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<a name="Large-Dense-Linear-Systems-TSQR"></a>
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<p>
Next: <a href="Large-Dense-Linear-Systems-Solution-Steps.html#Large-Dense-Linear-Systems-Solution-Steps" accesskey="n" rel="next">Large Dense Linear Systems Solution Steps</a>, Previous: <a href="Large-Dense-Linear-Systems-Normal-Equations.html#Large-Dense-Linear-Systems-Normal-Equations" accesskey="p" rel="previous">Large Dense Linear Systems Normal Equations</a>, Up: <a href="Large-Dense-Linear-Systems.html#Large-Dense-Linear-Systems" accesskey="u" rel="up">Large Dense Linear Systems</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Tall-Skinny-QR-_0028TSQR_0029-Approach"></a>
<h4 class="subsection">38.6.2 Tall Skinny QR (TSQR) Approach</h4>
<a name="index-large-linear-least-squares_002c-TSQR"></a>

<p>An algorithm which has better numerical stability for ill-conditioned
problems is known as the Tall Skinny QR (TSQR) method. This method
is based on computing the thin QR decomposition of the least squares
matrix <em>X = Q R</em>, where <em>Q</em> is an <em>n</em>-by-<em>p</em> matrix
with orthogonal columns, and <em>R</em> is a <em>p</em>-by-<em>p</em>
upper triangular matrix. Once these factors are calculated, the
residual becomes
</p><div class="example">
<pre class="example">\chi^2 = || Q^T y - R c ||^2 + \lambda^2 || c ||^2
</pre></div>
<p>which can be written as the matrix equation
</p><div class="example">
<pre class="example">[ R ; \lambda I ] c = [ Q^T b ; 0 ]
</pre></div>
<p>The matrix on the left hand side is now a much
smaller <em>2p</em>-by-<em>p</em> matrix which can
be solved with a standard SVD approach. The
<em>Q</em> matrix is just as large as the original
matrix <em>X</em>, however it does not need to be
explicitly constructed. The TSQR algorithm
computes only the <em>p</em>-by-<em>p</em> matrix
<em>R</em> and the <em>p</em>-by-1 vector <em>Q^T y</em>,
and updates these quantities as new blocks
are added to the system. Each time a new block of rows
(<em>X_i,y_i</em>) is added, the algorithm performs a QR decomposition
of the matrix
</p><div class="example">
<pre class="example">[ R_(i-1) ; X_i ]
</pre></div>
<p>where <em>R_{i-1}</em> is the upper triangular
<em>R</em> factor for the matrix
</p><div class="example">
<pre class="example">[ X_1 ; ... ; X_(i-1) ]
</pre></div>
<p>This QR decomposition is done efficiently taking into account
the sparse structure of <em>R_{i-1}</em>. See Demmel et al, 2008 for
more details on how this is accomplished. The number
of operations for this method is <em>O(2np^2 - {2 \over 3}p^3)</em>.
</p>
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Next: <a href="Large-Dense-Linear-Systems-Solution-Steps.html#Large-Dense-Linear-Systems-Solution-Steps" accesskey="n" rel="next">Large Dense Linear Systems Solution Steps</a>, Previous: <a href="Large-Dense-Linear-Systems-Normal-Equations.html#Large-Dense-Linear-Systems-Normal-Equations" accesskey="p" rel="previous">Large Dense Linear Systems Normal Equations</a>, Up: <a href="Large-Dense-Linear-Systems.html#Large-Dense-Linear-Systems" accesskey="u" rel="up">Large Dense Linear Systems</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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