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<H2><A NAME="SECTION000727000000000000000">
Sorting utilities -- </A>
<H2><A NAME="SECTION000726000000000000000">
psb_sizeof -- Memory occupation</A>
</H2>
<P>
<FONT SIZE="+1"><B>psb_msort -- Sorting by the Merge-sort
algorithm</B></FONT>
This function computes the memory occupation of a PSBLAS object.
<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)
isz = psb_sizeof(a)
isz = psb_sizeof(desc_a)
isz = psb_sizeof(prec)
</PRE>
<P>
These serial routines sort a sequence <IMG
WIDTH="19" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img115.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>
<DT><STRONG><B>On Entry</B></STRONG></DT>
<DD>
</DD>
<DT><STRONG>x</STRONG></DT>
<DD>The sequence to be sorted.
<DT><STRONG>a</STRONG></DT>
<DD>A sparse matrix
<IMG
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img1.png"
ALT="$A$">.
<BR>
Type:<B>required</B>.
Scope: <B>local</B>
<BR>
Specified as: an integer, real or complex array of rank 1.
Type: <B>required</B>
<BR>
Intent: <B>in</B>.
<BR>
Specified as: a structured data of type spdata<TT>psb_Tspmat_type</TT>.
</DD>
<DT><STRONG>ix</STRONG></DT>
<DD>A vector of indices.
<DT><STRONG>desc_a</STRONG></DT>
<DD>Communication descriptor.
<BR>
Type:<B>optional</B>.
Scope: <B>local</B>
<BR>
Specified as: an integer array of (at least) the same size as <IMG
WIDTH="19" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img115.png"
ALT="$X$">.
Type: <B>required</B>
<BR>
Intent: <B>in</B>.
<BR>
Specified as: a structured data of type descdata<TT>psb_desc_type</TT>.
</DD>
<DT><STRONG>dir</STRONG></DT>
<DD>The desired ordering.
<DT><STRONG>prec</STRONG></DT>
<DD>Scope: <B>local</B>
<BR>
Type:<B>optional</B>.
Type: <B>required</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="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img116.png"
ALT="$IX$">.
Intent: <B>in</B>.
<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>
Specified as: a preconditioner data structure precdata<TT>psb_prec_type</TT>.
</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.
<DT><STRONG>Function value</STRONG></DT>
<DD>The memory occupation of the object specified in
the calling sequence, in bytes.
<BR>
Type:<B>required</B>.
Scope: <B>local</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="13" ALIGN="BOTTOM" BORDER="0"
SRC="img20.png"
ALT="$x$">.
Returned as: an <code>integer(psb_long_int_k_)</code> number.
</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="181" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img117.png"
ALT="$flag = psb\_sort\_ovw\_idx\_$"> then the entries in <IMG
WIDTH="61" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img118.png"
ALT="$ix(1:n)$">
where <IMG
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img16.png"
ALT="$n$"> is the size of <IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img20.png"
ALT="$x$"> are initialized to <!-- MATH
$ix(i) \leftarrow
i$
-->
<IMG
WIDTH="67" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img119.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="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img120.png"
ALT="$ix(i)$"> the position that the item <IMG
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img121.png"
ALT="$x(i)$">
occupied in the original data sequence;
</LI>
<LI>If <!-- MATH
$flag = psb\_sort\_keep\_idx\_$
-->
<IMG
WIDTH="184" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img122.png"
ALT="$flag = psb\_sort\_keep\_idx\_$"> the routine will assume that
the entries in <IMG
WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img123.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="img124.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="46" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img125.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="img124.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>
<P>
<P>
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