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<hr size="1">
<a name="lapack-calling-conventions-1"></a>
<h3 class="subsection">D.10.1 <code>lapack</code> calling conventions</h3>

<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.
</p>
<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><li>
Some functions require a symmetric matrix as an input parameter.  Any
<a name="index-symmetric-matrices"></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"></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><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"><code>math</code></a>), and function names beginning
with <code>z</code> pertain to complex numbers (<a href="complex.html#complex"><code>complex</code></a>). The
specifications of similarly named functions are otherwise identical.
</li></ul>

<dl compact="compact">
<dt> <code>dgesvx</code></dt>
<dt> <code>zgesvx</code></dt>
<dd><p>These library functions take a pair <code>(<var>a</var>,<var>b</var>)</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 <code><var>a</var> <var>x</var> = <var>b</var></code>.
Otherwise they return an empty list.
</p></dd>
<dt> <code>dgelsd</code></dt>
<dt> <code>zgelsd</code></dt>
<dd><p>These functions generalize those above by taking a pair
<code>(<var>a</var>,<var>b</var>)</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 <code><var>b</var> - <var>a</var> <var>x</var></code>.
</p></dd>
<dt> <code>dgesdd</code></dt>
<dt> <code>zgesdd</code></dt>
<dd><p>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"></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
<code><var>min</var>(<var>m</var>,<var>n</var>)</code> but could be less if the input time
series aren&rsquo;t linearly independent. An empty list could be returned
due to lack of convergence.
</p></dd>
<dt> <code>dgeevx</code></dt>
<dt> <code>zgeevx</code></dt>
<dd><p>These functions take a non-symmetric square matrix and
return a pair <code>(<var>e</var>,<var>v</var>)</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"></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.
</p></dd>
<dt> <code>dpptrf</code></dt>
<dt> <code>zpptrf</code></dt>
<dd><p>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"></a>
of triangular matrices, each equal to the transpose of the other,
whose product is the original matrix. 
</p>
<ul>
<li>
If the input matrix is specified in lower triangular form, the lower
triangular Cholesky factor is returned.
</li><li>
If the input matrix is specified in square or upper triangular form,
the upper triangular Cholesky factor is returned.
</li><li>
In either case, the result is returned in triangular form.
</li></ul>
</dd>
<dt> <code>dggglm</code></dt>
<dt> <code>zggglm</code></dt>
<dd><p>The input is a pair of matrices and a vector
<a name="index-generalized-least-squares"></a>
<a name="index-least-squares-1"></a>
<code>((<var>A</var>,<var>B</var>),<var>d</var>)</code>. The output is a pair of vectors
<code>(<var>x</var>,<var>y</var>)</code> satisfying <code><var>A</var><var>x</var> +
<var>B</var><var>y</var> = <var>d</var></code> 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>.
</p></dd>
<dt> <code>dgglse</code></dt>
<dt> <code>zgglse</code></dt>
<dd><p>The input is of the form <code>((<var>A</var>,<var>c</var>),(<var>B</var>,<var>d</var>))</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
<code><var>A</var><var>x</var> - <var>c</var></code> subject to the constraint that
<code><var>B</var><var>x</var> = <var>d</var></code>. 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 <code><var>p</var> &lt;= <var>n</var> &lt;=
<var>m</var> + <var>p</var></code>.
</p></dd>
<dt> <code>dsyevr</code></dt>
<dd><p>This function takes a symmetric real matrix and returns a pair
<code>(<var>e</var>,<var>v</var>)</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>.
</p></dd>
<dt> <code>dspev</code></dt>
<dd><p>This function takes a symmetric real matrix and returns a pair
<code>(<var>e</var>,<var>v</var>)</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.
</p></dd>
<dt> <code>zheevr</code></dt>
<dd><p>This function takes a complex Hermitian matrix and returns a pair
<a name="index-Hermitian-matrix"></a>
<code>(<var>e</var>,<var>v</var>)</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.
</p>
<ul>
<li>
A Hermitian matrix has <var>Aij</var> equal to the complex conjugate of
<var>Aji</var>.
</li><li>
Although not exactly symmetric, a Hermitian matrix may nevertheless
be given in either upper or lower triangular form.
</li><li>
This function is faster but less storage efficient than <code>zhpev</code>,
and calls it automatically if it runs out of memory.
</li></ul>
</dd>
<dt> <code>zhpev</code></dt>
<dd><p>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.
</p></dd>
</dl>

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