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<title>avram - a virtual machine code interpreter: D.10.3 Additional lapack notes</title>

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<a name="Additional-lapack-notes"></a>
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<a name="Additional-lapack-notes-1"></a>
<h3 class="subsection">D.10.3 Additional <code>lapack</code> notes</h3>

<p>The functions <code>dgesdd</code> and <code>zgesdd</code> are an effective
dimensionality reduction technique for a large database of time
<a name="index-dimensionality-reduction"></a>
series.  A set of basis vectors can be computed once for the database,
and then any time series in the database can be expressed as a linear
combination thereof. To the extent that the data embody any redundant
information, an approximate reconstruction of an individual series
from the database will require fewer coefficients (maybe far fewer) in
terms of the basis than original length of the series.
</p>
<p>The library functions <code>dgelsd</code> and <code>zgelsd</code> are good for
<a name="index-least-squares-2"></a>
finding least squares fits to empirical data. If the matrix parameter
<var>a</var> is interpreted as a list of inputs and the vector parameter
<var>b</var> as the list of corresponding output data from some unknown
linear function of <var>n</var> variables <var>f</var>, then <var>x</var> is the list
of coefficients whereby <var>f</var> achieves the optimum fit to the data
in the least squares sense.
</p>
<p>These functions solve a special case of the problem solved by
<a name="index-generalized-least-squares-1"></a>
<a name="index-least-squares-3"></a>
<code>dggglm</code> and <code>zggglm</code> where the parameter <var>B</var> is the
identity matrix. For the latter functions, the output vector <var>y</var>
can be interpreted as a measure of the error, and <var>B</var> can be
chosen to express unequal costs for errors at different points in
the fitted function.
</p>
<p>Cholesky decompositions obtained by <code>dpptrf</code> and <code>zpptrf</code>
<a name="index-Cholesky-decomposition-1"></a>
are useful for generating correlated random numbers. A population of
vectors of uncorrelated standard normally distributed random numbers
can be made to exhibit any correlations to order by multiplying all of
<a name="index-correlation"></a>
the vectors by the lower Cholesky factor of the desired covariance
<a name="index-covariance-matrix-1"></a>
matrix.
</p>
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