Punctuation (missing comma after "however").

git-svn-id: https://scm.gforge.inria.fr/anonscm/svn/mpfr/trunk@14542 280ebfd0-de03-0410-8827-d642c229c3f4

2 files changed, 11 insertions, 11 deletions

diff --git a/doc/algorithms.tex b/doc/algorithms.tex
index 9b11d0491..e1ba990a7 100644
--- a/doc/algorithms.tex
+++ b/doc/algorithms.tex
@@ -954,7 +954,7 @@ which yields $X - \frac{1}{2} \cdot 2^{-n} \leq A^{-1/2} \leq
               X + 1.21 \cdot 2^{-n}$,
 thus the right bound is not a priori fulfilled.
 % (3*1.003*1.311^2+0.5)/8+0.5 <= 1.21
-However the only problematic case is $n=19$, which gives exactly
+However, the only problematic case is $n=19$, which gives exactly
 $(n+3)/2 = 11$, since for $12 \leq n \leq 18$,
 the error terms in $2^{-2h}$ are halved.
 An exhaustive search of all possible inputs for $h=11$ and $n=19$ gives
@@ -966,7 +966,7 @@ Thus as soon as $n \geq 2$, Eq.~(\ref{rec_sqrt_eq1}) is fulfilled.
 
 In summary, Algorithm $\texttt{ApproximateInverseSquareRoot}$ provides an
 approximation $X$ of $A^{-1/2}$ with an error of at most one ulp.
-However if the input $A$ was itself truncated at precision $\geq p$
+However, if the input $A$ was itself truncated at precision $\geq p$
 from an input $A_0$ ---
 for example when the output precision $p$ is less than the input precision ---
 then we have $|X - A^{-1/2}| \leq \ulp(X)$, and
@@ -3176,7 +3176,7 @@ $\rho_k = -a_k \sigma_{k-1} / (1 + a_k \sigma_{k-1})$,
 $\sigma_k = 1 / (1 + a_k \sigma_{k-1})$, and:
 \[ a_k = \frac{k (a-k)}{(x + 2k - 1 - a) (x + 2k + 1 - a)}. \]
 Gautschi proves that when  $x \geq 1/4$ and $a \leq x + 1/4$, then
-$|\rho_k| \leq 1$ for all $k \geq 1$. However this is not enough to bound
+$|\rho_k| \leq 1$ for all $k \geq 1$. However, this is not enough to bound
 the tail of the above series.
 
 We consider here the case where $a \leq -1$, thus $a_k \leq 0$
@@ -4511,7 +4511,7 @@ $$  \wprec = \precc + 1 + 4 + \lceil \log_2(4\kvar)\rceil + \cond - \Exp(\svar).
 The \texttt{mpfr\_get\_str} function with size $0$ and base $b$ chooses an
 output precision of $1 + \lceil e \log(2)/\log(b) \rceil$ for a precision
 of $e$ bits if $b$ is not a power of two \cite{Mat68}.
-However the code uses instead $1 + \lceil e y \rceil$, where $y$ is
+However, the code uses instead $1 + \lceil e y \rceil$, where $y$ is
 an upper $76$-bit approximation of $\log(2)/\log(b)$. When do both values
 differ?
 In the case $b=2^k$, the worst case is when the first output digit contains
diff --git a/doc/mpfr.texi b/doc/mpfr.texi
index 9a43535c6..6757c9f02 100644
--- a/doc/mpfr.texi
+++ b/doc/mpfr.texi
@@ -561,7 +561,7 @@ You should include that file in any program using the MPFR library:
 @end example
 
 @cindex @code{stdio.h}
-Note however that prototypes for MPFR functions with @code{FILE *} parameters
+Note, however, that prototypes for MPFR functions with @code{FILE *} parameters
 are provided only if @code{<stdio.h>} is included too (before @file{mpfr.h}):
 
 @example
@@ -3495,7 +3495,7 @@ the MPFR library (contents of the @file{PATCHES} file), separated by spaces.
 Note: If the program has been compiled with an older MPFR version and is
 dynamically linked with a new MPFR library version, the identifiers of the
 patches applied to the old (compile-time) MPFR version are not available
-(however this information should not have much interest in general).
+(however, this information should not have much interest in general).
 @end deftypefun
 
 @deftypefun int mpfr_buildopt_tls_p (void)
@@ -4000,7 +4000,7 @@ with @code{mpfr_custom_init_set} is undefined.
 @deftypefun void mpfr_custom_move (mpfr_t @var{x}, void *@var{new_position})
 Inform MPFR that the significand of @var{x} has moved due to a garbage collect
 and update its new position to @code{new_position}.
-However the application has to move the significand and the @code{mpfr_t}
+However, the application has to move the significand and the @code{mpfr_t}
 itself.
 The behavior of this function for any @code{mpfr_t} not initialized
 with @code{mpfr_custom_init_set} is undefined.
@@ -4145,7 +4145,7 @@ Warning!  A program assuming (intentionally or not) that
 it is built and run against MPFR@tie{}2.x.
 
 The rounding modes @code{GMP_RNDx} were renamed to @code{MPFR_RNDx}
-in MPFR@tie{}3.0. However the old names @code{GMP_RNDx} have been kept for
+in MPFR@tie{}3.0. However, the old names @code{GMP_RNDx} have been kept for
 compatibility (this might change in future versions), using:
 @example
 #define GMP_RNDN MPFR_RNDN
@@ -4154,7 +4154,7 @@ compatibility (this might change in future versions), using:
 #define GMP_RNDD MPFR_RNDD
 @end example
 The rounding mode ``round away from zero'' (@code{MPFR_RNDA}) was added in
-MPFR@tie{}3.0 (however no rounding mode @code{GMP_RNDA} exists).
+MPFR@tie{}3.0 (however, no rounding mode @code{GMP_RNDA} exists).
 Faithful rounding (@code{MPFR_RNDF}) was added in MPFR@tie{}4.0, but
 currently, it is partially supported.
 @c That's sufficient information for now. More should be said in future
@@ -4548,7 +4548,7 @@ In MPFR@tie{}2.x, if a macro @code{INTMAX_C} or @code{UINTMAX_C} was defined
 (e.g. when the @code{__STDC_CONSTANT_MACROS} macro had been defined
 before @code{<stdint.h>} or @code{<inttypes.h>} has been included),
 @code{intmax_t} was assumed to be defined.
-However this was not always the case (more precisely, @code{intmax_t}
+However, this was not always the case (more precisely, @code{intmax_t}
 can be defined only in the namespace @code{std}, as with Boost), so
 that compilations could fail.
 Thus the check for @code{INTMAX_C} or @code{UINTMAX_C} is now disabled for
@@ -4567,7 +4567,7 @@ namespace, though this is not clean.
 
 @end itemize
 
-The divide-by-zero exception is new in MPFR@tie{}3.1. However it should
+The divide-by-zero exception is new in MPFR@tie{}3.1. However, it should
 not introduce incompatible changes for programs that strictly follow
 the MPFR API since the exception can only be seen via new functions.