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@@ -16,7 +16,7 @@ href="userhtmlse11.html#tailuserhtmlse11.html" >prev-tail</a>] [<a
href="userhtmlse9.html#tailuserhtmlse12.html">tail</a>] [<a
href="userhtml.html# " >up</a>] </p></div>
<h3 class="sectionHead"><span class="titlemark">12 </span> <a
id="x19-14800012"></a>Extensions</h3>
id="x20-14900012"></a>Extensions</h3>
<!--l. 3--><p class="noindent" >The EXT, CUDA and RSB subdirectories contains a set of extensions to the base
library. The extensions provide additional storage formats beyond the ones already
contained in the base library, as well as interfaces to:
@@ -42,14 +42,14 @@ class="cmtt-10">http://sourceforge.net/projects/librsb/</span></a>, for comput
multicore parallel machines.</dd></dl>
<!--l. 14--><p class="noindent" >The infrastructure laid out in the base library to allow for these extensions is detailed in
the references&#x00A0;<span class="cite">[<a
href="userhtmlli2.html#XDesPat:11">21</a>,&#x00A0;<a
href="userhtmlli2.html#XCaFiRo:2014">22</a>,&#x00A0;<a
href="userhtmlli2.html#XSparse03">11</a>]</span>; the CUDA-specific data formats are described
href="userhtmlli3.html#XDesPat:11">21</a>,&#x00A0;<a
href="userhtmlli3.html#XCaFiRo:2014">22</a>,&#x00A0;<a
href="userhtmlli3.html#XSparse03">11</a>]</span>; the CUDA-specific data formats are described
in&#x00A0;<span class="cite">[<a
href="userhtmlli2.html#XOurTechRep">23</a>]</span>.
href="userhtmlli3.html#XOurTechRep">23</a>]</span>.
<!--l. 19--><p class="noindent" >
<h4 class="subsectionHead"><span class="titlemark">12.1 </span> <a
id="x19-14900012.1"></a>Using the extensions</h4>
id="x20-15000012.1"></a>Using the extensions</h4>
<!--l. 21--><p class="noindent" >A sample application using the PSBLAS extensions will contain the following
steps:
<ul class="itemize1">
@@ -142,31 +142,31 @@ speed of the sparse matrix-vector product with the various data structures inclu
in the library.
<!--l. 146--><p class="noindent" >
<h4 class="subsectionHead"><span class="titlemark">12.2 </span> <a
id="x19-15000012.2"></a>Extensions&#8217; Data Structures</h4>
id="x20-15100012.2"></a>Extensions&#8217; Data Structures</h4>
<!--l. 150--><p class="noindent" >Access to the facilities provided by the EXT library is mainly achieved through
the data types that are provided within. The data classes are derived from
the base classes in PSBLAS, through the Fortran&#x00A0;2003 mechanism of <span
class="pplri7t-">type</span>
<span
class="pplri7t-">extension</span>&#x00A0;<span class="cite">[<a
href="userhtmlli2.html#XMRC:11">18</a>]</span>.
href="userhtmlli3.html#XMRC:11">18</a>]</span>.
<!--l. 155--><p class="indent" > The data classes are divided between the general purpose CPU extensions, the
GPU interfaces and the RSB interfaces. In the description we will make use of the
notation introduced in Table&#x00A0;<a
href="#x19-150001r22">22<!--tex4ht:ref: tab:notation --></a>.
href="#x20-151001r22">22<!--tex4ht:ref: tab:notation --></a>.
<div class="table">
<!--l. 160--><p class="indent" > <a
id="x19-150001r22"></a><hr class="float"><div class="float"
id="x20-151001r22"></a><hr class="float"><div class="float"
>
<div class="caption"
><span class="id">Table&#x00A0;22: </span><span
class="content">Notation for parameters describing a sparse matrix</span></div><!--tex4ht:label?: x19-150001r22 -->
class="content">Notation for parameters describing a sparse matrix</span></div><!--tex4ht:label?: x20-151001r22 -->
<div class="center"
>
<!--l. 162--><p class="noindent" >
@@ -276,7 +276,7 @@ class="td11"> </td></tr></table>
<a
id="x19-150002r5"></a>
id="x20-151002r5"></a>
@@ -285,18 +285,18 @@ src="mat.png" alt="PIC"
width="147" height="147" >
<br /> <div class="caption"
><span class="id">Figure&#x00A0;5: </span><span
class="content">Example of sparse matrix</span></div><!--tex4ht:label?: x19-150002r5 -->
class="content">Example of sparse matrix</span></div><!--tex4ht:label?: x20-151002r5 -->
<!--l. 198--><p class="indent" > </div><hr class="endfigure">
<h4 class="subsectionHead"><span class="titlemark">12.3 </span> <a
id="x19-15100012.3"></a>CPU-class extensions</h4>
id="x20-15200012.3"></a>CPU-class extensions</h4>
<!--l. 203--><p class="noindent" >
<h5 class="likesubsubsectionHead"><a
id="x19-152000"></a>ELLPACK</h5>
id="x20-153000"></a>ELLPACK</h5>
<!--l. 205--><p class="noindent" >The ELLPACK/ITPACK format (shown in Figure&#x00A0;<a
href="#x19-152001r6">6<!--tex4ht:ref: fig:ell --></a>) comprises two 2-dimensional
href="#x20-153001r6">6<!--tex4ht:ref: fig:ell --></a>) comprises two 2-dimensional
arrays <span class="obeylines-h"><span class="verb"><span
class="cmtt-10">AS</span></span></span> and <span class="obeylines-h"><span class="verb"><span
class="cmtt-10">JA</span></span></span> with <span class="obeylines-h"><span class="verb"><span
@@ -317,7 +317,7 @@ row.
<a
id="x19-152001r6"></a>
id="x20-153001r6"></a>
@@ -327,13 +327,13 @@ width="233" height="233" >
<br /> <div class="caption"
><span class="id">Figure&#x00A0;6: </span><span
class="content">ELLPACK compression of matrix in Figure&#x00A0;<a
href="#x19-150002r5">5<!--tex4ht:ref: fig:dense --></a></span></div><!--tex4ht:label?: x19-152001r6 -->
href="#x20-151002r5">5<!--tex4ht:ref: fig:dense --></a></span></div><!--tex4ht:label?: x20-153001r6 -->
<!--l. 225--><p class="indent" > </div><hr class="endfigure">
<a
id="x19-152002r1"></a>
id="x20-153002r1"></a>
@@ -344,7 +344,7 @@ href="#x19-150002r5">5<!--tex4ht:ref: fig:dense --></a></span></div><!--tex4ht:l
<!--l. 231-->
<pre class="lstlisting" id="listing-221"><span class="label"><a
id="x19-152003r1"></a></span><span
id="x20-153003r1"></a></span><span
class="cmtt-9">&#x00A0;</span><span
class="cmtt-9">&#x00A0;</span><span
class="cmtt-9">&#x00A0;</span><span style="color:#000000"><span
@@ -353,7 +353,7 @@ class="cmtt-9">i</span></span><span style="color:#000000"><span
class="cmtt-9">=1,</span></span><span style="color:#000000"><span
class="cmtt-9">n</span></span>
<span class="label"><a
id="x19-152004r2"></a></span><span
id="x20-153004r2"></a></span><span
class="cmtt-9">&#x00A0;</span><span
class="cmtt-9">&#x00A0;</span><span
class="cmtt-9">&#x00A0;</span><span
@@ -362,7 +362,7 @@ class="cmtt-9">&#x00A0;</span><span style="color:#000000"><span
class="cmtt-9">t</span></span><span style="color:#000000"><span
class="cmtt-9">=0</span></span>
<span class="label"><a
id="x19-152005r3"></a></span><span
id="x20-153005r3"></a></span><span
class="cmtt-9">&#x00A0;</span><span
class="cmtt-9">&#x00A0;</span><span
class="cmtt-9">&#x00A0;</span><span
@@ -373,7 +373,7 @@ class="cmtt-9">j</span></span><span style="color:#000000"><span
class="cmtt-9">=1,</span></span><span style="color:#000000"><span
class="cmtt-9">maxnzr</span></span>
<span class="label"><a
id="x19-152006r4"></a></span><span
id="x20-153006r4"></a></span><span
class="cmtt-9">&#x00A0;</span><span
class="cmtt-9">&#x00A0;</span><span
class="cmtt-9">&#x00A0;</span><span
@@ -401,7 +401,7 @@ class="cmtt-9">,</span></span><span style="color:#000000"><span
class="cmtt-9">j</span></span><span style="color:#000000"><span
class="cmtt-9">))</span></span>
<span class="label"><a
id="x19-152007r5"></a></span><span
id="x20-153007r5"></a></span><span
class="cmtt-9">&#x00A0;</span><span
class="cmtt-9">&#x00A0;</span><span
class="cmtt-9">&#x00A0;</span><span
@@ -410,7 +410,7 @@ class="cmtt-9">&#x00A0;</span><span style="color:#000000"><span
class="cmtt-9">end</span></span><span style="color:#000000"> </span><span style="color:#000000"><span
class="cmtt-9">do</span></span>
<span class="label"><a
id="x19-152008r6"></a></span><span
id="x20-153008r6"></a></span><span
class="cmtt-9">&#x00A0;</span><span
class="cmtt-9">&#x00A0;</span><span
class="cmtt-9">&#x00A0;</span><span
@@ -423,7 +423,7 @@ class="cmtt-9">)</span></span><span style="color:#000000"> </span><span style="c
class="cmtt-9">=</span></span><span style="color:#000000"> </span><span style="color:#000000"><span
class="cmtt-9">t</span></span>
<span class="label"><a
id="x19-152009r7"></a></span><span
id="x20-153009r7"></a></span><span
class="cmtt-9">&#x00A0;</span><span
class="cmtt-9">&#x00A0;</span><span
class="cmtt-9">&#x00A0;</span><span style="color:#000000"><span
@@ -431,9 +431,9 @@ class="cmtt-9">end</span></span><span style="color:#000000"> </span><span style=
class="cmtt-9">do</span></span></pre>
<a
id="x19-152010r1"></a>
id="x20-153010r1"></a>
<a
id="x19-152011"></a>
id="x20-153011"></a>
<span
class="pplb7t-">Algorithm</span><span
class="pplb7t-">&#x00A0;1:</span>&#x00A0; Matrix-Vector product in ELL format
@@ -446,7 +446,7 @@ class="zplmr7m-">y </span><span
class="zplmr7t-">= </span><span
class="zplmr7m-">Ax </span>can be computed with the code shown in
Alg.&#x00A0;<a
href="#x19-152010r1">1<!--tex4ht:ref: alg:ell --></a>; it costs one memory write per outer iteration, plus three memory reads and
href="#x20-153010r1">1<!--tex4ht:ref: alg:ell --></a>; it costs one memory write per outer iteration, plus three memory reads and
two floating-point operations per inner iteration.
<!--l. 247--><p class="indent" > Unless all rows have exactly the same number of nonzeros, some of the
coefficients in the <span class="obeylines-h"><span class="verb"><span
@@ -455,12 +455,12 @@ overhead both in terms of memory space and redundant operations (multiplications
by zero). The overhead can be acceptable if:
<ol class="enumerate1" >
<li
class="enumerate" id="x19-152013x1">
class="enumerate" id="x20-153013x1">
<!--l. 253--><p class="noindent" >The maximum number of nonzeros per row is not much larger than the
average;
</li>
<li
class="enumerate" id="x19-152015x2">
class="enumerate" id="x20-153015x2">
<!--l. 255--><p class="noindent" >The regularity of the data structure allows for faster code, e.g. by allowing
vectorization, thereby offsetting the additional storage requirements.</li></ol>
<!--l. 259--><p class="noindent" >In the extreme case where the input matrix has one full row, the ELLPACK
@@ -488,7 +488,7 @@ class="cmtt-10">psb_T_ell_sparse_mat</span></span></span>:
</pre>
<!--l. 295--><p class="nopar" > </div></div>
<h5 class="likesubsubsectionHead"><a
id="x19-153000"></a>Hacked ELLPACK</h5>
id="x20-154000"></a>Hacked ELLPACK</h5>
@@ -564,7 +564,7 @@ format.
<a
id="x19-153001r7"></a>
id="x20-154001r7"></a>
@@ -574,7 +574,7 @@ width="248" height="248" >
<br /> <div class="caption"
><span class="id">Figure&#x00A0;7: </span><span
class="content">Hacked ELLPACK compression of matrix in Figure&#x00A0;<a
href="#x19-150002r5">5<!--tex4ht:ref: fig:dense --></a></span></div><!--tex4ht:label?: x19-153001r7 -->
href="#x20-151002r5">5<!--tex4ht:ref: fig:dense --></a></span></div><!--tex4ht:label?: x20-154001r7 -->
@@ -601,9 +601,9 @@ class="cmtt-10">psb_T_hll_sparse_mat</span></span></span>:
</pre>
<!--l. 388--><p class="nopar" > </div></div>
<h5 class="likesubsubsectionHead"><a
id="x19-154000"></a>Diagonal storage</h5>
id="x20-155000"></a>Diagonal storage</h5>
<!--l. 396--><p class="noindent" >The DIAgonal (DIA) format (shown in Figure&#x00A0;<a
href="#x19-154001r8">8<!--tex4ht:ref: fig:dia --></a>) has a 2-dimensional array <span class="obeylines-h"><span class="verb"><span
href="#x20-155001r8">8<!--tex4ht:ref: fig:dia --></a>) has a 2-dimensional array <span class="obeylines-h"><span class="verb"><span
class="cmtt-10">AS</span></span></span>
containing in each column the coefficients along a diagonal of the matrix, and an
integer array <span class="obeylines-h"><span class="verb"><span
@@ -614,7 +614,7 @@ are padded with zeros as necessary.
class="zplmr7m-">y </span><span
class="zplmr7t-">= </span><span
class="zplmr7m-">Ax </span>is shown in Alg.&#x00A0;<a
href="#x19-154003r2">2<!--tex4ht:ref: alg:dia --></a>; it
href="#x20-155003r2">2<!--tex4ht:ref: alg:dia --></a>; it
costs one memory read per outer iteration, plus three memory reads, one memory
write and two floating-point operations per inner iteration. The accesses to
<span class="obeylines-h"><span class="verb"><span
@@ -627,7 +627,7 @@ required.
<a
id="x19-154001r8"></a>
id="x20-155001r8"></a>
@@ -637,13 +637,13 @@ width="248" height="248" >
<br /> <div class="caption"
><span class="id">Figure&#x00A0;8: </span><span
class="content">DIA compression of matrix in Figure&#x00A0;<a
href="#x19-150002r5">5<!--tex4ht:ref: fig:dense --></a></span></div><!--tex4ht:label?: x19-154001r8 -->
href="#x20-151002r5">5<!--tex4ht:ref: fig:dense --></a></span></div><!--tex4ht:label?: x20-155001r8 -->
<!--l. 419--><p class="indent" > </div><hr class="endfigure">
<a
id="x19-154002r2"></a>
id="x20-155002r2"></a>
@@ -669,9 +669,9 @@ href="#x19-150002r5">5<!--tex4ht:ref: fig:dense --></a></span></div><!--tex4ht:l
</pre>
<!--l. 450--><p class="nopar" > </div></div>
<a
id="x19-154003r2"></a>
id="x20-155003r2"></a>
<a
id="x19-154004"></a>
id="x20-155004"></a>
<span
class="pplb7t-">Algorithm</span><span
class="pplb7t-">&#x00A0;2:</span>&#x00A0; Matrix-Vector product in DIA format
@@ -698,7 +698,7 @@ class="cmtt-10">psb_T_dia_sparse_mat</span></span></span>:
</pre>
<!--l. 486--><p class="nopar" > </div></div>
<h5 class="likesubsubsectionHead"><a
id="x19-155000"></a>Hacked DIA</h5>
id="x20-156000"></a>Hacked DIA</h5>
<!--l. 495--><p class="noindent" >Storage by DIAgonals is an attractive option for matrices whose coefficients are
located on a small set of diagonals, since they do away with storing explicitly the
indices and therefore reduce significantly memory traffic. However, having a few
@@ -749,7 +749,7 @@ class="pplri7t-">hackOffsets[k]</span>.
<a
id="x19-155001r9"></a>
id="x20-156001r9"></a>
@@ -759,7 +759,7 @@ width="248" height="248" >
<br /> <div class="caption"
><span class="id">Figure&#x00A0;9: </span><span
class="content">Hacked DIA compression of matrix in Figure&#x00A0;<a
href="#x19-150002r5">5<!--tex4ht:ref: fig:dense --></a></span></div><!--tex4ht:label?: x19-155001r9 -->
href="#x20-151002r5">5<!--tex4ht:ref: fig:dense --></a></span></div><!--tex4ht:label?: x20-156001r9 -->
@@ -804,7 +804,7 @@ class="cmtt-10">psb_T_hdia_sparse_mat</span></span></span>:
<h4 class="subsectionHead"><span class="titlemark">12.4 </span> <a
id="x19-15600012.4"></a>CUDA-class extensions</h4>
id="x20-15700012.4"></a>CUDA-class extensions</h4>
<!--l. 4--><p class="noindent" >For computing with CUDA we define a dual memorization strategy in which each
variable on the CPU (&#8220;host&#8221;) side has a GPU (&#8220;device&#8221;) side. When a GPU-type
variable is initialized, the data contained is (usually) the same on both sides. Each