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<title>GNU Scientific Library &ndash; Reference Manual: BLAS Support</title>

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<a name="BLAS-Support"></a>
<div class="header">
<p>
Next: <a href="Linear-Algebra.html#Linear-Algebra" accesskey="n" rel="next">Linear Algebra</a>, Previous: <a href="Sorting.html#Sorting" accesskey="p" rel="previous">Sorting</a>, Up: <a href="index.html#Top" accesskey="u" rel="up">Top</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="BLAS-Support-1"></a>
<h2 class="chapter">13 BLAS Support</h2>
<a name="index-linear-algebra_002c-BLAS"></a>
<a name="index-matrix_002c-operations"></a>
<a name="index-vector_002c-operations"></a>
<a name="index-BLAS"></a>
<a name="index-CBLAS"></a>
<a name="index-Basic-Linear-Algebra-Subroutines-_0028BLAS_0029"></a>

<p>The Basic Linear Algebra Subprograms (<small>BLAS</small>) define a set of fundamental
operations on vectors and matrices which can be used to create optimized
higher-level linear algebra functionality.
</p>
<p>The library provides a low-level layer which corresponds directly to the
C-language <small>BLAS</small> standard, referred to here as &ldquo;<small>CBLAS</small>&rdquo;, and a
higher-level interface for operations on GSL vectors and matrices.
Users who are interested in simple operations on GSL vector and matrix
objects should use the high-level layer described
in this chapter.  The functions are declared in the file
<samp>gsl_blas.h</samp> and should satisfy the needs of most users.  
</p>
<p>Note that GSL matrices are implemented using dense-storage so the
interface only includes the corresponding dense-storage <small>BLAS</small>
functions.  The full <small>BLAS</small> functionality for band-format and
packed-format matrices is available through the low-level <small>CBLAS</small>
interface.  Similarly, GSL vectors are restricted to positive strides,
whereas the low-level <small>CBLAS</small> interface supports negative
strides as specified in the <small>BLAS</small> standard.<a name="DOCF12" href="#FOOT12"><sup>12</sup></a>
</p> 
<p>The interface for the <code>gsl_cblas</code> layer is specified in the file
<samp>gsl_cblas.h</samp>.  This interface corresponds to the <small>BLAS</small> Technical
Forum&rsquo;s standard for the C interface to legacy <small>BLAS</small>
implementations. Users who have access to other conforming <small>CBLAS</small>
implementations can use these in place of the version provided by the
library.  Note that users who have only a Fortran <small>BLAS</small> library can
use a <small>CBLAS</small> conformant wrapper to convert it into a <small>CBLAS</small>
library.  A reference <small>CBLAS</small> wrapper for legacy Fortran
implementations exists as part of the <small>CBLAS</small> standard and can
be obtained from Netlib.  The complete set of <small>CBLAS</small> functions is
listed in an appendix (see <a href="GSL-CBLAS-Library.html#GSL-CBLAS-Library">GSL CBLAS Library</a>).
</p>
<p>There are three levels of <small>BLAS</small> operations,
</p>
<dl compact="compact">
<dt><b>Level 1</b></dt>
<dd><p>Vector operations, e.g. <em>y = \alpha x + y</em>
</p></dd>
<dt><b>Level 2</b></dt>
<dd><p>Matrix-vector operations, e.g. <em>y = \alpha A x + \beta y</em>
</p></dd>
<dt><b>Level 3</b></dt>
<dd><p>Matrix-matrix operations, e.g. <em>C = \alpha A B + C</em>
</p></dd>
</dl>

<p>Each routine has a name which specifies the operation, the type of
matrices involved and their precisions.  Some of the most common
operations and their names are given below,
</p>
<dl compact="compact">
<dt><b>DOT</b></dt>
<dd><p>scalar product, <em>x^T y</em>
</p></dd>
<dt><b>AXPY</b></dt>
<dd><p>vector sum, <em>\alpha x + y</em>
</p></dd>
<dt><b>MV</b></dt>
<dd><p>matrix-vector product, <em>A x</em>
</p></dd>
<dt><b>SV</b></dt>
<dd><p>matrix-vector solve, <em>inv(A) x</em>
</p></dd>
<dt><b>MM</b></dt>
<dd><p>matrix-matrix product, <em>A B</em>
</p></dd>
<dt><b>SM</b></dt>
<dd><p>matrix-matrix solve, <em>inv(A) B</em>
</p></dd>
</dl>

<p>The types of matrices are,
</p>
<dl compact="compact">
<dt><b>GE</b></dt>
<dd><p>general
</p></dd>
<dt><b>GB</b></dt>
<dd><p>general band
</p></dd>
<dt><b>SY</b></dt>
<dd><p>symmetric
</p></dd>
<dt><b>SB</b></dt>
<dd><p>symmetric band
</p></dd>
<dt><b>SP</b></dt>
<dd><p>symmetric packed
</p></dd>
<dt><b>HE</b></dt>
<dd><p>hermitian
</p></dd>
<dt><b>HB</b></dt>
<dd><p>hermitian band
</p></dd>
<dt><b>HP</b></dt>
<dd><p>hermitian packed
</p></dd>
<dt><b>TR</b></dt>
<dd><p>triangular 
</p></dd>
<dt><b>TB</b></dt>
<dd><p>triangular band
</p></dd>
<dt><b>TP</b></dt>
<dd><p>triangular packed
</p></dd>
</dl>

<p>Each operation is defined for four precisions,
</p>
<dl compact="compact">
<dt><b>S</b></dt>
<dd><p>single real
</p></dd>
<dt><b>D</b></dt>
<dd><p>double real
</p></dd>
<dt><b>C</b></dt>
<dd><p>single complex
</p></dd>
<dt><b>Z</b></dt>
<dd><p>double complex
</p></dd>
</dl>

<p>Thus, for example, the name <small>SGEMM</small> stands for &ldquo;single-precision
general matrix-matrix multiply&rdquo; and <small>ZGEMM</small> stands for
&ldquo;double-precision complex matrix-matrix multiply&rdquo;.
</p>
<p>Note that the vector and matrix arguments to BLAS functions must not
be aliased, as the results are undefined when the underlying arrays
overlap (see <a href="Aliasing-of-arrays.html#Aliasing-of-arrays">Aliasing of arrays</a>). 
</p>
<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top">&bull; <a href="GSL-BLAS-Interface.html#GSL-BLAS-Interface" accesskey="1">GSL BLAS Interface</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="BLAS-Examples.html#BLAS-Examples" accesskey="2">BLAS Examples</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="BLAS-References-and-Further-Reading.html#BLAS-References-and-Further-Reading" accesskey="3">BLAS References and Further Reading</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
</table>

<div class="footnote">
<hr>
<h4 class="footnotes-heading">Footnotes</h4>

<h3><a name="FOOT12" href="#DOCF12">(12)</a></h3>
<p>In the low-level
<small>CBLAS</small> interface, a negative stride accesses the vector elements
in reverse order, i.e. the <em>i</em>-th element is given by
<em>(N-i)*|incx|</em> for <em>incx &lt; 0</em>.</p>
</div>
<hr>
<div class="header">
<p>
Next: <a href="Linear-Algebra.html#Linear-Algebra" accesskey="n" rel="next">Linear Algebra</a>, Previous: <a href="Sorting.html#Sorting" accesskey="p" rel="previous">Sorting</a>, Up: <a href="index.html#Top" accesskey="u" rel="up">Top</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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