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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.
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