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<html>
<head>
<title>
LAWSON - Least Squares Routines
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
LAWSON <br> Least Squares Routines
</h1>
<hr>
<p>
<b>LAWSON</b>
is a FORTRAN90 library which
solves least squares
problems.
</p>
<p>
The most common least squares problems considers an overdetermined
M by N linear system A*X=B. A least squares solution X is sought
which has the property that, although it generally is not a solution
of the system, it is the best approximation to a solution, in the
sense that it minimizes the L2 norm of the residual R=A*X-B.
</p>
<p>
In some cases, a unique solution to the system A*X=B will exist,
and in that case the least squares solution will coincide with
what is ordinarily meant by a solution.
</p>
<p>
In underdetermined cases, where multiple solutions exist,
the least squares solution is usually taken to be that solution X
which has minimum L2 norm, that is, which minimizes ||X||.
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>LAWSON</b> is available in
<a href = "../../f77_src/lawson/lawson.html">a FORTRAN77 version</a> and
<a href = "../../f_src/lawson/lawson.html">a FORTRAN90 version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/brent/brent.html">
BRENT</a>,
a FORTRAN90 library which
contains Richard Brent's routines for finding the zero, local minimizer,
or global minimizer of a scalar function of a scalar argument, without
the use of derivative information.
</p>
<p>
<a href = "../../f_src/bvls/bvls.html">
BVLS</a>,
a FORTRAN90 library which
applies least squares methods to solve a linear system for which
lower and upper constraints may have been placed on every variable.
</p>
<p>
<a href = "../../f_src/dqed/dqed.html">
DQED</a>,
a FORTRAN90 library which
solves constrained least squares problems.
</p>
<p>
<a href = "../../m_src/entrust/entrust.html">
ENTRUST</a>,
a MATLAB program which
solves problems in scalar optimization or nonlinear least squares.
</p>
<p>
<a href = "../../f_src/minpack/minpack.html">
MINPACK</a>,
a FORTRAN90 library which
solves systems of nonlinear equations, or the least squares minimization of the
residual of a set of linear or nonlinear equations.
</p>
<p>
<a href = "../../f_src/nl2sol/nl2sol.html">
NL2SOL</a>,
a FORTRAN90 library which
implements an adaptive nonlinear least-squares algorithm.
</p>
<p>
<a href = "../../f_src/qr_solve/qr_solve.html">
QR_SOLVE</a>,
a FORTRAN90 library which
computes the least squares solution of a linear system A*x=b.
</p>
<p>
<a href = "../../f_src/praxis/praxis.html">
PRAXIS</a>,
a FORTRAN90 library which
minimizes a scalar function of several variables.
</p>
<p>
<a href = "../../f_src/test_opt/test_opt.html">
TEST_OPT</a>,
a FORTRAN90 library which
defines test problems
requiring the minimization of a scalar function of several variables.
</p>
<p>
<a href = "../../f_src/toms611/toms611.html">
TOMS611</a>,
a FORTRAN90 library which
seeks the minimizer of a scalar functional
of multiple variables.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Gene Golub, Christian Reinsch,<br>
Singular Value Decomposition and Least Squares Solutions,<br>
Numerische Mathematik,<br>
Volume 14, Number 5, April 1970, pages 403-420.
</li>
<li>
Charles Lawson, Richard Hanson,<br>
Solving Least Squares Problems,<br>
SIAM, 1995,<br>
ISBN: 0898713560,<br>
LC: QA275.L38.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "lawson.f90">lawson.f90</a>, the main source code.
</li>
<li>
<a href = "lawson.sh">lawson.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "lawson_prb.f90">lawson_prb.f90</a>, a sample problem.
</li>
<li>
<a href = "lawson_prb.dat">lawson_prb.dat</a>,
a data file used by the sample problem.
</li>
<li>
<a href = "lawson_prb.sh">lawson_prb.sh</a>,
commands to compile, link and run the sample problem.
</li>
<li>
<a href = "lawson_prb_output.txt">lawson_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>BNDACC</b> accumulates information for a banded least squares problem.
</li>
<li>
<b>BNDSOL</b> solves a banded least squares problem accumulated by BNDACC.
</li>
<li>
<b>DIFF</b> is used in tests that depend on machine precision.
</li>
<li>
<b>G1</b> computes an orthogonal rotation matrix.
</li>
<li>
<b>G2</b> applies a rotation matrix to a vector (X,Y).
</li>
<li>
<b>GEN</b> generates numbers for construction of test cases.
</li>
<li>
<b>H12</b> constructs or applies a Householder transformation.
</li>
<li>
<b>HFTI:</b> Householder forward triangulation with column interchanges.
</li>
<li>
<b>LDP</b> implements least distance programming
</li>
<li>
<b>MFEOUT</b> labeled matrix output for use with singular value analysis.
</li>
<li>
<b>NNLS</b> implements the nonnegative least squares algorithm.
</li>
<li>
<b>QRBD</b> uses the QR algorithm for the singular values of a bidiagonal matrix.
</li>
<li>
<b>SVA</b> carries out a singular value analysis.
</li>
<li>
<b>SVDRS:</b> singular value decomposition also treating right side vector.
</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 21 October 2008.
</i>
<!-- John Burkardt -->
</body>
</html>