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<H2><A NAME="SECTION000727000000000000000">
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Sorting utilities -- </A>
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</H2>
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<P>
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<BIG CLASS="LARGE"><B>psb_msort -- Sorting by the Merge-sort
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algorithm</B></BIG>
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<P>
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<BIG CLASS="LARGE"><B>psb_qsort -- Sorting by the Quicksort
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algorithm</B></BIG>
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<P>
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<BIG CLASS="LARGE"><B>psb_hsort -- Sorting by the Heapsort algorithm</B></BIG>
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<PRE>
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call psb_msort(x,ix,dir,flag)
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call psb_qsort(x,ix,dir,flag)
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call psb_hsort(x,ix,dir,flag)
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</PRE>
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<P>
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These serial routines sort a sequence <SPAN CLASS="MATH"><IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img123.png"
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ALT="$X$"></SPAN> into ascending or
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descending order. The argument meaning is identical for the three
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calls; the only difference is the algorithm used to accomplish the
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task (see Usage Notes below).
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<DL>
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<DT><STRONG>Type:</STRONG></DT>
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<DD>Asynchronous.
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</DD>
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<DT><STRONG><B>On Entry </B></STRONG></DT>
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<DD>
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</DD>
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<DT><STRONG>x</STRONG></DT>
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<DD>The sequence to be sorted.
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<BR>
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Type:<B>required</B>.
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<BR>
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Specified as: an integer, real or complex array of rank 1.
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</DD>
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<DT><STRONG>ix</STRONG></DT>
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<DD>A vector of indices.
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<BR>
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Type:<B>optional</B>.
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<BR>
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Specified as: an integer array of (at least) the same size as <SPAN CLASS="MATH"><IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img123.png"
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ALT="$X$"></SPAN>.
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</DD>
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<DT><STRONG>dir</STRONG></DT>
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<DD>The desired ordering.
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<BR>
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Type:<B>optional</B>.
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<BR>
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Specified as: an integer value: <DL>
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<DT><STRONG>Integer and real data:</STRONG></DT>
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<DD><code>psb_sort_up_</code>,
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<code>psb_sort_down_</code>, <code>psb_asort_up_</code>, <code>psb_asort_down_</code>;
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default <code>psb_sort_up_</code>.
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</DD>
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<DT><STRONG>Complex data:</STRONG></DT>
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<DD><code>psb_lsort_up_</code>,
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<code>psb_lsort_down_</code>, <code>psb_asort_up_</code>, <code>psb_asort_down_</code>;
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default <code>psb_lsort_up_</code>.
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</DD>
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</DL>
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</DD>
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<DT><STRONG>flag</STRONG></DT>
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<DD>Whether to keep the original values in <SPAN CLASS="MATH"><IMG
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WIDTH="27" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img124.png"
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ALT="$IX$"></SPAN>.
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<BR>
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Type:<B>optional</B>.
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<BR>
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Specified as: an integer value <code>psb_sort_ovw_idx_</code> or
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<code>psb_sort_keep_idx_</code>; default <code>psb_sort_ovw_idx_</code>.
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<P>
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</DD>
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</DL>
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<P>
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<DL>
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<DT><STRONG><B>On Return</B></STRONG></DT>
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<DD>
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</DD>
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<DT><STRONG>x</STRONG></DT>
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<DD>The sequence of values, in the chosen ordering.
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<BR>
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Type:<B>required</B>.
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<BR>
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Specified as: an integer, real or complex array of rank 1.
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</DD>
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<DT><STRONG>ix</STRONG></DT>
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<DD>A vector of indices.
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<BR>
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Type: <B>Optional</B>
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<BR>
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An integer array of rank 1, whose entries are moved to the same
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position as the corresponding entries in <SPAN CLASS="MATH"><IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img20.png"
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ALT="$x$"></SPAN>.
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</DD>
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</DL>
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<P>
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<P>
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<BIG CLASS="LARGE"><B>Notes</B></BIG>
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<OL>
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<LI>For integer or real data the sorting can be performed in the up/down direction, on the
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natural or absolute values;
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</LI>
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<LI>For complex data the sorting can be done in a lexicographic
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order (i.e.: sort on the real part with ties broken according to
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the imaginary part) or on the absolute values;
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</LI>
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<LI>The routines return the items in the chosen ordering; the
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output difference is the handling of ties (i.e. items with an
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equal value) in the original input. With the merge-sort algorithm
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ties are preserved in the same relative order as they had in the
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original sequence, while this is not guaranteed for quicksort or
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heapsort;
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</LI>
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<LI>If <!-- MATH
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$flag = psb\_sort\_ovw\_idx\_$
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-->
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<SPAN CLASS="MATH"><IMG
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WIDTH="180" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img125.png"
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ALT="$flag = psb\_sort\_ovw\_idx\_$"></SPAN> then the entries in <SPAN CLASS="MATH"><IMG
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WIDTH="62" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img126.png"
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ALT="$ix(1:n)$"></SPAN>
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where <SPAN CLASS="MATH"><IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img16.png"
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ALT="$n$"></SPAN> is the size of <SPAN CLASS="MATH"><IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img20.png"
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ALT="$x$"></SPAN> are initialized to <!-- MATH
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$ix(i) \leftarrow
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i$
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-->
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<SPAN CLASS="MATH"><IMG
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WIDTH="66" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img127.png"
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ALT="$ix(i) \leftarrow
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i$"></SPAN>; thus, upon return from the subroutine, for each
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index <SPAN CLASS="MATH"><IMG
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WIDTH="9" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$i$"></SPAN> we have in <SPAN CLASS="MATH"><IMG
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WIDTH="37" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img128.png"
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ALT="$ix(i)$"></SPAN> the position that the item <SPAN CLASS="MATH"><IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img129.png"
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ALT="$x(i)$"></SPAN>
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occupied in the original data sequence;
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</LI>
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<LI>If <!-- MATH
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$flag = psb\_sort\_keep\_idx\_$
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-->
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<SPAN CLASS="MATH"><IMG
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WIDTH="185" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img130.png"
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ALT="$flag = psb\_sort\_keep\_idx\_$"></SPAN> the routine will assume that
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the entries in <SPAN CLASS="MATH"><IMG
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WIDTH="35" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img131.png"
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ALT="$ix(:)$"></SPAN> have already been initialized by the user;
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</LI>
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<LI>The three sorting algorithms have a similar <SPAN CLASS="MATH"><IMG
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WIDTH="74" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img132.png"
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ALT="$O(n \log n)$"></SPAN> expected
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running time; in the average case quicksort will be the
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fastest and merge-sort the slowest. However note that:
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<OL>
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<LI>The worst case running time for quicksort is <SPAN CLASS="MATH"><IMG
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WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img133.png"
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ALT="$O(n^2)$"></SPAN>; the algorithm
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implemented here follows the well-known median-of-three heuristics,
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but the worst case may still apply;
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</LI>
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<LI>The worst case running time for merge-sort and heap-sort is
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<SPAN CLASS="MATH"><IMG
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WIDTH="74" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img132.png"
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ALT="$O(n \log n)$"></SPAN> as the average case;
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</LI>
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<LI>The merge-sort algorithm is implemented to take advantage of
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subsequences that may be already in the desired ordering prior to
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the subroutine call; this situation is relatively common when
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dealing with groups of indices of sparse matrix entries, thus
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merge-sort is the preferred choice when a sorting is needed
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by other routines in the library.
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</LI>
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</OL>
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</LI>
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</OL>
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