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<html>
<head>
<title>
TEST_APPROX - Test Data for Approximation Algorithms
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TEST_APPROX <br> Test Data for Approximation Algorithms
</h1>
<hr>
<p>
<b>TEST_APPROX</b>
is a C++ library which
provides sets of test data for approximation algorithms.
</p>
<p>
<b>TEST_APPROX</b> contains a number of vectors of data (X(1:N),Y(1:N))
for which no underlying functional relationship is given.
</p>
<p>
The task of interpolation software is to find, from some given
class of functions, the function G(X) which exactly matches the
known data values. That is, G(X(1:N)) = Y(1:N).
</p>
<p>
The task of approximation software is to find, from some given
class of functions, the function H(X) for which some approximation
error is minimized. There are many forms of error measurement.
For instance, the error might simply be the sum of the differences
of the function and the data at the data abscissas:
<blockquote>
l1(X) = sum ( 1 <= I <= N ) abs ( H(X(I)) - Y(I) )
</blockquote>
or the square root of the sum of squares
<blockquote>
l2(X) = sqrt ( sum ( 1 <= I <= N ) ( H(X(I)) - Y(I) )^2 )
</blockquote>
or the maximum pointwise error:
<blockquote>
l_inf(X) = max ( abs ( H(X(I)) - Y(I) ) )
</blockquote>
In cases where a functional form is given, the error might be
measured in terms of the integral of the absolute value of the
difference over some interval:
<blockquote>
L1(X,A,B) = integral ( A <= X <= B ) abs ( H(X) - F(X) ) dx
</blockquote>
and so on.
</p>
<p>
The problems available include:
<ol>
<li>
<a href = "p01_data.png">p01_data.png</a>: DeBoor example, Mars position data
</li>
<li>
<a href = "p02_data.png">p02_data.png</a>: DeBoor example, roughly linear data
</li>
<li>
<a href = "p03_data.png">p03_data.png</a>: The pulse data, 0 0 0 0 0 1 0 0 0 0 0;
</li>
<li>
<a href = "p04_data.png">p04_data.png</a>: The jump data, 0 0 0 0 0 1/2 1 1 1 1 1;
</li>
<li>
<a href = "p05_data.png">p05_data.png</a>: DeBoor's Titanium Property data;
</li>
<li>
<a href = "p06_data.png">p06_data.png</a>: The Sawtooth data;
</li>
<li>
<a href = "p07_data.png">p07_data.png</a>: Concavity test data;
</li>
<li>
<a href = "p08_data.png">p08_data.png</a>: Extrapolation test data;
</li>
<li>
<a href = "p09_data.png">p09_data.png</a>: Sunspot data, 1700-1960;
</li>
<li>
<a href = "p10_data.png">p10_data.png</a>: 100 samples of y=2+5x+10*N(0,1),
where N(0,1) is a random normal value with 0 mean and unit variance;
</li>
</ol>
</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>TEST_APPROX</b> is available in
<a href = "../../c_src/test_approx/test_approx.html">a C version</a> and
<a href = "../../cpp_src/test_approx/test_approx.html">a C++ version</a> and
<a href = "../../f77_src/test_approx/test_approx.html">a FORTRAN77 version</a> and
<a href = "../../f_src/test_approx/test_approx.html">a FORTRAN90 version</a> and
<a href = "../../m_src/test_approx/test_approx.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/chebyshev.html">
CHEBYSHEV</a>,
a C++ library which
computes the Chebyshev interpolant/approximant to a given function
over an interval.
</p>
<p>
<a href = "../../cpp_src/divdif/divdif.html">
DIVDIF</a>,
a C++ library which
includes many routines to construct and evaluate divided difference
interpolants.
</p>
<p>
<a href = "../../cpp_src/spline/spline.html">
SPLINE</a>,
a C++ library which
includes many routines to construct and evaluate spline
interpolants and approximants.
</p>
<p>
<a href = "../../datasets/test_approx/test_approx.html">
TEST_APPROX</a>,
a dataset directory which
contains sets of data (x,y) for which an approximating formula is desired.
</p>
<p>
<a href = "../../cpp_src/test_interp_1d/test_interp_1d.html">
TEST_INTERP_1D</a>,
a C++ library which
defines test problems for interpolation of data y(x),
depending on a 1D argument.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Samuel Conte, Carl deBoor,<br>
Elementary Numerical Analysis,<br>
Second Edition,<br>
McGraw Hill, 1972,<br>
ISBN: 07-012446-4,<br>
LC: QA297.C65.
</li>
<li>
Carl deBoor,<br>
A Practical Guide to Splines,<br>
Springer, 2001,<br>
ISBN: 0387953663,<br>
LC: QA1.A647.v27.
</li>
<li>
Max Waldmeier,<br>
The Sunspot-Activity in the Years 1610-1960,<br>
Shulthess, 1961,<br>
LC: QB525.W34.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "test_approx.cpp">test_approx.cpp</a>, the source code;
</li>
<li>
<a href = "test_approx.hpp">test_approx.hpp</a>, the include file;
</li>
<li>
<a href = "test_approx.sh">test_approx.sh</a>,
commands to compile the source code;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "test_approx_prb.cpp">test_approx_prb.cpp</a>, the
calling program;
</li>
<li>
<a href = "test_approx_prb.sh">test_approx_prb.sh</a>,
commands to compile, link and run the calling program;
</li>
<li>
<a href = "test_approx_prb_output.txt">test_approx_prb_output.txt</a>, the sample
output.
</li>
</ul>
</p>
<p>
<b>TEST07</b> creates data files for an Overhauser spline interpolant
to data set 7.
<ul>
<li>
<a href = "test07_data.txt">test07_data.txt</a>
the data.
</li>
<li>
<a href = "test07_approx.txt">test07_approx.txt</a>
the interpolant values.
</li>
<li>
<a href = "test07.png">test07.png</a>
a PNG image of the data and interpolant.
</li>
</ul>
</p>
<p>
<b>TEST08</b> creates data files for a cubic spline interpolant
to data set 7.
<ul>
<li>
<a href = "test08_data.txt">test08_data.txt</a>
the data.
</li>
<li>
<a href = "test08_approx.txt">test08_approx.txt</a>
the interpolant values.
</li>
<li>
<a href = "test08.png">test08.png</a>
a PNG image of the data and interpolant.
</li>
</ul>
</p>
<p>
<b>TEST10</b> creates data files for a B-spline approximant
to data set 7.
<ul>
<li>
<a href = "test10_data.txt">test10_data.txt</a>
the data.
</li>
<li>
<a href = "test10_approx.txt">test10_approx.txt</a>
the approximant values.
</li>
<li>
<a href = "test10.png">test10.png</a>
a PNG image of the data and approximant.
</li>
</ul>
</p>
<p>
<b>TEST11</b> creates data files for a beta spline approximant to
data set 7:
<ul>
<li>
<a href = "test11_data.txt">test11_data.txt</a>
the data.
</li>
<li>
<a href = "test11_approx.txt">test11_approx.txt</a>
the approximant values.
</li>
<li>
<a href = "test11.png">test11.png</a>
a PNG image of the data and approximant.
</li>
</ul>
</p>
<p>
<b>TEST12</b> creates data files for a Bernstein approximant
to data set 5.
<ul>
<li>
<a href = "test12_data.txt">test12_data.txt</a>
the data.
</li>
<li>
<a href = "test12_approx.txt">test12_approx.txt</a>
the interpolant values.
</li>
<li>
<a href = "test12.png">test12.png</a>
a PNG image of the data and approximant.
</li>
</ul>
</p>
<p>
<b>TEST13</b> creates data files for a cubic spline interpolant
to data set 5.
<ul>
<li>
<a href = "test13_data.txt">test13_data.txt</a>
the data.
</li>
<li>
<a href = "test13_approx.txt">test13_approx.txt</a>
the interpolant values.
</li>
<li>
<a href = "test13.png">test13.png</a>
a PNG image of the data and interpolant.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>P00_DAT</b> returns the data vector for any problem.
</li>
<li>
<b>P00_DATA_NUM</b> returns the dimension of the data vector for any problem.
</li>
<li>
<b>P00_PROB_NUM</b> returns the number of test problems.
</li>
<li>
<b>P00_STORY</b> prints the "story" for any problem.
</li>
<li>
<b>P00_TITLE</b> returns the title of any problem.
</li>
<li>
<b>P01_DAT</b> returns the data vector for problem 1.
</li>
<li>
<b>P01_DATA_NUM</b> returns the dimension of the data vector for problem 1.
</li>
<li>
<b>P01_STORY</b> prints the "story" for problem 1.
</li>
<li>
<b>P01_TITLE</b> returns the title of problem 1.
</li>
<li>
<b>P02_DAT</b> returns the data vector for problem 2.
</li>
<li>
<b>P02_DATA_NUM</b> returns the dimension of the data vector for problem 2.
</li>
<li>
<b>P02_STORY</b> prints the "story" for problem 2.
</li>
<li>
<b>P02_TITLE</b> returns the title of problem 2.
</li>
<li>
<b>P03_DAT</b> returns the data vector for problem 3.
</li>
<li>
<b>P03_DATA_NUM</b> returns the dimension of the data vector for problem 3.
</li>
<li>
<b>P03_STORY</b> prints the "story" for problem 3.
</li>
<li>
<b>P03_TITLE</b> returns the title of problem 3.
</li>
<li>
<b>P04_DAT</b> returns the data vector for problem 4.
</li>
<li>
<b>P04_DATA_NUM</b> returns the dimension of the data vector for problem 4.
</li>
<li>
<b>P04_STORY</b> prints the "story" for problem 4.
</li>
<li>
<b>P04_TITLE</b> returns the title of problem 4.
</li>
<li>
<b>P05_DAT</b> returns the data vector for problem 5.
</li>
<li>
<b>P05_DATA_NUM</b> returns the dimension of the data vector for problem 5.
</li>
<li>
<b>P05_STORY</b> prints the "story" for problem 5.
</li>
<li>
<b>P05_TITLE</b> returns the title of problem 5.
</li>
<li>
<b>P06_DAT</b> returns the data vector for problem 6.
</li>
<li>
<b>P06_DATA_NUM</b> returns the dimension of the data vector for problem 6.
</li>
<li>
<b>P06_STORY</b> prints the "story" for problem 6.
</li>
<li>
<b>P06_TITLE</b> returns the title of problem 6.
</li>
<li>
<b>P07_DAT</b> returns the data vector for problem 7.
</li>
<li>
<b>P07_DATA_NUM</b> returns the dimension of the data vector for problem 7.
</li>
<li>
<b>P07_STORY</b> prints the "story" for problem 7
</li>
<li>
<b>P07_TITLE</b> returns the title of problem 7.
</li>
<li>
<b>P08_DAT</b> returns the data vector for problem 8.
</li>
<li>
<b>P08_DATA_NUM</b> returns the dimension of the data vector for problem 8.
</li>
<li>
<b>P08_STORY</b> prints the "story" for problem 8.
</li>
<li>
<b>P08_TITLE</b> returns the title of problem 8.
</li>
<li>
<b>P09_DAT</b> returns the data vector for problem 9.
</li>
<li>
<b>P09_DATA_NUM</b> returns the dimension of the data vector for problem 9.
</li>
<li>
<b>P09_STORY</b> prints the "story" for problem 09
</li>
<li>
<b>P09_TITLE</b> returns the title of problem 09.
</li>
<li>
<b>P10_DAT</b> returns the data vector for problem 10.
</li>
<li>
<b>P10_DATA_NUM</b> returns the dimension of the data vector for problem 10.
</li>
<li>
<b>P10_STORY</b> prints the "story" for problem 10.
</li>
<li>
<b>P10_TITLE</b> returns the title of problem 10.
</li>
<li>
<b>R8VEC_COPY</b> copies an R8VEC.
</li>
<li>
<b>R8VEC2_PRINT</b> prints an R8VEC2.
</li>
<li>
<b>R8VEC2_WRITE</b> writes an R8VEC2 file.
</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 08 February 2012.
</i>
<!-- John Burkardt -->
</body>
</html>