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<h4 class="subsectionHead"><span class="titlemark">6.27 </span> <a
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id="x67-1040006.27"></a>Sorting utilities — </h4>
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<!--l. 1753--><p class="noindent" ><span
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class="cmbx-12">psb</span><span
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class="cmbx-12">_msort — Sorting by the Merge-sort algorithm</span>
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<!--l. 1756--><p class="noindent" ><span
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class="cmbx-12">psb</span><span
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class="cmbx-12">_qsort — Sorting by the Quicksort algorithm</span>
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<!--l. 1759--><p class="noindent" ><span
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class="cmbx-12">psb</span><span
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class="cmbx-12">_hsort — Sorting by the Heapsort algorithm</span>
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<pre class="verbatim" id="verbatim-60">
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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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<!--l. 1764--><p class="nopar" >
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<!--l. 1766--><p class="indent" > These serial routines sort a sequence <span
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class="cmmi-10">X </span>into ascending or descending order. The
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argument meaning is identical for the three calls; the only difference is the algorithm
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used to accomplish the task (see Usage Notes below).
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<dl class="description"><dt class="description">
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<span
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class="cmbx-10">Type:</span> </dt><dd
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class="description">Asynchronous.
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</dd><dt class="description">
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<span
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class="cmbx-10">On Entry</span> </dt><dd
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class="description">
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</dd><dt class="description">
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<span
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class="cmbx-10">x</span> </dt><dd
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class="description">The sequence to be sorted.<br
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class="newline" />Type:<span
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class="cmbx-10">required</span>.<br
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class="newline" />Specified as: an integer, real or complex array of rank 1.
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</dd><dt class="description">
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<span
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class="cmbx-10">ix</span> </dt><dd
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class="description">A vector of indices.<br
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class="newline" />Type:<span
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class="cmbx-10">optional</span>.<br
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class="newline" />Specified as: an integer array of (at least) the same size as <span
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class="cmmi-10">X</span>.
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</dd><dt class="description">
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<span
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class="cmbx-10">dir</span> </dt><dd
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class="description">The desired ordering.<br
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class="newline" />Type:<span
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class="cmbx-10">optional</span>.<br
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class="newline" />Specified as: an integer value:
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<dl class="description"><dt class="description">
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<span
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class="cmbx-10">Integer and real data:</span> </dt><dd
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class="description"><span class="obeylines-h"><span class="verb"><span
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class="cmtt-10">psb_sort_up_</span></span></span>, <span class="obeylines-h"><span class="verb"><span
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class="cmtt-10">psb_sort_down_</span></span></span>,
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<span class="obeylines-h"><span class="verb"><span
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class="cmtt-10">psb_asort_up_</span></span></span>, <span class="obeylines-h"><span class="verb"><span
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class="cmtt-10">psb_asort_down_</span></span></span>; default <span class="obeylines-h"><span class="verb"><span
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class="cmtt-10">psb_sort_up_</span></span></span>.
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</dd><dt class="description">
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<span
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class="cmbx-10">Complex data:</span> </dt><dd
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class="description"><span class="obeylines-h"><span class="verb"><span
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class="cmtt-10">psb_lsort_up_</span></span></span>, <span class="obeylines-h"><span class="verb"><span
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class="cmtt-10">psb_lsort_down_</span></span></span>, <span class="obeylines-h"><span class="verb"><span
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class="cmtt-10">psb_asort_up_</span></span></span>,
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<span class="obeylines-h"><span class="verb"><span
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class="cmtt-10">psb_asort_down_</span></span></span>; default <span class="obeylines-h"><span class="verb"><span
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class="cmtt-10">psb_lsort_up_</span></span></span>.</dd></dl>
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</dd><dt class="description">
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<span
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class="cmbx-10">flag</span> </dt><dd
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class="description">Whether to keep the original values in <span
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class="cmmi-10">IX</span>.<br
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class="newline" />Type:<span
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class="cmbx-10">optional</span>.<br
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class="newline" />Specified as: an integer value <span class="obeylines-h"><span class="verb"><span
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class="cmtt-10">psb_sort_ovw_idx_</span></span></span> or <span class="obeylines-h"><span class="verb"><span
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class="cmtt-10">psb_sort_keep_idx_</span></span></span>;
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default <span class="obeylines-h"><span class="verb"><span
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class="cmtt-10">psb_sort_ovw_idx_</span></span></span>.
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</dd></dl>
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<!--l. 1796--><p class="indent" >
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<dl class="description"><dt class="description">
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<span
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class="cmbx-10">On Return</span> </dt><dd
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class="description">
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</dd><dt class="description">
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<span
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class="cmbx-10">x</span> </dt><dd
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class="description">The sequence of values, in the chosen ordering.<br
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class="newline" />Type:<span
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class="cmbx-10">required</span>.<br
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class="newline" />Specified as: an integer, real or complex array of rank 1.
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</dd><dt class="description">
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<span
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class="cmbx-10">ix</span> </dt><dd
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class="description">A vector of indices.<br
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class="newline" />Type: <span
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class="cmbx-10">Optional </span><br
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class="newline" />An integer array of rank 1, whose entries are moved to the same position
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as the corresponding entries in <span
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class="cmmi-10">x</span>.</dd></dl>
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<!--l. 1807--><p class="noindent" ><span
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class="cmbx-12">Notes</span>
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<ol class="enumerate1" >
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<li
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class="enumerate" id="x67-104002x1">For integer or real data the sorting can be performed in the up/down
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direction, on the natural or absolute values;
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</li>
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<li
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class="enumerate" id="x67-104004x2">For complex data the sorting can be done in a lexicographic order (i.e.:
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sort on the real part with ties broken according to the imaginary part) or
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on the absolute values;
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</li>
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<li
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class="enumerate" id="x67-104006x3">The routines return the items in the chosen ordering; the output difference
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is the handling of ties (i.e. items with an equal value) in the original input.
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With the merge-sort algorithm ties are preserved in the same relative
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order as they had in the original sequence, while this is not guaranteed for
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quicksort or heapsort;
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</li>
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<li
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class="enumerate" id="x67-104008x4">If <span
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class="cmmi-10">flag </span>= <span
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class="cmmi-10">psb</span>_<span
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class="cmmi-10">sort</span>_<span
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class="cmmi-10">ovw</span>_<span
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class="cmmi-10">idx</span>_ then the entries in <span
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class="cmmi-10">ix</span>(1 : <span
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class="cmmi-10">n</span>) where <span
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class="cmmi-10">n </span>is the size
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of <span
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class="cmmi-10">x </span>are initialized to <span
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class="cmmi-10">ix</span>(<span
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class="cmmi-10">i</span>) <span
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class="cmsy-10">← </span><span
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class="cmmi-10">i</span>; thus, upon return from the subroutine,
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for each index <span
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class="cmmi-10">i </span>we have in <span
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class="cmmi-10">ix</span>(<span
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class="cmmi-10">i</span>) the position that the item <span
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class="cmmi-10">x</span>(<span
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class="cmmi-10">i</span>) occupied
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in the original data sequence;
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</li>
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<li
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class="enumerate" id="x67-104010x5">If <span
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class="cmmi-10">flag </span>= <span
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class="cmmi-10">psb</span>_<span
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class="cmmi-10">sort</span>_<span
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class="cmmi-10">keep</span>_<span
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class="cmmi-10">idx</span>_ the routine will assume that the entries in
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<span
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class="cmmi-10">ix</span>(:) have already been initialized by the user;
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</li>
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<li
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class="enumerate" id="x67-104012x6">The three sorting algorithms have a similar <span
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class="cmmi-10">O</span>(<span
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class="cmmi-10">n</span>log <span
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class="cmmi-10">n</span>) expected running time;
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in the average case quicksort will be the fastest and merge-sort the slowest.
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However note that:
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<ol class="enumerate2" >
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<li
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class="enumerate" id="x67-104014x1">The worst case running time for quicksort is <span
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class="cmmi-10">O</span>(<span
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class="cmmi-10">n</span><sup><span
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class="cmr-7">2</span></sup>); 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
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class="enumerate" id="x67-104016x2">The worst case running time for merge-sort and heap-sort is
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<span
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class="cmmi-10">O</span>(<span
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class="cmmi-10">n</span>log <span
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class="cmmi-10">n</span>) as the average case;
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</li>
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<li
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class="enumerate" id="x67-104018x3">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 dealing
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with groups of indices of sparse matrix entries, thus merge-sort is the
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preferred choice when a sorting is needed by other routines in the
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library.</li></ol>
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</li></ol>
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