===================================================================
RCS file: /home/cvs/OpenXM/src/asir-doc/parts/builtin/num.texi,v
retrieving revision 1.2
retrieving revision 1.13
diff -u -p -r1.2 -r1.13
--- OpenXM/src/asir-doc/parts/builtin/num.texi	1999/12/21 02:47:34	1.2
+++ OpenXM/src/asir-doc/parts/builtin/num.texi	2019/03/29 01:57:46	1.13
@@ -1,4 +1,4 @@
-@comment $OpenXM$
+@comment $OpenXM: OpenXM/src/asir-doc/parts/builtin/num.texi,v 1.12 2016/08/29 04:56:58 noro Exp $
 \BJP
 @node 数の演算,,, 組み込み函数
 @section 数の演算
@@ -13,17 +13,21 @@
 * fac::
 * igcd igcdcntl::
 * ilcm::
+* isqrt::
 * inv::
 * prime lprime::
 * random::
 * mt_save mt_load::
 * nm dn::
 * conj real imag::
-* eval::
+* eval deval::
 * pari::
-* setprec::
+* setbprec setprec::
 * setmod::
 * lrandom::
+* ntoint32 int32ton::
+* setround::
+* inttorat::
 @end menu
 
 \JP @node idiv irem,,, 数の演算
@@ -45,7 +49,7 @@
 @item return
 \JP 整数
 \EG integer
-@item i1,i2
+@item i1 i2
 \JP 整数
 \EG integer
 @end table
@@ -158,7 +162,7 @@ Returns 0 if the argument @var{i} is negative.
 @item return
 \JP 整数
 \EG integer
-@item i1,i2,i
+@item i1 i2 i
 \JP 整数
 \EG integer
 @end table
@@ -259,7 +263,7 @@ In most cases @code{3} is the fastest, but there are e
 @item return
 \JP 整数
 \EG integer
-@item i1,i2
+@item i1 i2
 \JP 整数
 \EG integer
 @end table
@@ -286,6 +290,26 @@ If one of argument is equal to 0, the return 0.
 @fref{igcd igcdcntl}, @fref{mt_save mt_load}.
 @end table
 
+\JP @node isqrt,,, 数の演算
+\EG @node isqrt,,, Numbers
+@subsection @code{isqrt}
+@findex isqrt
+
+@table @t
+@item isqrt(@var{n})
+\JP :: 平方根を越えない最大の整数を求める. 
+\EG :: The integer square root of @var{n}.
+@end table
+
+@table @var
+@item return
+\JP 非負整数
+\EG non-negative integer
+@item n
+\JP 非負整数
+\EG non-negative integer
+@end table
+
 \JP @node inv,,, 数の演算
 \EG @node inv,,, Numbers
 @subsection @code{inv}
@@ -301,7 +325,7 @@ If one of argument is equal to 0, the return 0.
 @item return
 \JP 整数
 \EG integer
-@item i,m
+@item i m
 \JP 整数
 \EG integer
 @end table
@@ -418,7 +442,7 @@ function
 @findex random
 
 @table @t
-@item radom([@var{seed}])
+@item random([@var{seed}])
 \JP :: 乱数を生成する. 
 @end table
 
@@ -481,7 +505,7 @@ one can trace a single random number sequence arcoss m
 @findex lrandom
 
 @table @t
-@item lradom(@var{bit})
+@item lrandom(@var{bit})
 \JP :: 多倍長乱数を生成する. 
 \EG :: Generates a long random number.
 @end table
@@ -692,14 +716,16 @@ These functions works also for polynomials with comple
 [2,11,(2-11*@@i)]
 @end example
 
-\JP @node eval,,, 数の演算
-\EG @node eval,,, Numbers
-@subsection @code{eval}
+\JP @node eval deval ,,, 数の演算
+\EG @node eval deval,,, Numbers
+@subsection @code{eval}, @code{deval}
 @findex eval
+@findex deval
 @cindex PARI
 
 @table @t
 @item eval(@var{obj}[,@var{prec}])
+@item deval(@var{obj})
 \JP :: @var{obj} の値の評価. 
 \EG :: Evaluate @var{obj} numerically.
 @end table
@@ -721,11 +747,18 @@ These functions works also for polynomials with comple
 @item
 @var{obj} に含まれる函数の値を可能な限り評価する. 
 @item
-計算は @b{PARI} (@xref{pari}) が行う. 
+@code{deval} は倍精度浮動小数を結果として 
+@code{eval} の場合, 有理数はそのまま残る. 
 @item
+@code{eval} においては, 計算は @b{MPFR} ライブラリが行う. 
+@code{deval} においては, 計算は C 数学ライブラリの関数を用いて行う. 
+@item
+@code{deval} は複素数は扱えない. 
+@item
+@code{eval} においては, 
 @var{prec} を指定した場合, 計算は, 10 進 @var{prec} 桁程度で行われる. 
 @var{prec} の指定がない場合, 現在設定されている精度で行われる. 
-(@xref{setprec})
+(@xref{setbprec setprec}.)
 @item
 @table @t
 @item 扱える函数は, 次の通り. 
@@ -740,7 +773,8 @@ These functions works also for polynomials with comple
 @code{exp}, @code{log}, @code{pow(a,b) (a^b)}
 @end table
 @item
-以下の記号を数として評価できる. 
+以下の記号を数として評価できる. ただし @code{@@i} を扱えるのは
+@code{eval}, @code{deval} のみである.
 @table @t
 @item @@i
 虚数単位
@@ -755,15 +789,21 @@ These functions works also for polynomials with comple
 Evaluates the value of the functions contained in @var{obj} as far as
 possible.
 @item
-The computation is done by @b{PARI} (@xref{pari}).
+@code{deval} returns
+double float. Rational numbers remain unchanged in results from @code{eval}.
 @item
+In @code{eval} the computation is done 
+by @b{MPFR} library. In @code{deval} the computation is
+done by the C math library.
+@item
+@code{deval} cannot handle complex numbers.
+@item
 When @var{prec} is specified, computation will be performed with a
 precision of about @var{prec}-digits.
 If @var{prec} is not specified, computation is performed with the
-precision set currently. (@xref{setprec})
+precision set currently. (@xref{setbprec setprec}.)
 @item
 Currently available numerical functions are listed below.
-Note they are only a small part of whole @b{PARI} functions.
 
 @table @t
 @code{sin}, @code{cos}, @code{tan},
@@ -776,7 +816,8 @@ Note they are only a small part of whole @b{PARI} func
 @code{exp}, @code{log}, @code{pow(a,b) (a^b)}
 @end table
 @item
-Symbols for special values are as the followings.
+Symbols for special values are as the followings. Note that
+@code{@@i} cannot be handled by @code{deval}.
 @table @t
 @item @@i
 unit of imaginary number
@@ -798,12 +839,16 @@ Napier's number (@t{exp}(1))
 0.86602540378443864674620506632
 [121] eval(sin(@@pi/3)-3^(1/2)/2,50);
 -2.78791084448179148471 E-58
+[122] eval(1/2);
+1/2
+[123] deval(sin(1)^2+cos(1)^2);
+1
 @end example
 
 @table @t
 \JP @item 参照
 \EG @item References
-@fref{ctrl}, @fref{setprec}, @fref{pari}.
+@fref{ctrl}, @fref{setbprec setprec}.
 @end table
 
 \JP @node pari,,, 数の演算
@@ -846,9 +891,7 @@ Napier's number (@t{exp}(1))
 函数値の評価を高速に行うことができる. @b{PARI} は他のプログラムから
 サブルーチンライブラリとして用いることができ, また, @samp{gp} という 
 @b{PARI}ライブラリのインタフェースにより UNIX のアプリケーションとして
-利用することもできる. 現在のバージョンは @b{2.0.17beta} でいくつかの ftp
-site (たとえば @code{ftp://megrez.ceremab.u-bordeaux.fr/pub/pari})
-から anonymous ftp できる. 
+利用することもできる. 
 @item
 最後の引数 @var{prec} で計算精度を指定できる.
 @var{prec} を省略した場合 @code{setprec()} で指定した精度となる. 
@@ -870,8 +913,6 @@ function evaluations as well as arithmetic operations 
 speed.  It can also be used from other external programs as a library.
 It provides a language interface named @samp{gp} to its library, which
 enables a user to use @b{PARI} as a calculator which runs on UNIX.
-The current version is @b{2.0.17beta}.  It can be obtained by several ftp
-sites. (For example, @code{ftp://megrez.ceremab.u-bordeaux.fr/pub/pari}.)
 @item
 The last argument (optional) @var{int} specifies the precision in digits
 for bigfloat operation.
@@ -960,7 +1001,6 @@ For details of individual functions, refer to the @b{P
 @code{lngamma},
 @code{logagm},
 @code{mat},
-@code{matinvr},
 @code{matrixqz2},
 @code{matrixqz3},
 @code{matsize},
@@ -1019,14 +1059,11 @@ For details of individual functions, refer to the @b{P
 
 \BJP
 @item
-@b{Asir} で用いているのは @b{PARI} のほんの一部の機能であるが, 今後
-より多くの機能が利用できるよう改良する予定である. 
+@b{Asir} で用いているのは @b{PARI} のほんの一部の機能である.
 \E
 \BEG
 @item
 @b{Asir} currently uses only a very small subset of @b{PARI}.
-We will improve @b{Asir} so that it can provide more functions of
-@b{PARI}.
 \E
 @end itemize
 
@@ -1045,19 +1082,20 @@ We will improve @b{Asir} so that it can provide more f
 @table @t
 \JP @item 参照
 \EG @item References
-@fref{setprec}.
+@fref{setbprec setprec}.
 @end table
 
-\JP @node setprec,,, 数の演算
-\EG @node setprec,,, Numbers
-@subsection @code{setprec}
+\JP @node setbprec setprec,,, 数の演算
+\EG @node setbprec setprec,,, Numbers
+@subsection @code{setbprec}, @code{setprec} 
+@findex setbprec
 @findex setprec
-@cindex PARI
 
 @table @t
-@item setprec([@var{n}])
-\JP :: @b{bigfloat} の桁数を @var{n} 桁に設定する. 
-\EG :: Sets the precision for @b{bigfloat} operations to @var{n} digits.
+@item setbprec([@var{n}])
+@itemx setprec([@var{n}])
+\JP :: @b{setbprec}, @b{setprec} は @b{bigfloat} の精度をそれぞれ 2 進, 10進 @var{n} 桁に設定する.
+\EG :: @b{setbprec}, @b{setprec} set the precision for @b{bigfloat} operations to @var{n} bits, @var{n} digits respectively.
 @end table
 
 @table @var
@@ -1075,7 +1113,7 @@ We will improve @b{Asir} so that it can provide more f
 引数がある場合, @b{bigfloat} の桁数を @var{n} 桁に設定する. 
 引数のあるなしにかかわらず, 以前に設定されていた値を返す. 
 @item
-@b{bigfloat} の計算は @b{PARI} (@xref{pari}) によって行われる. 
+@b{bigfloat} の計算は @b{MPFR} ライブラリによって行われる. 
 @item
 @b{bigfloat} での計算に対し有効である. 
 @b{bigfloat} の flag を on にする方法は, @code{ctrl} を参照. 
@@ -1085,13 +1123,13 @@ We will improve @b{Asir} so that it can provide more f
 \E
 \BEG
 @item
-When an argument is given, it
-sets the precision for @b{bigfloat} operations to @var{n} digits.
-The return value is always the previous precision in digits regardless of
+When an argument @var{n} is given, these functions 
+set the precision for @b{bigfloat} operations to @var{n} bits or @var{n} digits.
+The return value is always the previous precision regardless of
 the existence of an argument.
 
 @item
-@b{Bigfloat} operations are done by @b{PARI}. (@xref{pari})
+@b{Bigfloat} operations are done by @b{MPFR} library.
 @item
 This is effective for computations in @b{bigfloat}.
 Refer to @code{ctrl()} for turning on the `@b{bigfloat} flag.'
@@ -1104,18 +1142,101 @@ Therefore, it is safe to specify a larger value.
 
 @example
 [1] setprec();
-9
+15
 [2] setprec(100);
-9
+15
 [3] setprec(100);
-96
+99
+[4] setbprec();
+332
 @end example
 
 @table @t
 \JP @item 参照
-@fref{ctrl}, @fref{eval}, @fref{pari}.
+@fref{ctrl}, @fref{eval deval}.
 @end table
 
+\JP @node setround,,, 数の演算
+\EG @node setround,,, Numbers
+@subsection @code{setround}
+@findex setround
+
+@table @t
+@item setround([@var{mode}])
+\JP :: @b{bigfloat} の丸めモードを @var{mode} に設定する.
+\EG :: Sets the rounding mode @var{mode}.
+@end table
+
+@table @var
+@item return
+\JP 整数
+\EG integer
+@item mode
+\JP 整数
+\EG integer
+@end table
+
+@itemize @bullet
+\BJP
+@item
+引数がある場合, @b{bigfloat} の丸めモードを @var{mode} に設定する. 
+引数のあるなしにかかわらず, 以前に設定されていた値を返す. 
+丸めモードの意味は次のとおり.
+@table @code
+@item 0
+Round to nearest
+@item 1
+Round toward 0
+@item 2
+Round toward +infinity
+@item 3
+Round toward -infinity
+@end table
+@item
+@b{bigfloat} での計算に対し有効である. 
+@b{bigfloat} の flag を on にする方法は, @code{ctrl} を参照. 
+\E
+\BEG
+@item
+When an argument @var{mode} is given, these functions 
+set the rounding mode for @b{bigfloat} operations to @var{mode}.
+The return value is always the previous rounding mode regardless of
+the existence of an argument.
+The meanings of rounding modes are as follows
+@table @code
+@item 0
+Round to nearest
+@item 1
+Round toward 0
+@item 2
+Round toward +infinity
+@item 3
+Round toward -infinity
+@end table
+
+@item
+This is effective for computations in @b{bigfloat}.
+Refer to @code{ctrl()} for turning on the `@b{bigfloat} flag.'
+\E
+@end itemize
+
+@example
+[1] setprec();
+15
+[2] setprec(100);
+15
+[3] setprec(100);
+99
+[4] setbprec();
+332
+@end example
+
+@table @t
+\JP @item 参照
+@fref{ctrl}, @fref{eval deval}.
+@end table
+
+
 \JP @node setmod,,, 数の演算
 \EG @node setmod,,, Numbers
 @subsection @code{setmod}
@@ -1177,3 +1298,113 @@ return to toplevel
 \EG @fref{dp_mod dp_rat}, @fref{Types of numbers}.
 @end table
 
+\JP @node ntoint32 int32ton,,, 数の演算
+\EG @node ntoint32 int32ton,,, Numbers
+@subsection @code{ntoint32}, @code{int32ton}
+@findex ntoint32
+@findex int32ton
+
+@table @t
+@item ntoint32(@var{n})
+@itemx int32ton(@var{int32})
+\JP :: 非負整数と符号なし 32bit 整数の間の型変換. 
+\EG :: Type-conversion between a non-negative integer and an unsigned 32bit integer.
+@end table
+
+@table @var
+@item return
+\JP 符号なし 32bit 整数または非負整数
+\EG unsigned 32bit integer or non-negative integer
+@item n
+\JP 2^32 未満の非負整数
+\EG non-negative interger less than 2^32
+@item int32
+\JP 符号なし 32bit 整数
+\EG unsigned 32bit integer
+@end table
+
+@itemize @bullet
+\BJP
+@item 非負整数 (識別子 1) の符号なし 32bit 整数 (識別子 10) への変換, 
+またはその逆変換を行う. 
+@item 32bit 整数は @b{OpenXM} の基本構成要素であり, 整数をその型で送信
+する必要がある場合に用いる. 
+\E
+\BEG
+@item These functions do conversions between non-negative 
+integers (the type id 1) and unsigned 32bit integers (the type id 10).
+@item An unsigned 32bit integer is a fundamental construct of @b{OpenXM}
+and one often has to send an integer to a server as an unsigned 32bit
+integer. These functions are used in such a case.
+\E
+@end itemize
+
+@table @t
+\JP @item 参照
+\EG @item References
+\JP @fref{分散計算}, @fref{数の型}.
+\EG @fref{Distributed computation}, @fref{Types of numbers}.
+@end table
+
+\JP @node inttorat,,, 数の演算
+\EG @node inttorat,,, Numbers
+@subsection @code{inttorat}
+@findex inttorat
+
+@table @t
+@item inttorat(@var{a},@var{m},@var{b})
+\JP :: 整数-有理数変換を行う.
+\EG :: Perform the rational reconstruction.
+@end table
+
+@table @var
+@item return
+\JP リストまたは 0
+\EG list or 0
+@item a
+@itemx m
+@itemx b
+\JP 整数
+\EG integer
+@end table
+
+@itemize @bullet
+\BJP
+@item
+整数 @var{a} に対し, @var{xa=y} mod @var{m} を満たす正整数 @var{x}, 整数 @var{y}
+(@var{x}, @var{|y|} < @var{b}, @var{GCD(x,y)=1}) を求める.
+@item
+このような @var{x}, @var{y} が存在するなら @var{[y,x]} を返し, 存在しない場合には 0 を返す.
+@item
+@var{b} を @var{floor(sqrt(m/2))} と取れば, @var{x}, @var{y} は存在すれば一意である.
+@var{floor(sqrt(m/2))} は @code{isqrt(floor(m/2))} で計算できる.
+@end itemize
+\E
+\BEG
+@item
+For an integer @var{a}, find a positive integer @var{x} and an intger @var{y} satisfying
+@var{xa=y} mod @var{m}, @var{x}, @var{|y|} < @var{b} and @var{GCD(x,y)=1}.
+@item
+If such @var{x}, @var{y} exist then a list @var{[y,x]} is returned. Otherwise 0 is returned.
+@item
+If @var{b} is set to @var{floor(sqrt(M/2))}, then @var{x} and @var{y} are unique if they
+exist. @var{floor(sqrt(M/2))} can be computed by @code{floor} and @code{isqrt}.
+@end itemize
+\E
+
+@example
+[2121] M=lprime(0)*lprime(1);
+9996359931312779
+[2122] B=isqrt(floor(M/2));
+70697807
+[2123] A=234234829304;
+234234829304
+[2124] inttorat(A,M,B);
+[-20335178,86975031]
+@end example
+
+@table @t
+\JP @item 参照
+\EG @item References
+@fref{floor}, @fref{isqrt}.
+@end table