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<html>
<head>
<title>
INT_EXACTNESS - Exactness of One Dimensional Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
INT_EXACTNESS <br> Exactness of One Dimensional Quadrature Rules
</h1>
<hr>
<p>
<b>INT_EXACTNESS</b>
is a C++ program which
investigates the polynomial exactness of a one dimensional
quadrature rule defined on a finite interval.
</p>
<p>
The polynomial exactness of a quadrature rule is defined as the
highest degree <b>D</b> such that the quadrature rule is
guaranteed to integrate exactly all polynomials of degree
<b>DEGREE_MAX</b> or less, ignoring roundoff. The degree of a polynomial
is the maximum of the degrees of all its monomial terms. The degree
of a monomial term is the exponent. Thus, for instance,
the <b>DEGREE</b> of
<blockquote><b>
3*x<sup>5</sup> - 7*x<sup>9</sup> + 27
</b></blockquote>
is the maximum of 5, 9 and 0, so it is 9.
</p>
<p>
To be thorough, the program starts at <b>DEGREE</b> = 0, and then
proceeds to <b>DEGREE</b> = 1, 2, and so on up to a maximum degree
<b>DEGREE_MAX</b> specified by the user. At each value of <b>DEGREE</b>,
the program generates the corresponding monomial term, applies the
quadrature rule to it, and determines the quadrature error. The program
uses a scaling factor on each monomial so that the exact integral
should always be 1; therefore, each reported error can be compared
on a fixed scale.
</p>
<p>
The program is very flexible and interactive. The quadrature rule
is defined by three files, to be read at input, and the
maximum degree is specified by the user as well.
</p>
<p>
Note that the three files that define the quadrature rule
are assumed to have related names, of the form
<ul>
<li>
<i>prefix</i>_<b>x.txt</b>
</li>
<li>
<i>prefix</i>_<b>w.txt</b>
</li>
<li>
<i>prefix</i>_<b>r.txt</b>
</li>
</ul>
When running the program, the user only enters the common <i>prefix</i>
part of the file names, which is enough information for the program
to find all three files.
</p>
<p>
For information on the form of these files, see the
<b>QUADRATURE_RULES</b> directory listed below.
</p>
<p>
The exactness results are written to an output file with the
corresponding name:
<ul>
<li>
<i>prefix</i>_<b>exact.txt</b>
</li>
</ul>
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>int_exactness</b> <i>prefix</i> <i>degree_max</i>
</blockquote>
where
<ul>
<li>
<i>prefix</i> is the common prefix for the files containing the abscissa, weight
and region information of the quadrature rule;
</li>
<li>
<i>degree_max</i> is the maximum monomial degree to check. This would normally be
a relatively small nonnegative number, such as 5, 10 or 15.
</li>
</ul>
</p>
<p>
If the arguments are not supplied on the command line, the
program will prompt for them.
</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>INT_EXACTNESS</b> is available in
<a href = "../../cpp_src/int_exactness/int_exactness.html">a C++ version</a> and
<a href = "../../f_src/int_exactness/int_exactness.html">a FORTRAN90 version</a> and
<a href = "../../m_src/int_exactness/int_exactness.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/exactness/exactness.html">
EXACTNESS</a>,
a C++ library which
investigates the exactness of quadrature rules that estimate the
integral of a function with a density, such as 1, exp(-x) or
exp(-x^2), over an interval such as [-1,+1], [0,+oo) or (-oo,+oo).
</p>
<p>
<a href = "../../cpp_src/hermite_exactness/hermite_exactness.html">
HERMITE_EXACTNESS</a>,
a C++ program which
tests the polynomial exactness of Gauss-Hermite quadrature rules.
</p>
<p>
<a href = "../../cpp_src/int_exactness_chebyshev1/int_exactness_chebyshev1.html">
INT_EXACTNESS_CHEBYSHEV1</a>,
a C++ program which
tests the polynomial exactness of Gauss-Chebyshev type 1 quadrature rules.
</p>
<p>
<a href = "../../cpp_src/int_exactness_chebyshev2/int_exactness_chebyshev2.html">
INT_EXACTNESS_CHEBYSHEV2</a>,
a C++ program which
tests the polynomial exactness of Gauss-Chebyshev type 2 quadrature rules.
</p>
<p>
<a href = "../../cpp_src/int_exactness_gegenbauer/int_exactness_gegenbauer.html">
INT_EXACTNESS_GEGENBAUER</a>,
a C++ program which
tests the polynomial exactness of Gauss-Gegenbauer quadrature rules.
</p>
<p>
<a href = "../../cpp_src/int_exactness_gen_hermite/int_exactness_gen_hermite.html">
INT_EXACTNESS_GEN_HERMITE</a>,
a C++ program which
tests the polynomial exactness of a generalized Gauss-Hermite quadrature rule.
</p>
<p>
<a href = "../../cpp_src/int_exactness_gen_laguerre/int_exactness_gen_laguerre.html">
INT_EXACTNESS_GEN_LAGUERRE</a>,
a C++ program which
tests the polynomial exactness of a generalized Gauss-Laguerre quadrature rule.
</p>
<p>
<a href = "../../cpp_src/int_exactness_jacobi/int_exactness_jacobi.html">
INT_EXACTNESS_JACOBI</a>,
a C++ program which
tests the polynomial exactness of a Gauss-Jacobi quadrature rule.
</p>
<p>
<a href = "../../cpp_src/laguerre_exactness/laguerre_exactness.html">
LAGUERRE_EXACTNESS</a>,
a C++ program which
tests the polynomial exactness of Gauss-Laguerre quadrature rules
for integration over [0,+oo) with density function exp(-x).
</p>
<p>
<a href = "../../cpp_src/legendre_exactness/legendre_exactness.html">
LEGENDRE_EXACTNESS</a>,
a C++ program which
tests the monomial exactness of quadrature rules for the Legendre problem
of integrating a function with density 1 over the interval [-1,+1].
</p>
<p>
<a href = "../../cpp_src/nint_exactness/nint_exactness.html">
NINT_EXACTNESS</a>,
a C++ program which
tests the polynomial exactness of multidimensional integration rules.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<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>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "int_exactness.cpp">int_exactness.cpp</a>, the source code.
</li>
<li>
<a href = "int_exactness.sh">int_exactness.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>CC_D1_O2</b> is a Clenshaw-Curtis order 2 rule for 1D.
<ul>
<li>
<a href = "../../datasets/quadrature_rules/cc_d1_o2_x.txt">cc_d1_o2_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/cc_d1_o2_w.txt">cc_d1_o2_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/cc_d1_o2_r.txt">cc_d1_o2_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "cc_d1_o2_exact.txt">cc_d1_o2_exact.txt</a>,
the results of the exactness test, up to degree 5.
</li>
</ul>
</p>
<p>
<b>CC_D1_O3</b> is a Clenshaw-Curtis order 3 rule for 1D.
If you are paying attention, you may be surprised to see that
a Clenshaw Curtis rule of odd order has one more degree of
accuracy than you'd expect!
<ul>
<li>
<a href = "../../datasets/quadrature_rules/cc_d1_o3_x.txt">cc_d1_o3_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/cc_d1_o3_w.txt">cc_d1_o3_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/cc_d1_o3_r.txt">cc_d1_o3_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "cc_d1_o3_exact.txt">cc_d1_o3_exact.txt</a>,
the results of the exactness test, up to degree 5.
</li>
</ul>
</p>
<p>
<b>GL_D1_O3</b> is a Gauss-Legendre order 3 rule for 1D.
<ul>
<li>
<a href = "../../datasets/quadrature_rules/gl_d1_o3_x.txt">gl_d1_o3_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/gl_d1_o3_w.txt">gl_d1_o3_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/gl_d1_o3_r.txt">gl_d1_o3_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gl_d1_o3_exact.txt">gl_d1_o3_exact.txt</a>,
the results of the exactness test, up to degree 5.
</li>
</ul>
</p>
<p>
<b>NCC_D1_O5</b> is a Newton-Cotes Closed order 5 rule for 1D.
<ul>
<li>
<a href = "../../datasets/quadrature_rules/ncc_d1_o5_x.txt">ncc_d1_o5_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/ncc_d1_o5_w.txt">ncc_d1_o5_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/ncc_d1_o5_r.txt">ncc_d1_o5_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "ncc_d1_o5_exact.txt">ncc_d1_o5_exact.txt</a>,
the results of the exactness test, up to degree 7.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for INT_EXACTNESS.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is true if two characters are equal, disregarding case.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer value of a base 10 digit.
</li>
<li>
<b>DTABLE_DATA_READ</b> reads the data from a DTABLE file.
</li>
<li>
<b>DTABLE_HEADER_READ</b> reads the header from a DTABLE file.
</li>
<li>
<b>FILE_COLUMN_COUNT</b> counts the number of columns in the first line of a file.
</li>
<li>
<b>FILE_ROW_COUNT</b> counts the number of row records in a file.
</li>
<li>
<b>MONOMIAL_QUADRATURE</b> applies a quadrature rule to a monomial.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>S_LEN_TRIM</b> returns the length of a string to the last nonblank.
</li>
<li>
<b>S_TO_I4</b> reads an I4 from a string.
</li>
<li>
<b>S_TO_R8</b> reads an R8 from a string.
</li>
<li>
<b>S_TO_R8VEC</b> reads an R8VEC from a string.
</li>
<li>
<b>S_WORD_COUNT</b> counts the number of "words" in a string.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TIMESTRING</b> returns the current YMDHMS date as a string.
</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 revised on 11 February 2008.
</i>
<!-- John Burkardt -->
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