@comment $OpenXM: OpenXM/src/asir-doc/parts/builtin/num.texi,v 1.15 2020/09/01 09:25:32 noro Exp $
\BJP
@node 数の演算,,, 組み込み函数
@section 数の演算
\E
\BEG
@node Numbers,,, Built-in Function
@section Numbers
\E

@menu
* idiv irem::
* fac::
* igcd igcdcntl::
* ilcm::
* isqrt::
* inv::
* prime lprime::
* random::
* mt_save mt_load::
* nm dn::
* conj real imag::
* eval deval::
* pari::
* setbprec setprec::
* setmod::
* lrandom::
* ntoint32 int32ton::
* setround::
* inttorat::
@end menu

\JP @node idiv irem,,, 数の演算
\EG @node idiv irem,,, Numbers
@subsection @code{idiv}, @code{irem}
@findex idiv
@findex irem

@table @t
@item idiv(@var{i1},@var{i2})
\JP :: 整数除算による商. 
\EG :: Integer quotient of @var{i1} divided by @var{i2}.
@item irem(@var{i1},@var{i2})
\JP :: 整数除算による剰余. 
\EG :: Integer remainder of @var{i1} divided by @var{i2}.
@end table

@table @var
@item return
\JP 整数
\EG integer
@item i1 i2
\JP 整数
\EG integer
@end table

@itemize @bullet
\BJP
@item
@var{i1} の @var{i2} による整数除算による商, 剰余を求める. 
@item
@var{i2} は 0 であってはならない. 
@item
被除数が負の場合, 絶対値に対する値にマイナスをつけた値を返す. 
@item
@var{i1} @code{%} @var{i2} は, 結果が正に正規化されることを除けば
@code{irem()} の代わりに用いることができる. 
@item
多項式の場合は @code{sdiv}, @code{srem} を用いる. 
\E
\BEG
@item 
Integer quotient and remainder of @var{i1} divided by @var{i2}.
@item 
@var{i2} must not be 0.
@item 
If the dividend is negative, the results are obtained by changing the
sign of the results for absolute values of the dividend.
@item 
One can use
@var{i1} @code{%} @var{i2}
for replacement of @code{irem()} which only differs in the point that
the result is always normalized to non-negative values.
@item 
Use @code{sdiv()}, @code{srem()} for polynomial quotient.
\E
@end itemize

@example
[0] idiv(100,7);
14
[0] idiv(-100,7);
-14
[1] irem(100,7);
2
[1] irem(-100,7);
-2
@end example

@table @t
\JP @item 参照
\EG @item References
@fref{sdiv sdivm srem sremm sqr sqrm}, @fref{%}.
@end table

\JP @node fac,,, 数の演算
\EG @node fac,,, Numbers
@subsection @code{fac}
@findex fac

@table @t
@item fac(@var{i})
\JP :: @var{i} の階乗. 
\EG :: The factorial of @var{i}.
@end table

@table @var
@item return
\JP 整数
\EG integer
@item i
\JP 整数
\EG integer
@end table

@itemize @bullet
\BJP
@item
@var{i} の階乗を計算する. 
@item
@var{i} が負の場合は 0 を返す. 
\E
\BEG
@item 
The factorial of @var{i}.
@item 
Returns 0 if the argument @var{i} is negative.
\E
@end itemize

@example
[0] fac(50);
30414093201713378043612608166064768844377641568960512000000000000
@end example

\JP @node igcd igcdcntl,,, 数の演算
\EG @node igcd igcdcntl,,, Numbers
@subsection @code{igcd},@code{igcdcntl}
@findex igcd
@findex igcdcntl

@table @t
@item igcd(@var{i1},@var{i2})
\JP :: 整数の GCD (最大公約数)
\EG :: The integer greatest common divisor of @var{i1} and @var{i2}.
@item igcdcntl([@var{i}])
\JP :: 整数 GCDのアルゴリズム選択
\EG :: Selects an algorithm for integer GCD.
@end table

@table @var
@item return
\JP 整数
\EG integer
@item i1 i2 i
\JP 整数
\EG integer
@end table

@itemize @bullet
\BJP
@item
@code{igcd} は @var{i1} と @var{i2} の GCD を求める. 
@item
引数が整数でない場合は, エラーまたは無意味な結果を返す. 
@item
多項式の場合は, @code{gcd}, @code{gcdz} を用いる. 
@item
整数 GCD にはさまざまな方法があり, @code{igcdcntl} で設定できる. 

@table @code
@item 0
Euclid 互除法 (default)
@item 1
binary GCD
@item 2
bmod GCD
@item 3
accelerated integer GCD
@end table
@code{2}, @code{3} は @code{[Weber]} による. 

おおむね @code{3} が高速だが, 例外もある. 
\E
\BEG
@item 
Function @code{igcd()} returns the integer greatest common divisor of
the given two integers.
@item 
An error will result if the argument is not an integer; the result is
not valid even if one is returned.
@item 
Use @code{gcd()}, @code{gcdz()} for polynomial GCD.

@item
Various method of integer GCD computation are implemented
and they can be selected by @code{igcdcntl}.

@table @code
@item 0
Euclid algorithm (default)
@item 1
binary GCD
@item 2
bmod GCD
@item 3
accelerated integer GCD
@end table
@code{2}, @code{3} are due to @code{[Weber]}.

In most cases @code{3} is the fastest, but there are exceptions.
\E
@end itemize

@example
[0] A=lrandom(10^4)$
[1] B=lrandom(10^4)$
[2] C=lrandom(10^4)$
[3] D=A*C$
[4] E=A*B$
[5] cputime(1)$
[6] igcd(D,E)$
0.6sec + gc : 1.93sec(2.531sec)
[7] igcdcntl(1)$
[8] igcd(D,E)$  
0.27sec(0.2635sec)
[9] igcdcntl(2)$
[10] igcd(D,E)$  
0.19sec(0.1928sec)
[11] igcdcntl(3)$
[12] igcd(D,E)$  
0.08sec(0.08023sec)
@end example

@table @t
\JP @item 参照
\EG @item References
@fref{gcd gcdz}.
@end table

\JP @node ilcm,,, 数の演算
\EG @node ilcm,,, Numbers
@subsection @code{ilcm}
@findex ilcm

@table @t
@item ilcm(@var{i1},@var{i2})
\JP :: 最小公倍数を求める. 
\EG :: The integer least common multiple of @var{i1} and @var{i2}.
@end table

@table @var
@item return
\JP 整数
\EG integer
@item i1 i2
\JP 整数
\EG integer
@end table

@itemize @bullet
\BJP
@item
整数 @var{i1}, @var{i2} の最小公倍数を求める. 
@item
一方が 0 の場合 0 を返す. 
\E
\BEG
@item
This function computes the integer least common multiple of
@var{i1}, @var{i2}.
@item
If one of argument is equal to 0, the return 0.
\E
@end itemize

@table @t
\JP @item 参照
\EG @item References
@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}
@findex inv

@table @t
@item inv(@var{i},@var{m})
\JP :: @var{m} を法とする @var{i} の逆数
\EG :: the inverse (reciprocal) of @var{i} modulo @var{m}.
@end table

@table @var
@item return
\JP 整数
\EG integer
@item i m
\JP 整数
\EG integer
@end table

@itemize @bullet
\BJP
@item
@var{ia} @equiv{} 1 mod (@var{m}) なる整数 @var{a} を求める. 
@item
@var{i} と @var{m} は互いに素でなければならないが, @code{inv()} は
そのチェックは行わない. 
\E
\BEG
@item 
This function computes an integer such that
@var{ia} @equiv{} 1 mod (@var{m}).
@item 
The integer @var{i} and  @var{m} must be mutually prime.
However, @code{inv()} does not check it.
\E
@end itemize

@example
[71] igcd(1234,4321);  
1
[72] inv(1234,4321);
3239
[73] irem(3239*1234,4321);
1
@end example

@table @t
\JP @item 参照
\EG @item References
@fref{igcd igcdcntl}.
@end table

\JP @node prime lprime,,, 数の演算
\EG @node prime lprime,,, Numbers
@subsection @code{prime}, @code{lprime}
@findex prime
@findex lprime

@table @t
@item prime(@var{index})
@item lprime(@var{index})
\JP :: 素数を返す
\EG :: Returns a prime number.
@end table

@table @var
@item return
\JP 整数
\EG integer
@item index
\JP 整数
\EG integer
@end table

@itemize @bullet
\BJP
@item
@code{prime()}, @code{lprime()} いずれもシステムが内部に持つ
素数表の要素を返す. @code{index} は 0 以上の整数で, 素数表
のインデックスに用いられる. @code{prime()} は 16381 まで
の素数を小さい順に 1900 個, @code{lprime()} は, 10 進 8 桁で最大の
素数から大きい順に 999 個返す. それ以外のインデックスに対しては
0 を返す. 
@item
より一般的な素数生成函数としては,
@code{pari(nextprime,@var{number})}
がある. 
\E
\BEG
@item
The two functions, @code{prime()} and @code{lprime()}, returns
an element stored in the system table of prime numbers.
Here, @code{index} is a non-negative integer and be used as an index
for the prime tables.
The function @code{prime()} can return one of 1900 primes 
up to 16381 indexed so that the smaller one has smaller
index.  The function @code{lprime()} can return one of 999 primes which
are 8 digit sized and indexed so that the larger one has the smaller
index.
The two function always returns 0 for other indices.
@item
For more general function for prime generation, there is a @code{PARI}
function

@code{pari(nextprime,@var{number})}.
\E
@end itemize

@example
[95] prime(0);
2
[96] prime(1228);
9973
[97] lprime(0);
99999989
[98] lprime(999);
0
@end example

@table @t
\JP @item 参照
\EG @item References
@fref{pari}.
@end table

\JP @node random,,, 数の演算
\EG @node random,,, Numbers
@subsection @code{random}
@findex random

@table @t
@item random([@var{seed}])
\JP :: 乱数を生成する. 
@end table

@table @var
@item seed
@itemx return
\JP 自然数
\EG non-negative integer
@end table

@itemize @bullet
\BJP
@item
最大 2^32-1 の非負整数の乱数を生成する. 
@item
0 でない引数がある時, その値を seed として設定してから, 乱数を生成する. 
@item
default の seed は固定のため, 種を設定しなければ, 生成される乱数の
系列は起動毎に一定である. 
@item
松本眞-西村拓士による Mersenne Twister (http://www.math.keio.ac.jp/matsumoto/mt.html) アルゴリズムの, 彼ら自身による実装を用いている. 
@item
周期は 2^19937-1 と非常に長い. 
@item
@code{mt_save} により state をファイルに save できる. これを @code{mt_load}
で読み込むことにより, 異る Asir セッション間で一つの乱数の系列を辿ることが
できる. 
\E
\BEG
@item
Generates a random number which is a non-negative integer less than 2^32.
@item
If a non zero argument is specified, then after setting it as a random seed,
a random number is generated.
@item
As the default seed is fixed, the sequence of the random numbers is
always the same if a seed is not set.
@item
The algorithm is Mersenne Twister
(http://www.math.keio.ac.jp/matsumoto/mt.html) by M. Matsumoto and
T. Nishimura. The implementation is done also by themselves.
@item
The period of the random number sequence is 2^19937-1.
@item
One can save the state of the random number generator with @code{mt_save}.
By loading the state file with @code{mt_load},
one can trace a single random number sequence arcoss multiple sessions.
\E
@end itemize

@table @t
\JP @item 参照
\EG @item References
@fref{lrandom}, @fref{mt_save mt_load}.
@end table

\JP @node lrandom,,, 数の演算
\EG @node lrandom,,, Numbers
@subsection @code{lrandom}
@findex lrandom

@table @t
@item lrandom(@var{bit})
\JP :: 多倍長乱数を生成する. 
\EG :: Generates a long random number.
@end table

@table @var
@item bit
@item return
\JP 自然数
\EG integer
@end table

@itemize @bullet
\BJP
@item
高々 @var{bit} の非負整数の乱数を生成する. 
@item
@code{random} を複数回呼び出して結合し, 指定の bit 長にマスクしている. 
\E
\BEG
@item
Generates a non-negative integer of at most @var{bit} bits.
@item
The result is a concatination of outputs of @code{random}.
\E
@end itemize

@table @t
\JP @item 参照
\EG @item References
@fref{random}, @fref{mt_save mt_load}.
@end table

\JP @node mt_save mt_load,,, 数の演算
\EG @node mt_save mt_load,,, Numbers
@subsection @code{mt_save}, @code{mt_load}
@findex mt_save
@findex mt_load

@table @t
@item mt_save(@var{fname})
\JP :: 乱数生成器の現在の状態をファイルにセーブする. 
\EG :: Saves the state of the random number generator.
@item mt_load(@var{fname})
\JP :: ファイルにセーブされた乱数生成器の状態をロードする. 
\EG :: Loads a saved state of the random number generator.
@end table

@table @var
@item return
\JP 0 または 1
\EG 0 or 1
@item fname
\JP 文字列
\EG string
@end table

@itemize @bullet
\BJP
@item
ある状態をセーブし, その状態をロードすることで, 
一つの疑似乱数系列を, 新規の Asir セッションで続けてたどることが
できる. 
\E
\BEG
@item
One can save the state of the random number generator with @code{mt_save}.
By loading the state file with @code{mt_load},
one can trace a single random number sequence arcoss multiple 
@b{Asir} sessions.
\E
@end itemize

@example
[340] random();
3510405877
[341] mt_save("/tmp/mt_state");
1
[342] random();
4290933890
[343] quit;
% asir
This is Asir, Version 991108.
Copyright (C) FUJITSU LABORATORIES LIMITED.
3 March 1994. All rights reserved.
[340] mt_load("/tmp/mt_state");
1
[341] random();
4290933890
@end example

@table @t
\JP @item 参照
\EG @item References
@fref{random}, @fref{lrandom}.
@end table

\JP @node nm dn,,, 数の演算
\EG @node nm dn,,, Numbers
@subsection @code{nm}, @code{dn}
@findex nm
@findex dn

@table @t
@item nm(@var{rat})
\JP :: @var{rat} の分子. 
\EG :: Numerator of @var{rat}.
@item dn(@var{rat})
\JP :: @var{rat} の分母. 
\EG :: Denominator of @var{rat}.
@end table

@table @var
@item return
\JP 整数または多項式
\EG integer or polynomial
@item rat
\JP 有理数または有理式
\EG rational number or rational expression
@end table

@itemize @bullet
\BJP
@item
与えられた有理数また有理式の分子及び分母を返す. 
@item
有理数の場合, 分母は常に正で, 符号は分子が持つ. 
@item
有理式の場合, 単に分母, 分子を取り出すだけである. 
有理式に対しては, 約分は自動的には行われない. @code{red()}
を明示的に呼び出す必要がある. 
\E
\BEG
@item
Numerator and denominator of a given rational expression.
@item
For a rational number, they return its numerator and denominator,
respectively.  For a rational expression whose numerator and denominator
may contain rational numbers, they do not separate those rational
coefficients to numerators and denominators.
@item
For a rational number, the denominator is always kept positive, and
the sign is contained in the numerator.
@item
@b{Risa/Asir} does not cancel the common divisors unless otherwise explicitly
specified by the user.
Therefore, @code{nm()} and @code{dn()} return the numerator and the
denominator as it is, respectively.
\E
@end itemize

@example
[2] [nm(-43/8),dn(-43/8)];
[-43,8]
[3] dn((x*z)/(x*y));
y*x
[3] dn(red((x*z)/(x*y)));
y
@end example

@table @t
\JP @item 参照
\EG @item References
@fref{red}.
@end table

\JP @node conj real imag,,, 数の演算
\EG @node conj real imag,,, Numbers
@subsection @code{conj}, @code{real}, @code{imag}
@findex conj

@table @t
@item real(@var{comp})
\JP :: @var{comp} の実数部分. 
\EG :: Real part of @var{comp}.
@item imag(@var{comp})
\JP :: @var{comp} の虚数部分. 
\EG :: Imaginary part of @var{comp}.
@item conj(@var{comp})
\JP :: @var{comp} の共役複素数. 
\EG :: Complex conjugate of @var{comp}.
@end table

@table @var
@item return comp
\JP 複素数
\EG complex number
@end table

@itemize @bullet
\BJP
@item
複素数に対し, 実部, 虚部, 共役を求める. 
@item
これらは, 多項式に対しても働く. 
\E
\BEG
@item
Basic operations for complex numbers.
@item
These functions works also for polynomials with complex coefficients.
\E
@end itemize

@example
[111] A=(2+@@i)^3; 
(2+11*@@i)
[112] [real(A),imag(A),conj(A)];
[2,11,(2-11*@@i)]
@end example

\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

@table @var
@item return
\JP 数あるいは式
\EG number or expression
@item obj
\JP 一般の式
\EG general expression
@item prec
\JP 整数
\EG integer
@end table

@itemize @bullet
\BJP
@item
@var{obj} に含まれる函数の値を可能な限り評価する. 
@item
@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{setbprec setprec}.)
@item
@table @t
@item 扱える函数は, 次の通り. 
@code{sin}, @code{cos}, @code{tan},

@code{asin}, @code{acos}, @code{atan},

@code{sinh}, @code{cosh}, @code{tanh},

@code{asinh}, @code{acosh}, @code{atanh},

@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. 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
@item @@pi
円周率
@item @@e
自然対数の底
@end table
\E
\BEG
@item
Evaluates the value of the functions contained in @var{obj} as far as
possible.
@item
@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{setbprec setprec}.)
@item
Currently available numerical functions are listed below.

@table @t
@code{sin}, @code{cos}, @code{tan},

@code{asin}, @code{acos}, @code{atan},

@code{sinh}, @code{cosh}, @code{tanh},
@code{asinh}, @code{acosh}, @code{atanh},

@code{exp}, @code{log}, @code{pow(a,b) (a^b)}
@end table
@item
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
@item @@pi
the number pi,
the ratio of circumference to diameter
@item @@e
Napier's number (@t{exp}(1))
@end table
\E
@end itemize

@example
[118] eval(exp(@@pi*@@i));   
-1.0000000000000000000000000000
[119] eval(2^(1/2));
1.414213562373095048763788073031
[120] eval(sin(@@pi/3));
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{setbprec setprec}.
@end table

\JP @node pari,,, 数の演算
\EG @node pari,,, Numbers
@subsection @code{pari}
@findex pari
@cindex PARI

@table @t
@item pari(@var{func},@var{arg},@var{prec}) 
\JP :: @b{PARI} の函数 @var{func} を呼び出す. 
\EG :: Call @b{PARI} function @var{func}.
@end table

@table @var
@item return
\JP @var{func} 毎に異なる. 
\EG Depends on @var{func}.
@item func
\JP @b{PARI} の函数名
\EG Function name of @b{PARI}.
@item arg
\JP @var{func} の引数
\EG Arguments of @var{func}.
@item prec
\JP 整数
\EG integer
@end table

@itemize @bullet
\BJP
@item
@b{PARI} の函数を呼び出す. 

@item
@b{PARI} @code{[Batut et al.]} は Bordeaux 大学で開発されフ
リーソフトウェアとして公開されている. @b{PARI} は数式処理的な機能を有
してはいるが, 主なターゲットは整数論に関連した数 (@b{bignum},
@b{bigfloat}) の演算で, 四則演算に限らず@b{bigfloat} によるさまざまな
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. 
@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. 
@item
現時点で実行できる @b{PARI} の主な函数は次の通りである. いずれも
1 引数で @b{Asir} が対応できる型の引数をとる函数である. 
実行できる函数の全リストは asir-contrib の ox_pari マニュアルを参照.
なお各々の機能については @b{PARI} のマニュアルを参照のこと.
\E
\BEG
@item
This command connects @b{Asir} to @b{PARI} system so that several
functions of @b{PARI} can be conveniently used from @b{Risa/Asir}.
@item
@b{PARI} @code{[Batut et al.]} is developed at Bordeaux University, and
distributed as a free software.  Though it has a certain facility to computer
algebra, its major target is the operation of numbers (@b{bignum},
@b{bigfloat}) related to the number theory.  It facilitates various
function evaluations as well as arithmetic operations at a remarkable
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.
@item
The last argument (optional) @var{int} specifies the precision in digits
for bigfloat operation.
If the precision is not explicitly specified, operation will be performed
with the precision set by @code{setprec()}.
@item
Some of currently available functions of @b{PARI} system are as follows.
Note these are only a part of functions in @b{PARI} system.
A complete list of functions which can be called from asir
is in the ox_pari manual of the asir-contrib.
For details of individual functions, refer to the @b{PARI} manual.
(Some of them can be seen in the following example.)
\E

@code{abs},
@code{adj},
@code{arg},
@code{bigomega},
@code{binary},
@code{ceil},
@code{centerlift},
@code{cf},
@code{classno},
@code{classno2},
@code{conj},
@code{content},
@code{denom},
@code{det},
@code{det2},
@code{detr},
@code{dilog},
@code{disc},
@code{discf},
@code{divisors},
@code{eigen},
@code{eintg1},
@code{erfc},
@code{eta},
@code{floor},
@code{frac},
@code{galois},
@code{galoisconj},
@code{gamh},
@code{gamma},
@code{hclassno},
@code{hermite},
@code{hess},
@code{imag},
@code{image},
@code{image2},
@code{indexrank},
@code{indsort},
@code{initalg},
@code{isfund},
@code{isprime},
@code{ispsp},
@code{isqrt},
@code{issqfree},
@code{issquare},
@code{jacobi},
@code{jell},
@code{ker},
@code{keri},
@code{kerint},
@code{kerintg1},
@code{kerint2},
@code{kerr},
@code{length},
@code{lexsort},
@code{lift},
@code{lindep},
@code{lll},
@code{lllg1},
@code{lllgen},
@code{lllgram},
@code{lllgramg1},
@code{lllgramgen},
@code{lllgramint},
@code{lllgramkerim},
@iftex
@break
@end iftex
@code{lllgramkerimgen},
@code{lllint},
@code{lllkerim},
@code{lllkerimgen},
@code{lllrat},
@code{lngamma},
@code{logagm},
@code{mat},
@code{matrixqz2},
@code{matrixqz3},
@code{matsize},
@code{modreverse},
@code{mu},
@code{nextprime},
@code{norm},
@code{norml2},
@code{numdiv},
@code{numer},
@code{omega},
@code{order},
@code{ordred},
@code{phi},
@code{pnqn},
@code{polred},
@code{polred2},
@code{primroot},
@code{psi},
@code{quadgen},
@code{quadpoly},
@code{real},
@code{recip},
@code{redcomp},
@code{redreal},
@code{regula},
@code{reorder},
@code{reverse},
@code{rhoreal},
@code{roots},
@code{rootslong},
@code{round},
@code{sigma},
@code{signat},
@code{simplify},
@code{smalldiscf},
@code{smallfact},
@code{smallpolred},
@code{smallpolred2},
@code{smith},
@code{smith2},
@code{sort},
@code{sqr},
@code{sqred},
@code{sqrt},
@code{supplement},
@code{trace},
@code{trans},
@code{trunc},
@code{type},
@code{unit},
@code{vec},
@code{wf},
@code{wf2},
@code{zeta}

\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.
\E
\BEG
@item
@b{Asir} currently uses only a very small subset of @b{PARI}.
\E
@end itemize

@example
\JP /* 行列の固有ベクトルを求める. */
\EG /* Eigen vectors of a numerical matrix */
[0] pari(eigen,newmat(2,2,[[1,1],[1,2]]));
[ -1.61803398874989484819771921990 0.61803398874989484826 ]
[ 1 1 ]
\JP /* 1 変数多項式の根を求める. */
\EG /* Roots of a polynomial */
[1] pari(roots,t^2-2);
[ -1.41421356237309504876 1.41421356237309504876 ]
@end example

@table @t
\JP @item 参照
\EG @item References
@fref{setbprec setprec}.
@end table

\JP @node setbprec setprec,,, 数の演算
\EG @node setbprec setprec,,, Numbers
@subsection @code{setbprec}, @code{setprec} 
@findex setbprec
@findex setprec

@table @t
@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
@item return
\JP 整数
\EG integer
@item n
\JP 整数
\EG integer
@end table

@itemize @bullet
\BJP
@item
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{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. 
@item
設定できる桁数に上限はないが, 指定した桁数に設定されるとは
限らない. 大きめの値を設定するのが安全である. 
\E
\BEG
@item
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{MPFR} library.
@item
This is effective for computations in @b{bigfloat}.
Refer to @code{ctrl()} for turning on the `@b{bigfloat} flag.'
@item
There is no upper limit for precision digits.
It sets the precision to some digits around the specified precision.
Therefore, it is safe to specify a larger value.
\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 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}
@findex setmod

@table @t
@item setmod([@var{p}])
\JP :: 有限体を GF(@var{p}) に設定する. 
\EG :: Sets the ground field to GF(@var{p}).
@end table

@table @var
@item return
\JP 整数
\EG integer
@item n
\JP 2^27 未満の素数
\EG prime less than 2^27
@end table

@itemize @bullet
\BJP
@item
有限体を GF(@var{p}) に設定する. 設定値を返す. 
@item
有限体の元の型を持つ数は, それ自身はどの有限体に属するかの情報を持たず, 
現在設定されている素数 @var{p} により GF(@var{p}) 上での演算が適用される. 
@item
位数の大きな有限体に関しては @pxref{有限体に関する演算}.
\E
\BEG
@item
Sets the ground field to GF(@var{p}) and returns the value @var{p}.
@item
A member of a finite field does not have any information
about the field and the arithmetic operations over GF(@var{p}) are applied
with @var{p} set at the time.
@item
As for large finite fields, @pxref{Finite fields}.
\E
@end itemize

@example
[0] A=dp_mod(dp_ptod(2*x,[x]),3,[]);
(2)*<<1>>
[1] A+A;
addmi : invalid modulus
return to toplevel
[1] setmod(3);
3
[2] A+A;
(1)*<<1>>
@end example

@table @t
\JP @item 参照
\EG @item References
\JP @fref{dp_mod dp_rat}, @fref{数の型}.
\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{ceil floor rint dceil dfloor drint}, @fref{isqrt}.
@end table