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<H2><A NAME="SECTION000727000000000000000">
Sorting utilities -- </A>
</H2>

<P>
<FONT SIZE="+1"><B>psb_msort -- Sorting by the Merge-sort
  algorithm</B></FONT>

<P>
<FONT SIZE="+1"><B>psb_qsort -- Sorting by the Quicksort
  algorithm</B></FONT>

<P>
<FONT SIZE="+1"><B>psb_hsort -- Sorting by the Heapsort  algorithm</B></FONT>
<PRE>
call psb_msort(x,ix,dir,flag)
call psb_qsort(x,ix,dir,flag)
call psb_hsort(x,ix,dir,flag)
</PRE>

<P>
These  serial routines sort a sequence <IMG
 WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img114.png"
 ALT="$X$"> into ascending or
descending  order. The argument meaning is identical for the three
calls; the only difference is the algorithm used to accomplish the
task (see Usage Notes below). 
<DL>
<DT><STRONG>Type:</STRONG></DT>
<DD>Asynchronous.
</DD>
<DT><STRONG><B>On Entry </B></STRONG></DT>
<DD>
</DD>
<DT><STRONG>x</STRONG></DT>
<DD>The sequence to be sorted.
<BR>
Type:<B>required</B>.
<BR>
Specified as: an integer, real or complex  array of rank 1.
</DD>
<DT><STRONG>ix</STRONG></DT>
<DD>A vector of indices.
<BR>
Type:<B>optional</B>.
<BR>
Specified as: an integer array of (at least) the same size as <IMG
 WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img114.png"
 ALT="$X$">.
</DD>
<DT><STRONG>dir</STRONG></DT>
<DD>The desired ordering.
<BR>
Type:<B>optional</B>.
<BR>
Specified as: an integer value: <DL>
<DT><STRONG>Integer and real data:</STRONG></DT>
<DD><code>psb_sort_up_</code>,
<code>psb_sort_down_</code>, <code>psb_asort_up_</code>, <code>psb_asort_down_</code>;
default <code>psb_sort_up_</code>. 
</DD>
<DT><STRONG>Complex data:</STRONG></DT>
<DD><code>psb_lsort_up_</code>,
<code>psb_lsort_down_</code>, <code>psb_asort_up_</code>, <code>psb_asort_down_</code>;
default <code>psb_lsort_up_</code>. 
</DD>
</DL>
</DD>
<DT><STRONG>flag</STRONG></DT>
<DD>Whether to keep the original values in <IMG
 WIDTH="27" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img115.png"
 ALT="$IX$">.
<BR>
Type:<B>optional</B>.
<BR>
Specified as: an integer value <code>psb_sort_ovw_idx_</code> or
<code>psb_sort_keep_idx_</code>; default <code>psb_sort_ovw_idx_</code>.

<P>
</DD>
</DL>

<P>
<DL>
<DT><STRONG><B>On Return</B></STRONG></DT>
<DD>
</DD>
<DT><STRONG>x</STRONG></DT>
<DD>The sequence of values, in the chosen ordering.
<BR>
Type:<B>required</B>.
<BR>
Specified as: an integer, real or complex array of rank 1.
</DD>
<DT><STRONG>ix</STRONG></DT>
<DD>A vector of indices.
<BR>
Type: <B>Optional</B> 
<BR>
An integer array of rank 1, whose entries are moved to the same
position as the corresponding entries in <IMG
 WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img20.png"
 ALT="$x$">.
</DD>
</DL>

<P>

<P>
<FONT SIZE="+1"><B>Notes</B></FONT>

<OL>
<LI>For integer or real data the sorting can be performed in the up/down direction, on the
  natural or absolute values;
</LI>
<LI>For  complex data the sorting can  be done in a lexicographic
  order (i.e.: sort on the real part with ties broken according to
  the imaginary part) or on the absolute values; 
</LI>
<LI>The routines return the items in the chosen ordering; the
  output difference is the handling of  ties (i.e. items with an
  equal   value) in the original input. With the merge-sort algorithm
  ties are   preserved in the same relative order as they had in the
  original   sequence,  while this is not guaranteed for quicksort or
  heapsort; 
</LI>
<LI>If <!-- MATH
 $flag = psb\_sort\_ovw\_idx\_$
 -->
<IMG
 WIDTH="180" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img116.png"
 ALT="$flag = psb\_sort\_ovw\_idx\_$"> then the entries in <IMG
 WIDTH="62" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img117.png"
 ALT="$ix(1:n)$">
  where <IMG
 WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img16.png"
 ALT="$n$"> is the size of <IMG
 WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img20.png"
 ALT="$x$"> are initialized to <!-- MATH
 $ix(i) \leftarrow
  i$
 -->
<IMG
 WIDTH="66" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img118.png"
 ALT="$ix(i) \leftarrow
i$">; thus, upon return from the subroutine, for each
  index <IMG
 WIDTH="9" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="img4.png"
 ALT="$i$"> we have in <IMG
 WIDTH="37" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img119.png"
 ALT="$ix(i)$"> the position that the item <IMG
 WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img120.png"
 ALT="$x(i)$">
  occupied in the original data sequence;
</LI>
<LI>If <!-- MATH
 $flag = psb\_sort\_keep\_idx\_$
 -->
<IMG
 WIDTH="185" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img121.png"
 ALT="$flag = psb\_sort\_keep\_idx\_$"> the routine will assume that
  the entries in <IMG
 WIDTH="35" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img122.png"
 ALT="$ix(:)$"> have already been initialized by the user;
</LI>
<LI>The three sorting algorithms have a similar <IMG
 WIDTH="74" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img123.png"
 ALT="$O(n \log n)$"> expected
  running time; in the average case quicksort will be the
  fastest and merge-sort the slowest. However note that:

<OL>
<LI>The worst case running time for quicksort is <IMG
 WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img124.png"
 ALT="$O(n^2)$">; the algorithm
  implemented here follows the well-known median-of-three heuristics,
  but the worst case may still apply;
</LI>
<LI>The worst case running time for merge-sort and heap-sort is
  <IMG
 WIDTH="74" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img123.png"
 ALT="$O(n \log n)$"> as the average case;
</LI>
<LI>The merge-sort algorithm is implemented to take advantage of 
  subsequences that may be already in the desired ordering prior to
  the subroutine call; this situation is relatively common when
  dealing with groups of indices of sparse matrix entries, thus
  merge-sort is the preferred  choice when a sorting is needed
  by other routines in the library. 
</LI>
</OL>
</LI>
</OL>

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