[algorithms, numerics] Replace "exactly" with "at most" in Complexity

Author: frederick-vs-ja

source/algorithms.tex

@@ -4640,7 +4640,7 @@
 
 \pnum
 \complexity
-Applies \tcode{f} exactly \tcode{last - first} times.
+Applies \tcode{f} at most \tcode{last - first} times.
 
 \pnum
 \remarks
@@ -4671,7 +4671,7 @@
 
 \pnum
 \complexity
-Applies \tcode{f} exactly \tcode{last - first} times.
+Applies \tcode{f} at most \tcode{last - first} times.
 
 \pnum
 \remarks
@@ -4716,7 +4716,7 @@
 
 \pnum
 \complexity
-Applies \tcode{f} and \tcode{proj} exactly \tcode{last - first} times.
+Applies \tcode{f} and \tcode{proj} at most \tcode{last - first} times.
 
 \pnum
 \remarks
@@ -5368,7 +5368,7 @@
 \pnum
 \complexity
 For the non-parallel algorithm overloads,
-exactly \[ \min(\tcode{(i - first) + 1}, \ \tcode{(last - first) - 1}) \]
+at most \[ \min(\tcode{(i - first) + 1}, \ \tcode{(last - first) - 1}) \]
 applications of the corresponding predicate,
 where \tcode{i} is \tcode{adjacent_find}'s return value.
 For the parallel algorithm overloads,
@@ -5463,7 +5463,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{last - first} applications
+At most \tcode{last - first} applications
 of the corresponding predicate and any projection.
 \end{itemdescr}
 
@@ -5755,7 +5755,7 @@
 if \tcode{ForwardIterator1} and \tcode{Forward\-Iter\-ator2}
 meet the requirements of random access iterators and
 \tcode{last1 - first1 != last2 - first2}.
-Otherwise, exactly \tcode{last1 - first1} applications
+Otherwise, at most \tcode{last1 - first1} applications
 of the corresponding predicate
 if \tcode{equal(first1, last1, first2, last2, pred)} would return \tcode{true};
 otherwise, at worst \bigoh{N^2}, where $N$ has the value \tcode{last1 - first1}.
@@ -5805,7 +5805,7 @@
 \tcode{R1} and \tcode{R2} each model \libconcept{sized_range}, and
 \tcode{ranges::distance(r1) != ranges::distance(r2)}.
 \end{itemize}
-Otherwise, exactly \tcode{last1 - first1} applications
+Otherwise, at most \tcode{last1 - first1} applications
 of the corresponding predicate and projections
 if \tcode{ranges::equal(\brk{}first1, last1, first2, last2, pred, proj1, proj2)}
 would return \tcode{true};
@@ -6390,7 +6390,7 @@
 
 \pnum
 \complexity
-Exactly $N$ assignments.
+At most $N$ assignments.
 \end{itemdescr}
 
 \indexlibraryglobal{copy}%
@@ -6442,7 +6442,7 @@
 
 \pnum
 \complexity
-Exactly $N$ assignments.
+At most $N$ assignments.
 \end{itemdescr}
 
 \indexlibraryglobal{copy_n}%
@@ -6501,7 +6501,7 @@
 
 \pnum
 \complexity
-Exactly $N$ assignments.
+At most $N$ assignments.
 \end{itemdescr}
 
 \indexlibraryglobal{copy_if}%
@@ -6666,7 +6666,7 @@
 
 \pnum
 \complexity
-Exactly $N$ assignments.
+At most $N$ assignments.
 \end{itemdescr}
 
 \rSec2[alg.move]{Move}
@@ -6724,7 +6724,7 @@
 
 \pnum
 \complexity
-Exactly $N$ assignments.
+At most $N$ assignments.
 \end{itemdescr}
 
 \indexlibrary{\idxcode{move}!algorithm}%
@@ -6789,7 +6789,7 @@
 
 \pnum
 \complexity
-Exactly $N$ assignments.
+At most $N$ assignments.
 \end{itemdescr}
 
 \indexlibraryglobal{move_backward}%
@@ -6851,7 +6851,7 @@
 
 \pnum
 \complexity
-Exactly $N$ assignments.
+At most $N$ assignments.
 \end{itemdescr}
 
 \rSec2[alg.swap]{Swap}
@@ -6932,7 +6932,7 @@
 
 \pnum
 \complexity
-Exactly $M$ swaps.
+At most $M$ swaps.
 \end{itemdescr}
 
 \indexlibraryglobal{iter_swap}%
@@ -7119,7 +7119,7 @@
 
 \pnum
 \complexity
-Exactly $N$ applications of \tcode{op} or \tcode{binary_op}, and
+At most $N$ applications of \tcode{op} or \tcode{binary_op}, and
 any projections.
 This requirement also applies to the parallel algorithm overloads.
 
@@ -7231,7 +7231,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{last - first} applications
+At most \tcode{last - first} applications
 of the corresponding predicate and any projection.
 \end{itemdescr}
 
@@ -7389,7 +7389,7 @@
 
 \pnum
 \complexity
-Exactly $N$ applications
+At most $N$ applications
 of the corresponding predicate and any projection.
 \end{itemdescr}
 
@@ -7457,7 +7457,7 @@
 
 \pnum
 \complexity
-Exactly $N$ assignments.
+At most $N$ assignments.
 \end{itemdescr}
 
 \rSec2[alg.generate]{Generate}
@@ -7522,7 +7522,7 @@
 
 \pnum
 \complexity
-Exactly $N$ evaluations of \tcode{gen()} and assignments.
+At most $N$ evaluations of \tcode{gen()} and assignments.
 
 \pnum
 \remarks
@@ -7627,7 +7627,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{last - first} applications
+At most \tcode{last - first} applications
 of the corresponding predicate and any projection.
 
 \pnum
@@ -7878,7 +7878,7 @@
 
 \pnum
 \complexity
-For nonempty ranges, exactly \tcode{(last - first) - 1} applications
+For nonempty ranges, at most \tcode{(last - first) - 1} applications
 of the corresponding predicate and
 no more than twice as many applications of any projection.
 \end{itemdescr}
@@ -8086,7 +8086,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{(last - first)/2} swaps.
+At most \tcode{(last - first)/2} swaps.
 \end{itemdescr}
 
 \indexlibraryglobal{reverse_copy}%
@@ -8138,7 +8138,7 @@
 
 \pnum
 \complexity
-Exactly $N$ assignments.
+At most $N$ assignments.
 \end{itemdescr}
 
 \begin{itemdecl}
@@ -8178,7 +8178,7 @@
 
 \pnum
 \complexity
-Exactly $N$ assignments.
+At most $N$ assignments.
 \end{itemdescr}
 
 \rSec2[alg.rotate]{Rotate}
@@ -8310,7 +8310,7 @@
 
 \pnum
 \complexity
-Exactly $N$ assignments.
+At most $N$ assignments.
 \end{itemdescr}
 
 \begin{itemdecl}
@@ -8354,7 +8354,7 @@
 
 \pnum
 \complexity
-Exactly $N$ assignments.
+At most $N$ assignments.
 \end{itemdescr}
 
 \begin{itemdecl}
@@ -8518,7 +8518,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{(last - first) - 1} swaps.
+At most \tcode{(last - first) - 1} swaps.
 
 \pnum
 \remarks
@@ -9742,7 +9742,7 @@
 \begin{itemize}
 \item
   For the non-parallel algorithm overloads,
-  exactly $N$ applications of the predicate and projection.
+  at most $N$ applications of the predicate and projection.
   At most $N / 2$ swaps if the type of \tcode{first} meets
   the \oldconcept{BidirectionalIterator} requirements
   for the overloads in namespace \tcode{std} or
@@ -9830,7 +9830,7 @@
   For the non-parallel algorithm overloads,
   at most $N \log_2 N$ swaps,
   but only \bigoh{N} swaps if there is enough extra memory.
-  Exactly $N$ applications of the predicate and projection.
+  At most $N$ applications of the predicate and projection.
 \item
   For the parallel algorithm overloads,
   \bigoh{N \log N} swaps and \bigoh{N} applications of the predicate.
@@ -11307,7 +11307,7 @@
 
 \pnum
 \complexity
-Exactly one comparison and two applications of the projection, if any.
+At most one comparison and two applications of the projection, if any.
 
 \pnum
 \remarks
@@ -11355,7 +11355,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{ranges::distance(r) - 1} comparisons
+At most \tcode{ranges::distance(r) - 1} comparisons
 and twice as many applications of the projection, if any.
 
 \pnum
@@ -11390,7 +11390,7 @@
 
 \pnum
 \complexity
-Exactly one comparison and two applications of the projection, if any.
+At most one comparison and two applications of the projection, if any.
 
 \pnum
 \remarks
@@ -11438,7 +11438,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{ranges::distance(r) - 1} comparisons
+At most \tcode{ranges::distance(r) - 1} comparisons
 and twice as many applications of the projection, if any.
 
 \pnum
@@ -11475,7 +11475,7 @@
 
 \pnum
 \complexity
-Exactly one comparison and two applications of the projection, if any.
+At most one comparison and two applications of the projection, if any.
 
 \pnum
 \remarks
@@ -11590,7 +11590,7 @@
 
 \pnum
 \complexity
-Exactly $\max(\tcode{last - first - 1}, 0)$ comparisons and
+At most $\max(\tcode{last - first - 1}, 0)$ comparisons and
 twice as many projections.
 \end{itemdescr}
 
@@ -11647,7 +11647,7 @@
 
 \pnum
 \complexity
-Exactly $\max(\tcode{last - first - 1}, 0)$ comparisons and
+At most $\max(\tcode{last - first - 1}, 0)$ comparisons and
 twice as many projections.
 \end{itemdescr}
 
@@ -12703,7 +12703,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{(last - first) - 1} applications of the binary operation.
+At most \tcode{(last - first) - 1} applications of the binary operation.
 
 \pnum
 \remarks
@@ -13215,7 +13215,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{(last - first) - 1} applications of the binary operation.
+At most \tcode{(last - first) - 1} applications of the binary operation.
 
 \pnum
 \remarks
@@ -13250,7 +13250,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{last - first} increments and assignments.
+At most \tcode{last - first} increments and assignments.
 \end{itemdescr}
 
 \indexlibraryglobal{iota}%

source/numerics.tex

@@ -3122,7 +3122,7 @@
 
 \pnum
 \complexity
-Exactly $n \cdot \tcode{r}$ invocations
+At most $n \cdot \tcode{r}$ invocations
  of \tcode{e}.
 \end{itemdescr}
 
@@ -4447,7 +4447,7 @@
 
 \pnum
 \complexity
-Exactly $k$ invocations of \tcode{g} per attempt.
+At most $k$ invocations of \tcode{g} per attempt.
 
 \pnum
 \begin{note}