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

Author: frederick-vs-ja

source/algorithms.tex

@@ -3647,7 +3647,7 @@
 
 \pnum
 \complexity
-Applies \tcode{f} exactly \tcode{last - first} times.
+Applies \tcode{f} at most \tcode{last - first} times.
 
 \pnum
 \remarks
@@ -3678,7 +3678,7 @@
 
 \pnum
 \complexity
-Applies \tcode{f} exactly \tcode{last - first} times.
+Applies \tcode{f} at most \tcode{last - first} times.
 
 \pnum
 \remarks
@@ -3723,7 +3723,7 @@
 
 \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
@@ -4188,7 +4188,7 @@
 \pnum
 \complexity
 For the overloads with no \tcode{ExecutionPolicy},
-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 overloads with an \tcode{ExecutionPolicy},
@@ -4262,7 +4262,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{last - first} applications
+At most \tcode{last - first} applications
 of the corresponding predicate and any projection.
 \end{itemdescr}
 
@@ -4536,7 +4536,7 @@
 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}.
@@ -4586,7 +4586,7 @@
 \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};
@@ -5072,7 +5072,7 @@
 
 \pnum
 \complexity
-Exactly $N$ assignments.
+At most $N$ assignments.
 \end{itemdescr}
 
 \indexlibraryglobal{copy}%
@@ -5102,7 +5102,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{last - first} assignments.
+At most \tcode{last - first} assignments.
 \end{itemdescr}
 
 \indexlibraryglobal{copy_n}%
@@ -5148,7 +5148,7 @@
 
 \pnum
 \complexity
-Exactly $N$ assignments.
+At most $N$ assignments.
 \end{itemdescr}
 
 \indexlibraryglobal{copy_if}%
@@ -5218,7 +5218,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{last - first} applications
+At most \tcode{last - first} applications
 of the corresponding predicate and any projection.
 
 \pnum
@@ -5277,7 +5277,7 @@
 
 \pnum
 \complexity
-Exactly $N$ assignments.
+At most $N$ assignments.
 \end{itemdescr}
 
 \rSec2[alg.move]{Move}
@@ -5335,7 +5335,7 @@
 
 \pnum
 \complexity
-Exactly $N$ assignments.
+At most $N$ assignments.
 \end{itemdescr}
 
 \indexlibrary{\idxcode{move}!algorithm}%
@@ -5368,7 +5368,7 @@
 
 \pnum
 \complexity
-Exactly $N$ assignments.
+At most $N$ assignments.
 \end{itemdescr}
 
 \indexlibraryglobal{move_backward}%
@@ -5430,7 +5430,7 @@
 
 \pnum
 \complexity
-Exactly $N$ assignments.
+At most $N$ assignments.
 \end{itemdescr}
 
 \rSec2[alg.swap]{Swap}
@@ -5500,7 +5500,7 @@
 
 \pnum
 \complexity
-Exactly $M$ swaps.
+At most $M$ swaps.
 \end{itemdescr}
 
 \indexlibraryglobal{iter_swap}%
@@ -5642,7 +5642,7 @@
 
 \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 overload with an \tcode{ExecutionPolicy}.
 
@@ -5726,7 +5726,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{last - first} applications
+At most \tcode{last - first} applications
 of the corresponding predicate and any projection.
 \end{itemdescr}
 
@@ -5837,7 +5837,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{last - first} applications
+At most \tcode{last - first} applications
 of the corresponding predicate and any projection.
 \end{itemdescr}
 
@@ -5894,7 +5894,7 @@
 
 \pnum
 \complexity
-Exactly $N$ assignments.
+At most $N$ assignments.
 \end{itemdescr}
 
 \rSec2[alg.generate]{Generate}
@@ -5948,7 +5948,7 @@
 
 \pnum
 \complexity
-Exactly $N$ evaluations of \tcode{gen()} and assignments.
+At most $N$ evaluations of \tcode{gen()} and assignments.
 \end{itemdescr}
 
 \rSec2[alg.remove]{Remove}
@@ -6024,7 +6024,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{last - first} applications
+At most \tcode{last - first} applications
 of the corresponding predicate and any projection.
 
 \pnum
@@ -6134,7 +6134,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{last - first} applications
+At most \tcode{last - first} applications
 of the corresponding predicate and any projection.
 
 \pnum
@@ -6209,7 +6209,7 @@
 
 \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}
@@ -6324,7 +6324,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{last - first - 1} applications
+At most \tcode{last - first - 1} applications
 of the corresponding predicate
 and no more than twice as many applications of any projection.
 \end{itemdescr}
@@ -6367,7 +6367,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{(last - first)/2} swaps.
+At most \tcode{(last - first)/2} swaps.
 \end{itemdescr}
 
 \indexlibraryglobal{reverse_copy}%
@@ -6419,7 +6419,7 @@
 
 \pnum
 \complexity
-Exactly $N$ assignments.
+At most $N$ assignments.
 \end{itemdescr}
 
 \rSec2[alg.rotate]{Rotate}
@@ -6533,7 +6533,7 @@
 
 \pnum
 \complexity
-Exactly $N$ assignments.
+At most $N$ assignments.
 \end{itemdescr}
 
 \begin{itemdecl}
@@ -6680,7 +6680,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{(last - first) - 1} swaps.
+At most \tcode{(last - first) - 1} swaps.
 
 \pnum
 \remarks
@@ -7759,7 +7759,7 @@
 \begin{itemize}
 \item
   For the overload with no \tcode{ExecutionPolicy},
-  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
@@ -7835,7 +7835,7 @@
 \item
   For the overloads with no \tcode{ExecutionPolicy}, 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 overload with an \tcode{ExecutionPolicy},
   \bigoh{N \log N} swaps and \bigoh{N} applications of the predicate.
@@ -7912,7 +7912,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{last - first} applications of \tcode{pred} and \tcode{proj}.
+At most \tcode{last - first} applications of \tcode{pred} and \tcode{proj}.
 \end{itemdescr}
 
 \indexlibraryglobal{partition_point}%
@@ -8117,7 +8117,7 @@
 \begin{itemize}
 \item
   For the overloads with no \tcode{ExecutionPolicy}, and
-  if enough additional memory is available, exactly $N - 1$ comparisons.
+  if enough additional memory is available, at most $N - 1$ comparisons.
 \item
   Otherwise, \bigoh{N \log N} comparisons.
 \end{itemize}
@@ -8998,7 +8998,7 @@
 
 \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
@@ -9041,7 +9041,7 @@
 
 \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
@@ -9076,7 +9076,7 @@
 
 \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
@@ -9119,7 +9119,7 @@
 
 \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
@@ -9156,7 +9156,7 @@
 
 \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
@@ -9256,7 +9256,7 @@
 
 \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}
 
@@ -9303,7 +9303,7 @@
 
 \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}
 
@@ -10334,7 +10334,7 @@
 
 \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
@@ -10846,7 +10846,7 @@
 
 \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
@@ -10881,7 +10881,7 @@
 
 \pnum
 \complexity
-Exactly \tcode{last - first} increments and assignments.
+At most \tcode{last - first} increments and assignments.
 \end{itemdescr}
 
 \indexlibraryglobal{iota}%