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<html>
<head>
<title>
QR_SOLVE - Least Squares Solution of a Linear System A*x=b
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
QR_SOLVE <br> Least Squares Solution of a Linear System A*x=b
</h1>
<hr>
<p>
<b>QR_SOLVE</b>
is a C++ library which
computes a least squares solution of a linear system of the form A*x=b.
</p>
<p>
There are many possible cases that can arise with the matrix A.
Formally, we distinguish the cases M < N, M = N, and M > N, and
we expect trouble whenever M is not equal to N. Trouble may also
arise when M = N but the matrix is singular. Even if the matrix is,
mathematically speaking, non-singular, it may be so close to singularity
that an accurate solution is difficult to achieve.
</p>
<p>
When M > N, we are placing more conditions than we have degrees of freedom,
so we suppose that such a linear system cannot be solved. However, it is
possible that the extra conditions are illusory, being constructed from
linear combinations of a fundamental set of N conditions. Thus, a system
that we typically call "overdetermined" can have a solution in the ordinary
sense, that satisfies all the conditions, as long as the right hand side is
"consistent". Another way of saying this is that the system is solvable
if the right hand side lies in the column space of A...although that simply
says that it is a linear combination of the columns of A, which just says
A*x=b.
</p>
<p>
If A does not have full column rank, however, then even if the right hand
side lies in the column space of A, there will be more than one linear
combination of columns that produce b. Thus, the equations are consistent,
the system is solvable, but not uniquely so.
</p>
<p>
If M < N, then we are placing fewer conditions than we have degrees of
freedom. As long as the right hand side lies in the column space of A,
we can guarantee that there will be multiple solutions.
</p>
<p>
Thus, the question of a "solution" to the problem A*x=b is complicated
enough that it seems to defy a common algorithmic approach. Nonetheless,
there are some sensible, robust procedures for producing an answer that
corresponds to the classical solution, or solves the overdetermined problem
when the right hand side is consistent. This is the linear least squares
solution, which finds a vector x which minimizes the Euclidean norm of
the residual: ||r|| = ||A*x-b||. This solution is produced by computing
the QR factorization of the matrix A
</p>
<p>
When there are multiple solutions to the problem, the QR approach used here
produces a solution. A more satisfactory approach, using the pseudoinverse,
will produce a solution x which satisfies the additional constraint that
it has the minimum norm of the entire family of solutions. That pseudoinverse
approach is not implemented in this library. The singular value decomposition (SVD)
can also produce this minimal solution.
</p>
<p>
For comparison, a solver that applies the normal equations is included.
This approach requires M >= N, and that A have full column rank. It constructs
and solves the NxN system A'*A*x=A'*b. This system has a condition number that
is the square of the original system, so it also suffers from a significant
loss in accuracy.
</p>
<p>
We also include an SVD solver, which uses the pseudoinverse approach.
First compute A = U * S * V', where U and V are orthogonal, and S is
MxN diagonal, then to solve A*x=b write x = V * S^ * U' * b, where
S^ is the matrix formed by transposing S and then replacing each nonzero
diagonal element s by 1/s. (However, very small elements should probably
be zeroed instead of inverted.) This procedure will also produce a vector
x which minimizes the Euclidean norm. However, it has one feature that the
QR solver does not: in cases where the solution x is not unique (because
A does not have full column rank) the SVD solver returns the unique vector
x of minimum Euclidean norm.
</p>
<p>
<b>QR_SOLVE</b> needs the R8LIB library. The test program also needs
the TEST_LS library.
</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>QR_SOLVE</b> is available in
<a href = "../../c_src/qr_solve/qr_solve.html">a C version</a> and
<a href = "../../cpp_src/qr_solve/qr_solve.html">a C++ version</a> and
<a href = "../../f77_src/qr_solve/qr_solve.html">a FORTRAN77 version</a> and
<a href = "../../f_src/qr_solve/qr_solve.html">a FORTRAN90 version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/linpack_d/linpack_d.html">
LINPACK_D</a>,
a C++ library which
solves linear systems using double precision real arithmetic;
</p>
<p>
<a href = "../../cpp_src/llsq/llsq.html">
LLSQ</a>,
a C++ library which
solves the simple linear least squares problem of finding the formula
of a straight line y=a*x+b which minimizes the root-mean-square error
to a set of N data points.
</p>
<p>
<a href = "../../cpp_src/minpack/minpack.html">
MINPACK</a>,
a C++ 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 = "../../cpp_src/r8lib/r8lib.html">
R8LIB</a>,
a C++ library which
contains many utility routines using double precision real (R8) arithmetic.
</p>
<p>
<a href = "../../cpp_src/test_ls/test_ls.html">
TEST_LS</a>,
a C++ library which
implements linear least squares test problems of the form A*x=b.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,<br>
LINPACK User's Guide,<br>
SIAM, 1979,<br>
ISBN13: 978-0-898711-72-1,<br>
LC: QA214.L56.
</li>
<li>
David Kahaner, Cleve Moler, Steven Nash,<br>
Numerical Methods and Software,<br>
Prentice Hall, 1989,<br>
ISBN: 0-13-627258-4,<br>
LC: TA345.K34.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "qr_solve.cpp">qr_solve.cpp</a>, the source code.
</li>
<li>
<a href = "qr_solve.hpp">qr_solve.hpp</a>, the include file.
</li>
<li>
<a href = "qr_solve.sh">qr_solve.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "qr_solve_prb.cpp">qr_solve_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "qr_solve_prb.sh">qr_solve_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "qr_solve_prb_output.txt">qr_solve_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>DAXPY</b> computes constant times a vector plus a vector.
</li>
<li>
<b>DDOT</b> forms the dot product of two vectors.
</li>
<li>
<b>DNRM2</b> returns the euclidean norm of a vector.
</li>
<li>
<b>DQRANK</b> computes the QR factorization of a rectangular matrix.
</li>
<li>
<b>DQRDC</b> computes the QR factorization of a real rectangular matrix.
</li>
<li>
<b>DQRLS</b> factors and solves a linear system in the least squares sense.
</li>
<li>
<b>DQRLSS</b> solves a linear system in a least squares sense.
</li>
<li>
<b>DQRSL</b> computes transformations, projections, and least squares solutions.
</li>
<li>
<b>DROT</b> applies a plane rotation.
</li>
<li>
<b>DROTG</b> constructs a Givens plane rotation.
</li>
<li>
<b>DSCAL</b> scales a vector by a constant.
</li>
<li>
<b>DSVDC</b> computes the singular value decomposition of a real rectangular matrix.
</li>
<li>
<b>DSWAP</b> interchanges two vectors.
</li>
<li>
<b>NORMAL_SOLVE</b> solves a linear system using the normal equations.
</li>
<li>
<b>QR_SOLVE</b> solves a linear system in the least squares sense.
</li>
<li>
<b>SVD_SOLVE</b> solves a linear system in the least squares sense.
</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 04 October 2012.
</i>
<!-- John Burkardt -->
</body>
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</html>