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<html>
<head>
<title>
TABLE_ORTHONORMALIZE - Orthonormalize a Set of Vectors
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TABLE_ORTHONORMALIZE <br> Orthonormalize a Set of Vectors
</h1>
<hr>
<p>
<b>TABLE_ORTHONORMALIZE</b>
is a FORTRAN90 program which
can read in a set of vectors and
orthonormalize them.
</p>
<p>
<i>Caution</i>: it may not be clear from the documentation, but
the program implicitly assumes that each vector represents a
velocity field, and hence can be thought of as a quantity
V(M/2,2), where "M" is the appropriate dimension if we think of
the quantity as a vector of scalar values. Moreover, each
vector is written to a file as M/2 records of pairs of values.
</p>
<p>
For instance, the program can read three (or in general <b>N</b>)
files, each containing one of the M-dimensional vectors
<b>v<sub>1</sub></b>, <b>v<sub>2</sub></b> and <b>v<sub>3</sub></b>.
The program then outputs three new vectors <b>q<sub>1</sub></b>,
<b>q<sub>2</sub></b> and <b>q<sub>3</sub></b>, with the properties that
<ul>
<li>
The Euclidean norm of <b>q<sub>i</sub></b> is 1, for all <b>i</b>;
</li>
<li>
The dot product <b>q<sub>i</sub>•q<sub>j</sub></b> is 0,
for <b>i</b> and <b>j</b> distinct;
</li>
<li>
The vectors <b>q<sub>i</sub></b> span the same space as the
original vectors <b>v<sub>i</sub></b>;
</li>
</ul>
</p>
<p>
The algorithm collects the <b>N</b> input vectors <b>v<sub>j</sub></b>
into an <b>M</b> by <b>N</b> matrix <b>V</b>. It then
calls the appropriate LAPACK routines (<b>SGEQRF</b> and <b>SORGQR</b>)
to compute the QR factorization:
<pre><b>
V = Q * R
</b></pre>
and outputs the individual columns of <b>Q</b> as the new vectors
<b>q<sub>j</sub></b>.
</p>
<p>
This routine can be useful as a postprocessing tool for the output
from
<a href = "../cvt_basis/cvt_basis.html">CVT_BASIS</a> for example.
However, orthonormalization would probably <i>not</i> be appropriate
if <b>CVT_BASIS</b> had done its computations using
mass matrix preconditioning. In that case, the output vectors
have a special orthogonality property under an L2 inner product,
which would be lost if the vectors were processed by this program.
</p>
<p>
Some effort has been made to relieve the user of the tedious
necessity of specifying how many input files there are (the number
<b>N</b>), and what their dimensionality is (the number <b>M</b>).
It is assumed that the input data files have "consecutive" names,
such as <i>file01.txt</i>, <i>file02.txt</i>, ... and so on.
This is enough to figure out <b>N</b>. And the program counts
the number of lines in the first file, and assumes this value
of <b>M</b> may safely be applied to all the files.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<dl>
<dt>
<b>table_orthonormalize</b> <i>input_file_01</i> <i>output_file_01</i>
</dt>
<dd>
reads <i>input_file_01</i>, orthonormalizes the vectors, and
writes them to <i>output_file_01</i>.
</dd>
</dl>
</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">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/lapack/lapack.html">
LAPACK</a>,
a FORTRAN90 library which
supplies the routines used to orthonormalize the vectors.
</p>
<p>
<a href = "../../f_src/table_barplot_ppma/table_barplot_ppma.html">
TABLE_BARPLOT_PPMA</a>,
a FORTRAN90 program which
reads a table file and creates a
PPMA bargraph of the data.
</p>
<p>
<a href = "../../f_src/table_border/table_border.html">
TABLE_BORDER</a>,
a FORTRAN90 program which
can be used to add a border
(of zero values) to a table file.
</p>
<p>
<a href = "../../f_src/table_columns/table_columns.html">
TABLE_COLUMNS</a>,
a FORTRAN90 program which
can extract specific columns of data
from a table file.
</p>
<p>
<a href = "../../f_src/table_columns_permute/table_columns_permute.html">
TABLE_COLUMNS_PERMUTE</a>,
a FORTRAN90 program which
permutes the columns of a table file.
</p>
<p>
<a href = "../../f_src/table_delaunay/table_delaunay.html">
TABLE_DELAUNAY</a>,
a FORTRAN90 program which
computes the Delaunay triangulation of a set
of points.
</p>
<p>
<a href = "../../f_src/table_histogram/table_histogram.html">
TABLE_HISTOGRAM</a>,
a FORTRAN90 program which
can make a histogram of a set of points
stored in a table file.
</p>
<p>
<a href = "../../f_src/table_io/table_io.html">
TABLE_IO</a>,
a FORTRAN90 library which
supplies the routines used to read the TABLE file.
</p>
<p>
<a href = "../../f_src/table_latinize/table_latinize.html">
TABLE_LATINIZE</a>,
a FORTRAN90 program which
reads a file of
points and creates a "latinized" version by adjusting the data.
</p>
<p>
<a href = "../../f_src/table_merge/table_merge.html">
TABLE_MERGE</a>,
a FORTRAN90 program which
reads a file of
points, and removes duplicates, and points that are close to each other.
</p>
<p>
<a href = "../../f_src/table_quality/table_quality.html">
TABLE_QUALITY</a>,
a FORTRAN90 program which
reads a file of
points and computes the quality of dispersion.
</p>
<p>
<a href = "../../f_src/table_record_match/table_record_match.html">
TABLE_RECORD_MATCH</a>,
a FORTRAN90 program which
can be used to find close records in a table file.
</p>
<p>
<a href = "../../f_src/table_scale/table_scale.html">
TABLE_SCALE</a>,
a FORTRAN90 program which
can be used to multiply the entries of a table file
by a scale vector.
</p>
<p>
<a href = "../../f_src/table_shift/table_shift.html">
TABLE_SHIFT</a>,
a FORTRAN90 program which
can be used to shift the entries of a table file
by a shift vector.
</p>
<p>
<a href = "../../f_src/table_stats/table_stats.html">
TABLE_STATS</a>,
a FORTRAN90 program which
can read a table file and compute certain statistics.
</p>
<p>
<a href = "../../f_src/table_tet_mesh/table_tet_mesh.html">
TABLE_TET_MESH</a>,
a FORTRAN90 program which
can read a table file of 3D data, and compute a
tetrahedral mesh.
</p>
<p>
<a href = "../../f_src/table_top/table_top.html">
TABLE_TOP</a>,
a FORTRAN90 program which
can read a table file of M-dimensional data and
make a table of plots of all pairs of coordinates.
</p>
<p>
<a href = "../../f_src/table_unborder/table_unborder.html">
TABLE_UNBORDER</a>,
a FORTRAN90 program which
can be used to remove the border from a table file.
</p>
<p>
<a href = "../../f_src/table_uniform_noise/table_uniform_noise.html">
TABLE_UNIFORM_NOISE</a>,
a FORTRAN90 program which
can be used to add a uniform noise term to the data
in a table file.
</p>
<p>
<a href = "../../f_src/table_voronoi/table_voronoi.html">
TABLE_VORONOI</a>,
a FORTRAN90 program which
can be used to compute information about
the Voronoi diagram of the points.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Edward Anderson, Zhaojun Bai, Christian Bischof, Susan Blackford,
James Demmel, Jack Dongarra, Jeremy DuCroz, Anne Greenbaum,
Sven Hammarling, Alan McKenney, Danny Sorensen,<br>
LAPACK User's Guide,<br>
Third Edition,<br>
SIAM, 1999,<br>
QA76.73.F25L36
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "table_orthonormalize.f90">table_orthonormalize.f90</a>,
the source code.
</li>
<li>
<a href = "table_orthonormalize.sh">table_orthonormalize.sh</a>,
commands to compile and load the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
A set of sample input files you may copy, with M = 8 and N = 3:
<ul>
<li>
<a href = "v_01.txt">v_01.txt</a>, input file #1.
</li>
<li>
<a href = "v_02.txt">v_02.txt</a>, input file #2.
</li>
<li>
<a href = "v_03.txt">v_03.txt</a>, input file #3.
</li>
</ul>
</p>
<p>
A set
of output files corresponding to the sample input files,
with M = 8 and N = 3, including:
<ul>
<li>
<a href = "q_01.txt">q_01.txt</a>, output file #1.
</li>
<li>
<a href = "q_02.txt">q_02.txt</a>, output file #2.
</li>
<li>
<a href = "q_03.txt">q_03.txt</a>, output file #3.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for TABLE_ORTHONORMALIZE.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is a case insensitive comparison of two characters for equality.
</li>
<li>
<b>CH_IS_DIGIT</b> returns TRUE if a character is a decimal digit.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer value of a base 10 digit.
</li>
<li>
<b>DIGIT_INC</b> increments a decimal digit.
</li>
<li>
<b>DIGIT_TO_CH</b> returns the character representation of a decimal digit.
</li>
<li>
<b>FILE_NAME_INC</b> generates the next filename in a series.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>Q_FACTOR</b> determines the Q factor of a matrix.
</li>
<li>
<b>R8MAT_PRINT</b> prints an R8MAT.
</li>
<li>
<b>R8MAT_PRINT_SOME</b> prints some of an R8MAT.
</li>
<li>
<b>S_INPUT</b> prints a prompt string and reads a string from the user.
</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>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TIMESTRING</b> writes the current YMDHMS date into a string.
</li>
<li>
<b>XY_READ_DATA</b> reads data from an XY file.
</li>
<li>
<b>XY_READ_HEADER</b> determines the number of pairs of data in an XY file.
</li>
<li>
<b>XY_WRITE</b> writes an XY file.
</li>
<li>
<b>XY_WRITE_DATA</b> writes the data of an XY file.
</li>
<li>
<b>XY_WRITE_HEADER</b> writes the header of an XY file.
</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 31 October 2006.
</i>
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