAGUERRE_COMPUTE_POINTS:</b> Generalized Laguerre quadrature points.
</li>
<li>
<b>GEN_LAGUERRE_COMPUTE_POINTS_NP:</b> Generalized Laguerre quadrature points.
</li>
<li>
<b>GEN_LAGUERRE_COMPUTE_WEIGHTS:</b> Generalized Laguerre quadrature weights.
</li>
<li>
<b>GEN_LAGUERRE_COMPUTE_WEIGHTS_NP:</b> Generalized Laguerre quadrature weights.
</li>
<li>
<b>GEN_LAGUERRE_INTEGRAL</b> evaluates a monomial Generalized Laguerre integral.
</li>
<li>
<b>GEN_LAGUERRE_SS_COMPUTE</b> computes a Generalized Laguerre quadrature rule.
</li>
<li>
<b>GEN_LAGUERRE_SS_RECUR</b> evaluates a Generalized Laguerre polynomial.
</li>
<li>
<b>GEN_LAGUERRE_SS_ROOT</b> improves a root of a Generalized Laguerre polynomial.
</li>
<li>
<b>HC_COMPUTE_WEIGHTS_FROM_POINTS:</b> Hermite-Cubic weights, user-supplied points.
</li>
<li>
<b>HCC_COMPUTE</b> computes a Hermite-Cubic-Chebyshev-Spacing quadrature rule.
</li>
<li>
<b>HCC_COMPUTE_NP</b> computes a Hermite-Cubic-Chebyshev-Spacing quadrature rule.
</li>
<li>
<b>HCC_COMPUTE_POINTS</b> computes Hermite-Cubic-Chebyshev-Spacing quadrature points.
</li>
<li>
<b>HCC_COMPUTE_POINTS_NP:</b> Hermite-Cubic-Chebyshev-Spacing quadrature points.
</li>
<li>
<b>HCC_COMPUTE_WEIGHTS:</b> Hermite-Cubic-Chebyshev-Spacing quadrature weights.
</li>
<li>
<b>HCC_COMPUTE_WEIGHTS_NP:</b> Hermite-Cubic-Chebyshev-Spacing quadrature weights.
</li>
<li>
<b>HCE_COMPUTE</b> computes a Hermite-Cubic-Equal-Spacing quadrature rule.
</li>
<li>
<b>HCE_COMPUTE_NP</b> computes a Hermite-Cubic-Equal-Spacing quadrature rule.
</li>
<li>
<b>HCE_COMPUTE_POINTS</b> computes Hermite-Cubic-Equal-Spacing quadrature points.
</li>
<li>
<b>HCE_COMPUTE_POINTS_NP:</b> Hermite-Cubic-Equal-Spacing quadrature points.
</li>
<li>
<b>HCE_COMPUTE_WEIGHTS:</b> Hermite-Cubic-Equal-Spacing quadrature weights.
</li>
<li>
<b>HCE_COMPUTE_WEIGHTS_NP:</b> Hermite-Cubic-Equal-Spacing quadrature weights.
</li>
<li>
<b>HERMITE_COMPUTE</b> computes a Gauss-Hermite quadrature rule.
</li>
<li>
<b>HERMITE_COMPUTE_NP</b> computes a Hermite quadrature rule.
</li>
<li>
<b>HERMITE_COMPUTE_POINTS</b> computes Hermite quadrature points.
</li>
<li>
<b>HERMITE_COMPUTE_POINTS_NP</b> computes Hermite quadrature points.
</li>
<li>
<b>HERMITE_COMPUTE_WEIGHTS</b> computes Hermite quadrature weights.
</li>
<li>
<b>HERMITE_COMPUTE_WEIGHTS_NP</b> computes Hermite quadrature weights.
</li>
<li>
<b>HERMITE_GENZ_KEISTER_LOOKUP</b> looks up a Genz-Keister Hermite rule.
</li>
<li>
<b>HERMITE_GENZ_KEISTER_LOOKUP_POINTS</b> looks up Genz-Keister Hermite abscissas.
</li>
<li>
<b>HERMITE_GENZ_KEISTER_LOOKUP_POINTS_NP</b> looks up Genz-Keister Hermite abscissas.
</li>
<li>
<b>HERMITE_GENZ_KEISTER_LOOKUP_WEIGHTS</b> looks up Genz-Keister Hermite weights.
</li>
<li>
<b>HERMITE_GENZ_KEISTER_LOOKUP_WEIGHTS_NP</b> looks up Genz-Keister Hermite weights.
</li>
<li>
<b>HERMITE_GK18_LOOKUP_POINTS:</b> abscissas of a Hermite Genz-Keister 18 rule.
</li>
<li>
<b>HERMITE_GK22_LOOKUP_POINTS</b> looks up Hermite Genz-Keister 22 points.
</li>
<li>
<b>HERMITE_GK24_LOOKUP_POINTS</b> looks up Hermite Genz-Keister 24 points.
</li>
<li>
<b>HERMITE_INTEGRAL</b> evaluates a monomial Hermite integral.
</li>
<li>
<b>HERMITE_INTERPOLANT</b> sets up a divided difference table from Hermite data.
</li>
<li>
<b>HERMITE_INTERPOLANT_RULE:</b> quadrature rule for a Hermite interpolant.
</li>
<li>
<b>HERMITE_INTERPOLANT_VALUE</b> evaluates the Hermite interpolant polynomial.
</li>
<li>
<b>HERMITE_LOOKUP</b> looks up abscissas and weights for Gauss-Hermite quadrature.
</li>
<li>
<b>HERMITE_LOOKUP_POINTS</b> looks up abscissas for Hermite quadrature.
</li>
<li>
<b>HERMITE_LOOKUP_WEIGHTS</b> looks up weights for Hermite quadrature.
</li>
<li>
<b>HERMITE_SS_COMPUTE</b> computes a Hermite quadrature rule.
</li>
<li>
<b>HERMITE_SS_RECUR</b> finds the value and derivative of a Hermite polynomial.
</li>
<li>
<b>HERMITE_SS_ROOT</b> improves an approximate root of a Hermite polynomial.
</li>
<li>
<b>I4_CHOOSE</b> computes the binomial coefficient C(N,K).
</li>
<li>
<b>I4_LOG_2</b> returns the integer part of the logarithm base 2 of an I4.
</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>I4_POWER</b> returns the value of I^J.
</li>
<li>
<b>I4MAT_COPY</b> copies one I4MAT to another.
</li>
<li>
<b>I4MAT_COPY_NEW</b> copies an I4MAT to a "new" I4MAT.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT</b> prints an I4MAT, transposed.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT_SOME</b> prints some of an I4MAT, transposed.
</li>
<li>
<b>I4MAT_WRITE</b> writes an I4MAT file.
</li>
<li>
<b>I4VEC_ADD_NEW</b> computes C = A + B for I4VEC's.
</li>
<li>
<b>I4VEC_ANY_LT:</b> ( any ( A < B ) ) for I4VEC's.
</li>
<li>
<b>I4VEC_COPY</b> copies an I4VEC.
</li>
<li>
<b>I4VEC_COPY_NEW</b> copies an I4VEC to a "new" I4VEC.
</li>
<li>
<b>I4VEC_MIN_MV</b> determines U(1:N) /\ V for vectors U and a single vector V.
</li>
<li>
<b>I4VEC_PRINT</b> prints an I4VEC.
</li>
<li>
<b>I4VEC_PRODUCT</b> multiplies the entries of an I4VEC.
</li>
<li>
<b>I4VEC_SUM</b> sums the entries of an I4VEC.
</li>
<li>
<b>I4VEC_ZERO</b> zeroes an I4VEC.
</li>
<li>
<b>I4VEC_ZERO_NEW</b> creates and zeroes an I4VEC.
</li>
<li>
<b>IMTQLX</b> diagonalizes a symmetric tridiagonal matrix.
</li>
<li>
<b>JACOBI_COMPUTE:</b> Elhay-Kautsky method for Gauss-Jacobi quadrature rule.
</li>
<li>
<b>JACOBI_COMPUTE_NP</b> computes a Jacobi quadrature rule.
</li>
<li>
<b>JACOBI_COMPUTE_POINTS</b> computes Jacobi quadrature points.
</li>
<li>
<b>JACOBI_COMPUTE_POINTS_NP</b> computes Jacobi quadrature points.
</li>
<li>
<b>JACOBI_COMPUTE_WEIGHTS</b> computes Jacobi quadrature weights.
</li>
<li>
<b>JACOBI_COMPUTE_WEIGHTS_NP</b> computes Jacobi quadrature weights.
</li>
<li>
<b>JACOBI_INTEGRAL</b> integrates a monomial with Jacobi weight.
</li>
<li>
<b>JACOBI_SS_COMPUTE</b> computes a Jacobi quadrature rule.
</li>
<li>
<b>JACOBI_SS_RECUR</b> evaluates a Jacobi polynomial.
</li>
<li>
<b>JACOBI_SS_ROOT</b> improves an approximate root of a Jacobi polynomial.
</li>
<li>
<b>LAGUERRE_COMPUTE:</b> Laguerre quadrature rule by the Elhay-Kautsky method.
</li>
<li>
<b>LAGUERRE_COMPUTE_NP</b> computes a Laguerre quadrature rule.
</li>
<li>
<b>LAGUERRE_COMPUTE_POINTS</b> computes Laguerre quadrature points.
</li>
<li>
<b>LAGUERRE_COMPUTE_POINTS_NP</b> computes Laguerre quadrature points.
</li>
<li>
<b>LAGUERRE_COMPUTE_WEIGHTS</b> computes Laguerre quadrature weights.
</li>
<li>
<b>LAGUERRE_COMPUTE_WEIGHTS_NP</b> computes Laguerre quadrature weights.
</li>
<li>
<b>LAGUERRE_INTEGRAL</b> evaluates a monomial Laguerre integral.
</li>
<li>
<b>LAGUERRE_LOOKUP</b> looks up abscissas and weights for Laguerre quadrature.
</li>
<li>
<b>LAGUERRE_LOOKUP_POINTS</b> looks up abscissas for Laguerre quadrature.
</li>
<li>
<b>LAGUERRE_LOOKUP_WEIGHTS</b> looks up weights for Laguerre quadrature.
</li>
<li>
<b>LAGUERRE_SS_COMPUTE</b> computes a Laguerre quadrature rule.
</li>
<li>
<b>LAGUERRE_SS_RECUR</b> evaluates a Laguerre polynomial.
</li>
<li>
<b>LAGUERRE_SS_ROOT</b> improves a root of a Laguerre polynomial.
</li>
<li>
<b>LEGENDRE_COMPUTE:</b> Legendre quadrature rule by the Elhay-Kautsky method.
</li>
<li>
<b>LEGENDRE_COMPUTE_NP</b> computes a Legendre quadrature rule.
</li>
<li>
<b>LEGENDRE_COMPUTE_POINTS</b> computes Legendre quadrature points.
</li>
<li>
<b>LEGENDRE_COMPUTE_POINTS_NP</b> computes Legendre quadrature points.
</li>
<li>
<b>LEGENDRE_COMPUTE_WEIGHTS</b> computes Legendre quadrature weights.
</li>
<li>
<b>LEGENDRE_COMPUTE_WEIGHTS_NP</b> computes Legendre quadrature weights.
</li>
<li>
<b>LEGENDRE_DR_COMPUTE</b> computes a Legendre quadrature rule.
</li>
<li>
<b>LEGENDRE_INTEGRAL</b> evaluates a monomial Legendre integral.
</li>
<li>
<b>LEGENDRE_LOOKUP</b> looks up abscissas and weights for Gauss-Legendre quadrature.
</li>
<li>
<b>LEGENDRE_LOOKUP_POINTS</b> looks up abscissas for Gauss-Legendre quadrature.
</li>
<li>
<b>LEGENDRE_LOOKUP_WEIGHTS</b> looks up weights for Gauss-Legendre quadrature.
</li>
<li>
<b>LEGENDRE_ZEROS</b> returns the zeros of the Legendre polynomial of degree N.
</li>
<li>
<b>LEVEL_GROWTH_TO_ORDER:</b> convert Level and Growth to Order.
</li>
<li>
<b>LEVEL_TO_ORDER_DEFAULT:</b> default growth.
</li>
<li>
<b>LEVEL_TO_ORDER_EXPONENTIAL:</b> exponential growth.
</li>
<li>
<b>LEVEL_TO_ORDER_EXPONENTIAL_SLOW:</b> slow exponential growth;
</li>
<li>
<b>LEVEL_TO_ORDER_LINEAR:</b> linear growth.
</li>
<li>
<b>LEVEL_TO_ORDER_EXP_CC</b> is used for Clenshaw-Curtis type rules.
</li>
<li>
<b>LEVEL_TO_ORDER_EXP_F2</b> is used for Fejer 2 type rules.
</li>
<li>
<b>LEVEL_TO_ORDER_EXP_GAUSS</b> is used for Gauss type rules.
</li>
<li>
<b>LEVEL_TO_ORDER_EXP_GP</b> is used for Gauss-Patterson type rules.
</li>
<li>
<b>LEVEL_TO_ORDER_EXP_HGK</b> is used for Hermite Genz-Keister type rules.
</li>
<li>
<b>LEVEL_TO_ORDER_LINEAR_NN</b> is used for non-nested Gauss type rules.
</li>
<li>
<b>LEVEL_TO_ORDER_LINEAR_WN</b> is used for weakly-nested Gauss type rules.
</li>
<li>
<b>NC_COMPUTE</b> computes a Newton-Cotes quadrature rule.
</li>
<li>
<b>NC_COMPUTE_NEW</b> computes a Newton-Cotes quadrature rule.
</li>
<li>
<b>NCC_COMPUTE_POINTS:</b> points of a Newton-Cotes Closed quadrature rule.
</li>
<li>
<b>NCC_COMPUTE_WEIGHTS:</b> weights of a Newton-Cotes Closed quadrature rule.
</li>
<li>
<b>NCO_COMPUTE_POINTS:</b> points for a Newton-Cotes Open quadrature rule.
</li>
<li>
<b>NCO_COMPUTE_WEIGHTS:</b> weights for a Newton-Cotes Open quadrature rule.
</li>
<li>
<b>NCOH_COMPUTE_POINTS</b> computes points for a Newton-Cotes "open half" quadrature rule.
</li>
<li>
<b>NCOH_COMPUTE_WEIGHTS</b> computes weights for a Newton-Cotes "open half" quadrature rule.
</li>
<li>
<b>PATTERSON_LOOKUP</b> looks up Patterson quadrature points and weights.
</li>
<li>
<b>PATTERSON_LOOKUP_POINTS</b> looks up Patterson quadrature points.
</li>
<li>
<b>PATTERSON_LOOKUP_POINTS_NP</b> looks up Patterson quadrature points.
</li>
<li>
<b>PATTERSON_LOOKUP_WEIGHTS</b> looks up Patterson quadrature weights.
</li>
<li>
<b>PATTERSON_LOOKUP_WEIGHTS_NP</b> looks up Patterson quadrature weights.
</li>
<li>
<b>POINT_RADIAL_TOL_UNIQUE_COUNT</b> counts the tolerably unique points.
</li>
<li>
<b>POINT_RADIAL_TOL_UNIQUE_COUNT_INC1</b> counts the tolerably unique points.
</li>
<li>
<b>POINT_RADIAL_TOL_UNIQUE_COUNT_INC2</b> counts the tolerably unique points.
</li>
<li>
<b>POINT_RADIAL_TOL_UNIQUE_INDEX</b> indexes the tolerably unique points.