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<h4 class="subsection">D.10.1 <code>lapack</code> calling conventions</h4>

<p>A table describing the inputs and outputs to the <code>lapack</code> library
functions listed by their function names is given in this
section. Some general points related to most of the functions are
mentioned first.

     <ul>
<li>References to vectors, matrices, and packed matrices should be
understood as their list representations explained in <a href="Type-Conversions.html#Type-Conversions">Type Conversions</a>. Although <code>LAPACK</code> internally uses column order
arrays, the virtual code library interface exhibits a matrix as a list
of lists with one inner list for each row. 
<li>Some functions require a symmetric matrix as an input parameter.  Any
<a name="index-symmetric-matrices-750"></a>input parameter that is required to be a symmetric matrix may be
specified optionally either in square form or in triangular form as
<a name="index-triangular-matrices-751"></a>described in <a href="Two-dimensional-arrays.html#Two-dimensional-arrays">Two dimensional arrays</a>. If a square matrix form is
used, symmetry is not checked and the lower triangular portion is
ignored. 
<li>Some function names are listed in pairs differing only in the first
letter.  Function names beginning with <code>d</code> pertain to vectors or
matrices of real numbers (<a href="math.html#math">math</a>), and function names beginning
with <code>z</code> pertain to complex numbers (<a href="complex.html#complex">complex</a>). The
specifications of similarly named functions are otherwise identical. 
</ul>

     <dl>
<dt><code>dgesvx</code><br><dt><code>zgesvx</code><dd>These library functions take a pair <code>(</code><var>a</var><code>,</code><var>b</var><code>)</code> where
<var>a</var> is an <var>n</var> by <var>n</var> matrix and <var>b</var> is a vector of
length <var>n</var>. If <var>a</var> is non-singular, they return a
vector <var>x</var> such that <var>a</var> <var>x</var><code> = </code><var>b</var>. 
Otherwise they return an empty list. 
<br><dt><code>dgelsd</code><br><dt><code>zgelsd</code><dd>These functions generalize those above by taking a pair
<code>(</code><var>a</var><code>,</code><var>b</var><code>)</code> where <var>a</var> is an <var>m</var> by <var>n</var> matrix
and <var>b</var> is a vector of length <var>m</var>, with <var>m</var> greater than
<var>n</var>. They return a vector <var>x</var> of length <var>n</var> to minimize
the magnitude of <var>b</var><code> - </code><var>a</var> <var>x</var>. 
<br><dt><code>dgesdd</code><br><dt><code>zgesdd</code><dd>These functions take a list of <var>m</var> time series (i.e., vectors)
each of length <var>n</var> and return a list of basis vectors each of
length <var>n</var>. The basis vectors span the set of time series in the
<a name="index-singular-value-decomposition-752"></a>given list according to the singular value decomposition (i.e., with
the basis vectors forming a series in order of decreasing
significance). The number of basis vectors is at most
<var>min</var><code>(</code><var>m</var><code>,</code><var>n</var><code>)</code> but could be less if the input time
series aren't linearly independent. An empty list could be returned
due to lack of convergence. 
<br><dt><code>dgeevx</code><br><dt><code>zgeevx</code><dd>These functions take a non-symmetric square matrix and
return a pair <code>(</code><var>e</var><code>,</code><var>v</var><code>)</code> where <var>e</var> is a list of
eigenvectors and <var>v</var> is a list of eigenvalues, both of which will
<a name="index-eigenvectors-753"></a>contain only complex numbers. (N.B., both functions return complex
results even though <code>dgeevx</code> takes real input.) They could also
return <code>nil</code> due to a lack of convergence. 
<br><dt><code>dpptrf</code><br><dt><code>zpptrf</code><dd>These functions take a symmetric square matrix and
return one of the Cholesky factors. The Cholesky factors are a pair
<a name="index-Cholesky-decomposition-754"></a>of triangular matrices, each equal to the transpose of the other,
whose product is the original matrix.

          <ul>
<li>If the input matrix is specified in lower triangular form, the lower
triangular Cholesky factor is returned. 
<li>If the input matrix is specified in square or upper triangular form,
the upper triangular Cholesky factor is returned. 
<li>In either case, the result is returned in triangular form. 
</ul>
     <br><dt><code>dggglm</code><br><dt><code>zggglm</code><dd>The input is a pair of matrices and a vector
<a name="index-generalized-least-squares-755"></a><a name="index-least-squares-756"></a><code>((</code><var>A</var><code>,</code><var>B</var><code>),</code><var>d</var><code>)</code>. The output is a pair of vectors
<code>(</code><var>x</var><code>,</code><var>y</var><code>)</code> satisfying <var>A</var><var>x</var><code> +
</code><var>B</var><var>y</var><code> = </code><var>d</var> for which the magnitude of <var>y</var> is
minimal. The dimensions all have to be consistent, which means
the number of rows in <var>A</var> and <var>B</var> is the length of <var>d</var>,
the number of columns in <var>A</var> is the length of <var>x</var>, and
the number of columns in <var>B</var> is the length of <var>y</var>. 
<br><dt><code>dgglse</code><br><dt><code>zgglse</code><dd>The input is of the form <code>((</code><var>A</var><code>,</code><var>c</var><code>),(</code><var>B</var><code>,</code><var>d</var><code>))</code>
where <var>A</var> and <var>B</var> are matrices and <var>c</var> and <var>d</var> are
vectors. The output is a vector <var>x</var> to minimize the magnitude of
<var>A</var><var>x</var><code> - </code><var>c</var> subject to the constraint that
<var>B</var><var>x</var><code> = </code><var>d</var>. The dimensions have to be consistent,
which means <var>A</var> has <var>m</var> rows, <var>c</var> has length <var>m</var>,
<var>B</var> has <var>p</var> rows, <var>d</var> has length <var>p</var>, both <var>A</var> and
<var>B</var> have <var>n</var> columns, and the output <var>x</var> has length
<var>n</var>. It is also a requirement that <var>p</var><code> &lt;= </code><var>n</var><code> &lt;=
</code><var>m</var><code> + </code><var>p</var>. 
<br><dt><code>dsyevr</code><dd>This function takes a symmetric real matrix and returns a pair
<code>(</code><var>e</var><code>,</code><var>v</var><code>)</code> where <var>e</var> is a list of eigenvectors and
<var>v</var> is a list of eigenvalues. Both contain only real numbers. 
This function is fast and accurate but not as storage efficient as
possible. If there is insufficient memory, it automatically invokes
<code>dspev</code>. 
<br><dt><code>dspev</code><dd>This function takes a symmetric real matrix and returns a pair
<code>(</code><var>e</var><code>,</code><var>v</var><code>)</code> where <var>e</var> is a list of eigenvectors and
<var>v</var> is a list of eigenvalues. Both contain only real numbers.  It
uses roughly half the memory of <code>dsyevr</code> but is not as fast or
accurate. 
<br><dt><code>zheevr</code><dd>This function takes a complex Hermitian matrix and returns a pair
<a name="index-Hermitian-matrix-757"></a><code>(</code><var>e</var><code>,</code><var>v</var><code>)</code> where <var>e</var> is a list of eigenvectors and
<var>v</var> is a list of eigenvalues. The eigenvectors are complex but the
eigenvalues are real.

          <ul>
<li>A Hermitian matrix has <var>Aij</var> equal to the complex conjugate of
<var>Aji</var>. 
<li>Although not exactly symmetric, a Hermitian matrix may nevertheless
be given in either upper or lower triangular form. 
<li>This function is faster but less storage efficient than <code>zhpev</code>,
and calls it automatically if it runs out of memory. 
</ul>
     <br><dt><code>zhpev</code><dd>This function has the same inputs and approximate outputs as
<code>zheevr</code> but is slower and more memory efficient because it uses
only packed matrices. 
</dl>

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