Additional-lapack-notes.html 6.7 KB

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  57. <a name="Additional-lapack-notes-1"></a>
  58. <h3 class="subsection">D.10.3 Additional <code>lapack</code> notes</h3>
  59. <p>The functions <code>dgesdd</code> and <code>zgesdd</code> are an effective
  60. dimensionality reduction technique for a large database of time
  61. <a name="index-dimensionality-reduction"></a>
  62. series. A set of basis vectors can be computed once for the database,
  63. and then any time series in the database can be expressed as a linear
  64. combination thereof. To the extent that the data embody any redundant
  65. information, an approximate reconstruction of an individual series
  66. from the database will require fewer coefficients (maybe far fewer) in
  67. terms of the basis than original length of the series.
  68. </p>
  69. <p>The library functions <code>dgelsd</code> and <code>zgelsd</code> are good for
  70. <a name="index-least-squares-2"></a>
  71. finding least squares fits to empirical data. If the matrix parameter
  72. <var>a</var> is interpreted as a list of inputs and the vector parameter
  73. <var>b</var> as the list of corresponding output data from some unknown
  74. linear function of <var>n</var> variables <var>f</var>, then <var>x</var> is the list
  75. of coefficients whereby <var>f</var> achieves the optimum fit to the data
  76. in the least squares sense.
  77. </p>
  78. <p>These functions solve a special case of the problem solved by
  79. <a name="index-generalized-least-squares-1"></a>
  80. <a name="index-least-squares-3"></a>
  81. <code>dggglm</code> and <code>zggglm</code> where the parameter <var>B</var> is the
  82. identity matrix. For the latter functions, the output vector <var>y</var>
  83. can be interpreted as a measure of the error, and <var>B</var> can be
  84. chosen to express unequal costs for errors at different points in
  85. the fitted function.
  86. </p>
  87. <p>Cholesky decompositions obtained by <code>dpptrf</code> and <code>zpptrf</code>
  88. <a name="index-Cholesky-decomposition-1"></a>
  89. are useful for generating correlated random numbers. A population of
  90. vectors of uncorrelated standard normally distributed random numbers
  91. can be made to exhibit any correlations to order by multiplying all of
  92. <a name="index-correlation"></a>
  93. the vectors by the lower Cholesky factor of the desired covariance
  94. <a name="index-covariance-matrix-1"></a>
  95. matrix.
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