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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 FORTRAN90 library which
generates a variety of quadrature rules of various orders and types.
</p>
<p>
This library is used, in turn, by several other libraries, including
<b>SPARSE_GRID_MIXED</b> and <b>SGMGA</b>. This means that a program
that calls any 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>
<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 = "../../f_src/quadrule/quadrule.html">
QUADRULE</a>,
a FORTRAN90 library which
defines quadrature rules for 1-dimensional domains.
</p>
<p>
<a href = "../../f_src/r8lib/r8lib.html">
R8LIB</a>,
a FORTRAN90 library which
contains many utility routines, using "R8" or
"double precision real" arithmetic.
</p>
<p>
<a href = "../../f_src/sgmga/sgmga.html">
SGMGA</a>,
a FORTRAN90 library which
creates sparse grids based on a mixture of 1D quadrature rules,
allowing anisotropic weights for each dimension.
</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.f90">sandia_rules.f90</a>, the source code.
</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.f90">sandia_rules_prb.f90</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 rule.
</li>
<li>
<b>CCN_COMPUTE_POINTS:</b> compute nested Clenshaw Curtis points.
</li>
<li>
<b>CCN_COMPUTE_POINTS_NP:</b> abscissas of a nested Clenshaw Curtis rule.
</li>
<li>
<b>CCN_COMPUTE_WEIGHTS:</b> weights for nested Clenshaw Curtis rule.
</li>
<li>
<b>CCN_COMPUTE_WEIGHTS_NP</b> computes nested Clenshaw Curtis weights.
</li>
<li>
<b>CHEBYSHEV1_COMPUTE</b> computes a Chebyshev type 1 quadrature rule.
</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_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_POINTS:</b> abscissas of a Clenshaw Curtis rule.
</li>
<li>
<b>CLENSHAW_CURTIS_COMPUTE_POINTS_NP:</b> abscissas of a Clenshaw Curtis rule.
</li>
<li>
<b>CLENSHAW_CURTIS_COMPUTE_WEIGHTS</b> computes Clenshaw Curtis weights.
</li>
<li>
<b>CLENSHAW_CURTIS_COMPUTE_WEIGHTS_NP</b> computes Clenshaw Curtis weights.
</li>
<li>
<b>COMP_NEXT</b> computes the compositions of the integer N into K parts.
</li>
<li>
<b>DIF_TO_R8POLY</b> converts a divided difference table to a standard polynomial.
</li>
<li>
<b>DIF_VALS</b> evaluates a divided difference polynomial at a set of points.
</li>
<li>
<b>FEJER2_COMPUTE</b> computes a Fejer Type 2 quadrature rule.
</li>
<li>
<b>FEJER2_COMPUTE_POINTS</b> returns the abscissas of a Fejer type 2 rule.
</li>
<li>
<b>FEJER2_COMPUTE_POINTS_NP</b> returns the abscissas of a Fejer type 2 rule.
</li>
<li>
<b>FEJER2_COMPUTE_WEIGHTS</b> computes weights for a Fejer type 2 quadrature rule.
</li>
<li>
<b>FEJER2_COMPUTE_WEIGHTS_NP:</b> weights for a Fejer type 2 quadrature rule.
</li>
<li>
<b>GEGENBAUER_COMPUTE</b> computes a Gegenbauer quadrature rule.
</li>
<li>
<b>GEGENBAUER_INTEGRAL</b> integrates a monomial with Gegenbauer weight.
</li>
<li>
<b>GEGENBAUER_RECUR</b> finds the value and derivative of 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_POINTS:</b> abscissas of a Generalized Hermite rule.
</li>
<li>
<b>GEN_HERMITE_COMPUTE_POINTS_NP:</b> abscissas of a Generalized Hermite rule.
</li>
<li>
<b>GEN_HERMITE_COMPUTE_WEIGHTS:</b> weights of a Generalized Hermite rule.
</li>
<li>
<b>GEN_HERMITE_COMPUTE_WEIGHTS_NP:</b> weights of a Generalized Hermite rule.
</li>
<li>
<b>GEN_HERMITE_DR_COMPUTE</b> computes a Generalized Hermite 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_POINTS:</b> points of a Generalized Laguerre rule.
</li>
<li>
<b>GEN_LAGUERRE_COMPUTE_POINTS_NP:</b> points of a Generalized Laguerre rule.
</li>
<li>
<b>GEN_LAGUERRE_COMPUTE_WEIGHTS:</b> weights of a Generalized Laguerre rule.
</li>
<li>
<b>GEN_LAGUERRE_COMPUTE_WEIGHTS_NP:</b> weights of a Generalized Laguerre rule.
</li>
<li>
<b>GEN_LAGUERRE_INTEGRAL</b> evaluates a monomial genearlized 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> seeks roots of a Generalized Laguerre polynomial.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</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> abscissas of a Hermite-Cubic-Chebyshev-Spacing rule.
</li>
<li>
<b>HCC_COMPUTE_POINTS_NP:</b> abscissas of a Hermite-Cubic-Chebyshev-Spacing rule.
</li>
<li>
<b>HCC_COMPUTE_WEIGHTS</b> computes Hermite-Cubic-Chebyshev-Spacing weights.
</li>
<li>
<b>HCC_COMPUTE_WEIGHTS_NP</b> computes Hermite-Cubic-Chebyshev-Spacing 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> abscissas of a Hermite-Cubic-Equal-Spacing rule.
</li>
<li>
<b>HCE_COMPUTE_POINTS_NP:</b> abscissas of a Hermite-Cubic-Equal-Spacing rule.
</li>
<li>
<b>HCE_COMPUTE_WEIGHTS</b> computes Hermite-Cubic-Equal-Spacing weights.
</li>
<li>
<b>HCE_COMPUTE_WEIGHTS_NP</b> computes Hermite-Cubic-Equal-Spacing weights.
</li>
<li>
<b>HDATA_TO_DIF</b> sets up a divided difference table from Hermite data.
</li>
<li>
<b>HERMITE_COMPUTE</b> computes a Gauss-Hermite quadrature rule.
</li>
<li>
<b>HERMITE_COMPUTE_POINTS</b> computes points of a Hermite quadrature rule.
</li>
<li>
<b>HERMITE_COMPUTE_POINTS_NP</b> computes points of a Hermite quadrature rule.
</li>
<li>
<b>HERMITE_COMPUTE_WEIGHTS</b> computes weights of a Hermite quadrature rule.
</li>
<li>
<b>HERMITE_COMPUTE_WEIGHTS_NP</b> computes weights of a Hermite quadrature rule.
</li>
<li>
<b>HERMITE_GENZ_KEISTER_LOOKUP</b> returns a Genz-Keister rule for Hermite problems.
</li>
<li>
<b>HERMITE_GENZ_KEISTER_LOOKUP_POINTS:</b> abscissas of a Genz-Keister Hermite rule.
</li>
<li>
<b>HERMITE_GENZ_KEISTER_LOOKUP_POINTS_NP:</b> Genz-Keister Hermite abscissas.
</li>
<li>
<b>HERMITE_GENZ_KEISTER_LOOKUP_WEIGHTS:</b> weights for Genz-Keister Hermite rule.
</li>
<li>
<b>HERMITE_GENZ_KEISTER_LOOKUP_WEIGHTS_NP</b> sets weights for a Patterson rule.
</li>
<li>
<b>HERMITE_GK18_LOOKUP_POINTS:</b> abscissas of a Hermite Genz-Keister 18 rule.
</li>
<li>
<b>HERMITE_GK22_LOOKUP_POINTS:</b> abscissas of a Genz-Keister 22 Hermite rule.
</li>
<li>
<b>HERMITE_GK24_LOOKUP_POINTS:</b> abscissas of a Genz-Keister 24 Hermite rule.
</li>
<li>
<b>HERMITE_INTEGRAL</b> evaluates a monomial Hermite integral.
</li>
<li>
<b>HERMITE_INTERPOLANT_RULE:</b> quadrature rule for a Hermite interpolant.
</li>
<li>
<b>HERMITE_LOOKUP_POINTS</b> returns the abscissas of a Hermite rule.
</li>
<li>
<b>HERMITE_LOOKUP_WEIGHTS</b> returns weights for Hermite quadrature rules.
</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) as an I4.
</li>
<li>
<b>I4_LOG_2</b> returns the integer part of the logarithm base 2 of an I4.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT</b> prints an I4MAT, transposed.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT_SOME</b> prints some of the transpose of an I4MAT.
</li>
<li>
<b>I4MAT_WRITE</b> writes an I4MAT file.
</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>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_POINTS</b> returns the points of a Jacobi rule.
</li>
<li>
<b>JACOBI_COMPUTE_POINTS_NP</b> returns the points of a Jacobi rule.
</li>
<li>
<b>JACOBI_COMPUTE_WEIGHTS</b> returns the weights of a Jacobi rule.
</li>
<li>
<b>JACOBI_COMPUTE_WEIGHTS_NP</b> returns the weights of a Jacobi rule.
</li>
<li>
<b>JACOBI_INTEGRAL</b> evaluates the integral of a monomial with Jacobi weight.
</li>
<li>
<b>JACOBI_SS_COMPUTE</b> computes a Jacobi quadrature rule.
</li>
<li>
<b>JACOBI_SS_RECUR</b> finds the value and derivative of 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_POINTS</b> computes points of a Laguerre quadrature rule.
</li>
<li>
<b>LAGUERRE_COMPUTE_POINTS_NP</b> computes points of a Laguerre quadrature rule.
</li>
<li>
<b>LAGUERRE_COMPUTE_WEIGHTS</b> computes weights of a Laguerre quadrature rule.
</li>
<li>
<b>LAGUERRE_COMPUTE_WEIGHTS_NP</b> computes weights of a Laguerre quadrature rule.
</li>
<li>
<b>LAGUERRE_INTEGRAL</b> evaluates a monomial Laguerre integral.
</li>
<li>
<b>LAGUERRE_LOOKUP_POINTS</b> returns the abscissas of a Laguerre rule.
</li>
<li>
<b>LAGUERRE_LOOKUP_WEIGHTS</b> returns weights for Laguerre quadrature rules.
</li>
<li>
<b>LAGUERRE_SS_COMPUTE</b> computes a Laguerre quadrature rule.
</li>
<li>
<b>LAGUERRE_SS_RECUR</b> finds the value and derivative of a Laguerre polynomial.
</li>
<li>
<b>LAGUERRE_SS_ROOT</b> improves an approximate root of a Laguerre polynomial.
</li>
<li>
<b>LEGENDRE_COMPUTE:</b> Legendre quadrature rule by the Elhay-Kautsky method.
</li>
<li>
<b>LEGENDRE_COMPUTE_POINTS</b> computes abscissas of a Legendre quadrature rule.
</li>
<li>
<b>LEGENDRE_COMPUTE_POINTS_NP</b> computes abscissas of a Legendre quadrature rule.
</li>
<li>
<b>LEGENDRE_COMPUTE_WEIGHTS</b> computes weights of a Legendre quadrature rule.
</li>
<li>
<b>LEGENDRE_COMPUTE_WEIGHTS_NP</b> computes weights of a Legendre quadrature rule.
</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_ZEROS</b> computes 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>NC_COMPUTE</b> computes a Newton-Cotes quadrature rule.
</li>
<li>
<b>NCC_COMPUTE_POINTS:</b> Newton-Cotes Closed points
</li>
<li>
<b>NCC_COMPUTE_WEIGHTS:</b> Newton-Cotes Closed weights.
</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> points for a Newton-Cotes Open Half quadrature rule.
</li>
<li>
<b>NCOH_COMPUTE_WEIGHTS:</b> weights for a Newton-Cotes Open Half quadrature rule.
</li>
<li>
<b>PATTERSON_LOOKUP</b> returns the abscissas and weights of a Patterson rule.
</li>
<li>
<b>PATTERSON_LOOKUP_POINTS</b> returns the abscissas of a Patterson rule.
</li>
<li>
<b>PATTERSON_LOOKUP_POINTS_NP</b> returns the abscissas of a Patterson rule.
</li>
<li>
<b>PATTERSON_LOOKUP_WEIGHTS</b> sets weights for a Patterson rule.
</li>
<li>
<b>PATTERSON_LOOKUP_WEIGHTS_NP</b> sets weights for a Patterson rule.
</li>
<li>
<b>POINT_RADIAL_TOL_UNIQUE_COUNT</b> counts the tolerably unique points.
</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.
</li>
<li>
<b>POINT_RADIAL_TOL_UNIQUE_INDEX_OLD</b> indexes the tolerably unique points.
</li>
<li>
<b>POINT_RADIAL_TOL_UNIQUE_INDEX_INC1</b> indexes the tolerably unique points.
</li>
<li>
<b>POINT_RADIAL_TOL_UNIQUE_INDEX_INC2</b> indexes unique temporary points.
</li>
<li>
<b>POINT_RADIAL_TOL_UNIQUE_INDEX_INC3</b> merges index data.
</li>
<li>
<b>POINT_UNIQUE_INDEX</b> indexes unique points.
</li>
<li>
<b>PRODUCT_MIXED_WEIGHT</b> computes the weights of a mixed product rule.
</li>
<li>
<b>R8_CEILING</b> rounds an R8 "up" (towards +oo) to the next integer.
</li>
<li>
<b>R8_CHOOSE</b> computes the binomial coefficient C(N,K) as an R8.
</li>
<li>
<b>R8_EPSILON</b> returns the R8 roundoff unit.
</li>
<li>
<b>R8_FACTORIAL</b> computes the factorial function.
</li>
<li>
<b>R8_FACTORIAL2</b> computes the double factorial function.
</li>
<li>
<b>R8_FLOOR</b> rounds an R8 "down" (towards -infinity) to the next integer.
</li>
<li>
<b>R8_GAMMA</b> evaluates Gamma(X) for a real argument.
</li>
<li>
<b>R8_HUGE</b> returns a very large R8.
</li>
<li>
<b>R8_HYPER_2F1</b> evaluates the hypergeometric function F(A,B,C,X).
</li>
<li>
<b>R8_MOP</b> returns the I-th power of -1 as an R8 value.
</li>
<li>
<b>R8_PSI</b> evaluates the function Psi(X).
</li>
<li>
<b>R8COL_COMPARE</b> compares columns in an R8COL.
</li>
<li>
<b>R8COL_SORT_HEAP_A</b> ascending heapsorts an R8COL.
</li>
<li>
<b>R8COL_SORT_HEAP_INDEX_A</b> does an indexed heap ascending sort of an R8COL.
</li>
<li>
<b>R8COL_SORTED_TOL_UNDEX</b> indexes tolerably unique entries in a sorted R8COL.
</li>
<li>
<b>R8COL_SORTED_TOL_UNIQUE_COUNT:</b> tolerably unique elements in a sorted R8COL.
</li>
<li>
<b>R8COL_SORTED_UNIQUE_COUNT</b> counts unique elements in a sorted R8COL.
</li>
<li>
<b>R8COL_SWAP</b> swaps columns I and J of an R8COL.
</li>
<li>
<b>R8COL_TOL_UNDEX</b> indexes tolerably unique entries of an R8COL.
</li>
<li>
<b>R8COL_TOL_UNIQUE_COUNT</b> counts tolerably unique entries in an R8COL.
</li>
<li>
<b>R8COL_UNDEX</b> returns unique sorted indexes for an R8COL.
</li>
<li>
<b>R8COL_UNIQUE_INDEX</b> indexes the unique occurrence of values in an R8COL.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT</b> prints an R8MAT, transposed.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of an R8MAT, transposed.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</li>
<li>
<b>R8POLY_ANT_VAL</b> evaluates the antiderivative of a polynomial in standard form.
</li>
<li>
<b>R8VEC_CHEBYSHEV</b> creates a vector of Chebyshev spaced values.
</li>
<li>
<b>R8VEC_COMPARE</b> compares two R8VEC's.
</li>
<li>
<b>R8VEC_DIRECT_PRODUCT2</b> creates a direct product of R8VEC's.
</li>
<li>
<b>R8VEC_INDEX_SORTED_RANGE:</b> search index sorted vector for elements in a range.
</li>
<li>
<b>R8VEC_LEGENDRE</b> creates a vector of Legendre-spaced values.
</li>
<li>
<b>R8VEC_LINSPACE</b> creates a vector of linearly spaced values.
</li>
<li>
<b>R8VEC_MIN_POS</b> returns the minimum positive value of an R8VEC.
</li>
<li>
<b>R8VEC_PRINT</b> prints an R8VEC.
</li>
<li>
<b>R8VEC_SORT_HEAP_INDEX_A</b> does an indexed heap ascending sort of an R8VEC.
</li>
<li>
<b>R8VEC_UNIFORM_01</b> returns a unit pseudorandom R8VEC.
</li>
<li>
<b>SORT_HEAP_EXTERNAL</b> externally sorts a list of items into ascending order.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>VEC_COLEX_NEXT3</b> generates vectors in colex order.
</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 revised on 17 October 2011.
</i>
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