diff --git a/docs/html/userhtmlse6.html b/docs/html/userhtmlse6.html
index 7d23f86f..f5be71d9 100644
--- a/docs/html/userhtmlse6.html
+++ b/docs/html/userhtmlse6.html
@@ -3872,72 +3872,76 @@ class="cmtt-10">integer(psb_long_int_k_) number.
id="x12-1050006.27">Sorting utilities —
psb_msort — Sorting by the Merge-sort algorithm
+class="pplb7t-x-x-120">_isort — Sorting by the Insertion-sort algorithm
psb_qsort — Sorting by the Quicksort algorithm
+class="pplb7t-x-x-120">_msort — Sorting by the Merge-sort algorithm
psb_qsort — Sorting by the Quicksort algorithm
+
psb_hsort — Sorting by the Heapsort algorithm
+call psb_isort(x,ix,dir,flag)
call psb_msort(x,ix,dir,flag)
call psb_qsort(x,ix,dir,flag)
call psb_hsort(x,ix,dir,flag)
-
-
These serial routines sort a sequence
+
These serial routines sort a sequence 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).
+argument meaning is identical for all calls; the only difference is the algorithm used
+to accomplish the task (see Usage Notes below).
-
-
+
Type:
-
-
Asynchronous.
+
Asynchronous.
-
-
+
On Entry
-
-
+
-
-
+
x
-
-
The sequence to be sorted.
The sequence to be sorted.
Type:required.
Specified as: an integer, real or complex array of rank 1.
-
-
+
ix
-
-
A vector of indices.
A vector of indices.
Type:optional.
Specified as: an integer array of (at least) the same size as X.
-
-
+
dir
-
-
The desired ordering.
The desired ordering.
Type:optional.
Specified as: an integer value:
-
-
+
Integer and real data:
-
-
psb_sort_up_, psb_sort_down_, psb_asort_up_,
@@ -3948,11 +3952,11 @@ class="cmtt-10">psb_sort_up_
.
-
-
+
Complex data:
-
-
psb_lsort_up_, psb_lsort_down_, psb_asort_up_,
@@ -3960,11 +3964,11 @@ class="cmtt-10">psb_asort_up_
,
class="cmtt-10">psb_asort_down_
; default psb_lsort_up_.
-
+
flag
- Whether to keep the original values in Whether to keep the original values in IX.
Type:optional.
psb_sort_keep_idx_
;
default psb_sort_ovw_idx_.
-
+
-
-
+
On Return
-
-
+
-
-
+
x
-
-
The sequence of values, in the chosen ordering.
The sequence of values, in the chosen ordering.
Type:required.
Specified as: an integer, real or complex array of rank 1.
-
-
+
ix
-
-
A vector of indices.
A vector of indices.
Type: Optional
An integer array of rank 1, whose entries are moved to the same position
as the corresponding entries in x.
-Notes
-
-
For integer or real data the sorting can be performed in the up/down
+
For integer or real data the sorting can be performed in the up/down
direction, on the natural or absolute values;
-
-
For complex data the sorting can be done in a lexicographic order (i.e.:
+
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;
@@ -4020,15 +4024,15 @@ class="pplb7t-x-x-120">Notes
- The routines return the items in the chosen ordering; the output
+
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;
+ original input. With the insertion-sort or merge-sort algorithms ties are
+ preserved in the same relative order as they had in the original sequence,
+ while this is not guaranteed for quicksort or heapsort;
- If If flag = psb_) occupied
- If If flag = psb_(:) have already been initialized by the user;
- The three sorting algorithms have a similar Three sorting algorithms have a similar O(n log n) expected running time;
+class="zplmr7t-">)
expected running time,
+ whereas insertion sort has an expected running time O(n2); of the other three,
in the average case quicksort will be the fastest and merge-sort the slowest.
However note that:
-
-
The worst case running time
+
The the best case running time for insertion sort is Ω(n) while the
+ average and worst case are O(n2); however for very short input
+ sequences this is likely to be the fastest method;
+
+-
+
The worst case running time
for quicksort is O(); the algorithm implemented here follows the
apply;
-
-
The worst case running time for merge-sort and heap-sort is
+ class="enumerate" id="x12-105018x3">
+
The worst case running time for merge-sort and heap-sort is
O(n log n) as the average case;
+
+
+
-
-
The merge-sort algorithm is implemented to take advantage of
+ class="enumerate" id="x12-105020x4">
+
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.
diff --git a/docs/psblas-3.9.pdf b/docs/psblas-3.9.pdf
index 6f6505c6..a0d16c37 100644
--- a/docs/psblas-3.9.pdf
+++ b/docs/psblas-3.9.pdf
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diff --git a/docs/src/toolsrout.tex b/docs/src/toolsrout.tex
index d026f585..d58faadc 100644
--- a/docs/src/toolsrout.tex
+++ b/docs/src/toolsrout.tex
@@ -1780,6 +1780,9 @@ Returned as: an \verb|integer(psb_long_int_k_)| number.
\clearpage\subsection{Sorting utilities --- }
%\addcontentsline{toc}{subsection}{Sorting utilities}
+{\par\noindent\large\bfseries psb\_isort --- Sorting by the Insertion-sort
+ algorithm}
+
{\par\noindent\large\bfseries psb\_msort --- Sorting by the Merge-sort
algorithm}
@@ -1788,13 +1791,14 @@ Returned as: an \verb|integer(psb_long_int_k_)| number.
{\par\noindent\large\bfseries psb\_hsort --- Sorting by the Heapsort algorithm}
\begin{verbatim}
+call psb_isort(x,ix,dir,flag)
call psb_msort(x,ix,dir,flag)
call psb_qsort(x,ix,dir,flag)
call psb_hsort(x,ix,dir,flag)
\end{verbatim}
These serial routines sort a sequence $X$ into ascending or
-descending order. The argument meaning is identical for the three
+descending order. The argument meaning is identical for all
calls; the only difference is the algorithm used to accomplish the
task (see Usage Notes below).
\begin{description}
@@ -1843,7 +1847,7 @@ position as the corresponding entries in $x$.
the imaginary part) or on the absolute values;
\item 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
+ equal value) in the original input. With the insertion-sort or merge-sort algorithms
ties are preserved in the same relative order as they had in the
original sequence, while this is not guaranteed for quicksort or
heapsort;
@@ -1854,11 +1858,15 @@ position as the corresponding entries in $x$.
occupied in the original data sequence;
\item If $flag = psb\_sort\_keep\_idx\_$ the routine will assume that
the entries in $ix(:)$ have already been initialized by the user;
-\item The three sorting algorithms have a similar $O(n \log n)$ expected
- running time; in the average case quicksort will be the
+\item Three sorting algorithms have a similar $O(n \log n)$ expected
+ running time, whereas insertion sort has an expected running time
+ $O(n^2)$; of the other three, in the average case quicksort will be the
fastest and merge-sort the slowest. However note that:
-\begin{enumerate}
-\item The worst case running time for quicksort is $O(n^2)$; the algorithm
+ \begin{enumerate}
+ \item The the best case running time for insertion sort is $\Omega(n)$ while the average
+ and worst case are $O(n^2)$; however for very short input sequences this is
+ likely to be the fastest method;
+ \item The worst case running time for quicksort is $O(n^2)$; the algorithm
implemented here follows the well-known median-of-three heuristics,
but the worst case may still apply;
\item The worst case running time for merge-sort and heap-sort is