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version 1.1.1.1, 1999/12/08 05:47:44

version 1.14, 2020/08/27 11:05:27

Line 1

@comment $OpenXM: OpenXM/src/asir-doc/parts/builtin/num.texi,v 1.13 2019/03/29 01:57:46 noro Exp $

\BJP

@node $B?t$N1i;;(B,,, $BAH$_9~$_H!?t(B

@section $B?t$N1i;;(B

\BEG

@node Numbers,,, Built-in Function

@section Numbers

@menu

* idiv irem::

* 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

@node idiv irem,,, $B?t$N1i;;(B

\JP @node idiv irem,,, $B?t$N1i;;(B

\EG @node idiv irem,,, Numbers

@subsection @code{idiv}, @code{irem}

@findex idiv

@findex irem

@table @t

@item idiv(@var{i1},@var{i2})

:: $B@0?t=|;;$K$h$k>&(B.

\JP :: $B@0?t=|;;$K$h$k>&(B.

\EG :: Integer quotient of @var{i1} divided by @var{i2}.

@item irem(@var{i1},@var{i2})

:: $B@0?t=|;;$K$h$k>jM>(B.

\JP :: $B@0?t=|;;$K$h$k>jM>(B.

\EG :: Integer remainder of @var{i1} divided by @var{i2}.

@end table

@table @var

@item return

$B@0?t(B

\JP $B@0?t(B

@item i1,i2

\EG integer

$B@0?t(B

@item i1 i2

\JP $B@0?t(B

\EG integer

@end table

@itemize @bullet

\BJP

@item

@var{i1} $B$N(B @var{i2} $B$K$h$k@0?t=|;;$K$h$k>&(B, $B>jM>$r5a$a$k(B.

@item

Line 50

Line 67

@code{irem()} $B$NBe$o$j$KMQ$$$k$3$H$,$G$-$k(B.

@item

$BB?9`<0$N>l9g$O(B @code{sdiv}, @code{srem} $B$rMQ$$$k(B.

\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.

@end itemize

@example

Line 64

Line 98

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{sdiv sdivm srem sremm sqr sqrm}, @fref{%}.

@end table

@node fac,,, $B?t$N1i;;(B

\JP @node fac,,, $B?t$N1i;;(B

\EG @node fac,,, Numbers

@subsection @code{fac}

@findex fac

@table @t

@item fac(@var{i})

:: @var{i} $B$N3,>h(B.

\JP :: @var{i} $B$N3,>h(B.

\EG :: The factorial of @var{i}.

@end table

@table @var

@item return

$B@0?t(B

\JP $B@0?t(B

\EG integer

@item i

$B@0?t(B

\JP $B@0?t(B

\EG integer

@end table

@itemize @bullet

\BJP

@item

@var{i} $B$N3,>h$r7W;;$9$k(B.

@item

@var{i} $B$,Ii$N>l9g$O(B 0 $B$rJV$9(B.

\BEG

@item

The factorial of @var{i}.

@item

Returns 0 if the argument @var{i} is negative.

@end itemize

@example

Line 96

Line 143

30414093201713378043612608166064768844377641568960512000000000000

@end example

@node igcd igcdcntl,,, $B?t$N1i;;(B

\JP @node igcd igcdcntl,,, $B?t$N1i;;(B

\EG @node igcd igcdcntl,,, Numbers

@subsection @code{igcd},@code{igcdcntl}

@findex igcd

@findex igcdcntl

@table @t

@item igcd(@var{i1},@var{i2})

:: $B@0?t$N(B GCD ($B:GBg8xLs?t(B)

\JP :: $B@0?t$N(B GCD ($B:GBg8xLs?t(B)

\EG :: The integer greatest common divisor of @var{i1} and @var{i2}.

@item igcdcntl([@var{i}])

:: $B@0?t(B GCD$B$N%"%k%4%j%:%`A*Br(B

\JP :: $B@0?t(B GCD$B$N%"%k%4%j%:%`A*Br(B

\EG :: Selects an algorithm for integer GCD.

@end table

@table @var

@item return

$B@0?t(B

\JP $B@0?t(B

@item i1,i2,i

\EG integer

$B@0?t(B

@item i1 i2 i

\JP $B@0?t(B

\EG integer

@end table

@itemize @bullet

\BJP

@item

@code{igcd} $B$O(B @var{i1} $B$H(B @var{i2} $B$N(B GCD $B$r5a$a$k(B.

@item

Line 135 bmod GCD

Line 188 bmod GCD

@item 3

accelerated integer GCD

@end table

2, 3 $B$O(B K. Weber, ACM TOMS, Vol.21, No. 1 (1995), pp. 111-122 $B$K$h$k(B.

@code{2}, @code{3} $B$O(B @code{[Weber]} $B$K$h$k(B.

$B$*$*$`$M(B 3 $B$,9bB.$@$,(B, $BNc30$b$"$k(B.

$B$*$*$`$M(B @code{3} $B$,9bB.$@$,(B, $BNc30$b$"$k(B.

\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.

@end itemize

@example

Line 161 accelerated integer GCD

Line 243 accelerated integer GCD

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{gcd gcdz}.

@end table

@node ilcm,,, $B?t$N1i;;(B

\JP @node ilcm,,, $B?t$N1i;;(B

\EG @node ilcm,,, Numbers

@subsection @code{ilcm}

@findex ilcm

@table @t

@item ilcm(@var{i1},@var{i2})

:: $B:G>.8xG\?t$r5a$a$k(B.

\JP :: $B:G>.8xG\?t$r5a$a$k(B.

\EG :: The integer least common multiple of @var{i1} and @var{i2}.

@end table

@table @var

@item return

$B@0?t(B

\JP $B@0?t(B

@item i1,i2

\EG integer

$B@0?t(B

@item i1 i2

\JP $B@0?t(B

\EG integer

@end table

@itemize @bullet

\BJP

@item

$B@0?t(B @var{i1}, @var{i2} $B$N:G>.8xG\?t$r5a$a$k(B.

@item

$B0lJ}$,(B 0 $B$N>l9g(B 0 $B$rJV$9(B.

\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.

@end itemize

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{igcd igcdcntl}, @fref{mt_save mt_load}.

@end table

@node inv,,, $B?t$N1i;;(B

\JP @node isqrt,,, $B?t$N1i;;(B

\EG @node isqrt,,, Numbers

@subsection @code{isqrt}

@findex isqrt

@table @t

@item isqrt(@var{n})

\JP :: $BJ?J}:,$r1[$($J$$:GBg$N@0?t$r5a$a$k(B.

\EG :: The integer square root of @var{n}.

@end table

@table @var

@item return

\JP $BHsIi@0?t(B

\EG non-negative integer

@item n

\JP $BHsIi@0?t(B

\EG non-negative integer

@end table

\JP @node inv,,, $B?t$N1i;;(B

\EG @node inv,,, Numbers

@subsection @code{inv}

@findex inv

@table @t

@item inv(@var{i},@var{m})

:: @var{m} $B$rK!$H$9$k(B @var{i} $B$N5U?t(B

\JP :: @var{m} $B$rK!$H$9$k(B @var{i} $B$N5U?t(B

\EG :: the inverse (reciprocal) of @var{i} modulo @var{m}.

@end table

@table @var

@item return

$B@0?t(B

\JP $B@0?t(B

@item i,m

\EG integer

$B@0?t(B

@item i m

\JP $B@0?t(B

\EG integer

@end table

@itemize @bullet

\BJP

@item

@var{ia} @equiv{} 1 mod (@var{m}) $B$J$k@0?t(B @var{a} $B$r5a$a$k(B.

@item

@var{i} $B$H(B @var{m} $B$O8_$$$KAG$G$J$1$l$P$J$i$J$$$,(B, @code{inv()} $B$O(B

$B$=$N%A%'%C%/$O9T$o$J$$(B.

\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.

@end itemize

@example

Line 228 accelerated integer GCD

Line 358 accelerated integer GCD

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{igcd igcdcntl}.

@end table

@node prime lprime,,, $B?t$N1i;;(B

\JP @node prime lprime,,, $B?t$N1i;;(B

\EG @node prime lprime,,, Numbers

@subsection @code{prime}, @code{lprime}

@findex prime

@findex lprime

Line 240 accelerated integer GCD

Line 372 accelerated integer GCD

@table @t

@item prime(@var{index})

@item lprime(@var{index})

:: $BAG?t$rJV$9(B

\JP :: $BAG?t$rJV$9(B

\EG :: Returns a prime number.

@end table

@table @var

@item return

$B@0?t(B

\JP $B@0?t(B

\EG integer

@item index

$B@0?t(B

\JP $B@0?t(B

\EG integer

@end table

@itemize @bullet

\BJP

@item

@code{prime()}, @code{lprime()} $B$$$:$l$b%7%9%F%`$,FbIt$K;}$D(B

$BAG?tI=$NMWAG$rJV$9(B. @code{index} $B$O(B 0 $B0J>e$N@0?t$G(B, $BAG?tI=(B

Line 259 accelerated integer GCD

Line 395 accelerated integer GCD

$BAG?t$+$iBg$-$$=g$K(B 999 $B8DJV$9(B. $B$=$l0J30$N%$%s%G%C%/%9$KBP$7$F$O(B

0 $B$rJV$9(B.

@item

$B$h$j0lHLE*$JAG?t@8@.H!?t$H$7$F$O(B, @code{pari(nextprime,@var{number})}

$B$h$j0lHLE*$JAG?t@8@.H!?t$H$7$F$O(B,

@code{pari(nextprime,@var{number})}

$B$,$"$k(B.

\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})}.

@end itemize

@example

Line 275 accelerated integer GCD

Line 431 accelerated integer GCD

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{pari}.

@end table

@node random,,, $B?t$N1i;;(B

\JP @node random,,, $B?t$N1i;;(B

\EG @node random,,, Numbers

@subsection @code{random}

@findex random

@table @t

@item radom([@var{seed}])

@item random([@var{seed}])

:: $BMp?t$r@8@.$9$k(B.

\JP :: $BMp?t$r@8@.$9$k(B.

@end table

@table @var

@item seed

@item return

@itemx return

$B<+A3?t(B

\JP $B<+A3?t(B

\EG non-negative integer

@end table

@itemize @bullet

\BJP

@item

$B:GBg(B 2^32-1 $B$NHsIi@0?t$NMp?t$r@8@.$9$k(B.

@item

Line 310 default $B$N(B seed $B$O8GDj$N$?$a(B, $B<o$r@_Dj$

Line 470 default $B$N(B seed $B$O8GDj$N$?$a(B, $B<o$r@_Dj$

@code{mt_save} $B$K$h$j(B state $B$r%U%!%$%k$K(B save $B$G$-$k(B. $B$3$l$r(B @code{mt_load}

$B$GFI$_9~$`$3$H$K$h$j(B, $B0[$k(B Asir $B%;%C%7%g%s4V$G0l$D$NMp?t$N7ONs$rC)$k$3$H$,(B

$B$G$-$k(B.

\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.

@end itemize

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{lrandom}, @fref{mt_save mt_load}.

@end table

@node lrandom,,, $B?t$N1i;;(B

\JP @node lrandom,,, $B?t$N1i;;(B

\EG @node lrandom,,, Numbers

@subsection @code{lrandom}

@findex lrandom

@table @t

@item lradom(@var{bit})

@item lrandom(@var{bit})

:: $BB?G\D9Mp?t$r@8@.$9$k(B.

\JP :: $BB?G\D9Mp?t$r@8@.$9$k(B.

\EG :: Generates a long random number.

@end table

@table @var

@item bit

@item return

$B<+A3?t(B

\JP $B<+A3?t(B

\EG integer

@end table

@itemize @bullet

\BJP

@item

$B9b!9(B @var{bit} $B$NHsIi@0?t$NMp?t$r@8@.$9$k(B.

@item

@code{random} $B$rJ#?t2s8F$S=P$7$F7k9g$7(B, $B;XDj$N(B bit $BD9$K%^%9%/$7$F$$$k(B.

\BEG

@item

Generates a non-negative integer of at most @var{bit} bits.

@item

The result is a concatination of outputs of @code{random}.

@end itemize

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{random}, @fref{mt_save mt_load}.

@end table

@node mt_save mt_load,,, $B?t$N1i;;(B

\JP @node mt_save mt_load,,, $B?t$N1i;;(B

\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})

:: $BMp?t@8@.4o$N8=:_$N>uBV$r%U%!%$%k$K%;!<%V$9$k(B.

\JP :: $BMp?t@8@.4o$N8=:_$N>uBV$r%U%!%$%k$K%;!<%V$9$k(B.

\EG :: Saves the state of the random number generator.

@item mt_load(@var{fname})

:: $B%U%!%$%k$K%;!<%V$5$l$?Mp?t@8@.4o$N>uBV$r%m!<%I$9$k(B.

\JP :: $B%U%!%$%k$K%;!<%V$5$l$?Mp?t@8@.4o$N>uBV$r%m!<%I$9$k(B.

\EG :: Loads a saved state of the random number generator.

@end table

@table @var

@item return

0 $B$^$?$O(B 1

\JP 0 $B$^$?$O(B 1

\EG 0 or 1

@item fname

$BJ8;zNs(B

\JP $BJ8;zNs(B

\EG string

@end table

@itemize @bullet

@item $B$"$k>uBV$r%;!<%V$7(B, $B$=$N>uBV$r%m!<%I$9$k$3$H$G(B,

\BJP

@item

$B$"$k>uBV$r%;!<%V$7(B, $B$=$N>uBV$r%m!<%I$9$k$3$H$G(B,

$B0l$D$N5?;wMp?t7ONs$r(B, $B?75,$N(B Asir $B%;%C%7%g%s$GB3$1$F$?$I$k$3$H$,(B

$B$G$-$k(B.

\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.

@end itemize

@example

Line 388 Copyright (C) FUJITSU LABORATORIES LIMITED.

Line 597 Copyright (C) FUJITSU LABORATORIES LIMITED.

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{random}, @fref{lrandom}.

@end table

@node nm dn,,, $B?t$N1i;;(B

\JP @node nm dn,,, $B?t$N1i;;(B

\EG @node nm dn,,, Numbers

@subsection @code{nm}, @code{dn}

@findex nm

@findex dn

@table @t

@item nm(@var{rat})

:: @var{rat} $B$NJ,;R(B.

\JP :: @var{rat} $B$NJ,;R(B.

\EG :: Numerator of @var{rat}.

@item dn(@var{rat})

:: @var{rat} $B$NJ,Jl(B.

\JP :: @var{rat} $B$NJ,Jl(B.

\EG :: Denominator of @var{rat}.

@end table

@table @var

@item return

$B@0?t$^$?$OB?9`<0(B

\JP $B@0?t$^$?$OB?9`<0(B

\EG integer or polynomial

@item rat

$BM-M}?t$^$?$OM-M}<0(B

\JP $BM-M}?t$^$?$OM-M}<0(B

\EG rational number or rational expression

@end table

@itemize @bullet

\BJP

@item

$BM?$($i$l$?M-M}?t$^$?M-M}<0$NJ,;R5Z$SJ,Jl$rJV$9(B.

@item

Line 420 Copyright (C) FUJITSU LABORATORIES LIMITED.

Line 636 Copyright (C) FUJITSU LABORATORIES LIMITED.

$BM-M}<0$N>l9g(B, $BC1$KJ,Jl(B, $BJ,;R$r<h$j=P$9$@$1$G$"$k(B.

$BM-M}<0$KBP$7$F$O(B, $BLsJ,$O<+F0E*$K$O9T$o$l$J$$(B. @code{red()}

$B$rL@<(E*$K8F$S=P$9I,MW$,$"$k(B.

\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.

@end itemize

@example

Line 432 y

Line 666 y

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{red}.

@end table

@node conj real imag,,, $B?t$N1i;;(B

\JP @node conj real imag,,, $B?t$N1i;;(B

\EG @node conj real imag,,, Numbers

@subsection @code{conj}, @code{real}, @code{imag}

@findex conj

@table @t

@item real(@var{comp})

:: @var{comp} $B$N<B?tItJ,(B.

\JP :: @var{comp} $B$N<B?tItJ,(B.

\EG :: Real part of @var{comp}.

@item imag(@var{comp})

:: @var{comp} $B$N5u?tItJ,(B.

\JP :: @var{comp} $B$N5u?tItJ,(B.

\EG :: Imaginary part of @var{comp}.

@item conj(@var{comp})

:: @var{comp} $B$N6&LrJ#AG?t(B.

\JP :: @var{comp} $B$N6&LrJ#AG?t(B.

\EG :: Complex conjugate of @var{comp}.

@end table

@table @var

@item return comp

$BJ#AG?t(B

\JP $BJ#AG?t(B

\EG complex number

@end table

@itemize @bullet

\BJP

@item

$BJ#AG?t$KBP$7(B, $B<BIt(B, $B5uIt(B, $B6&Lr$r5a$a$k(B.

@item

$B$3$l$i$O(B, $BB?9`<0$KBP$7$F$bF/$/(B.

\BEG

@item

Basic operations for complex numbers.

@item

These functions works also for polynomials with complex coefficients.

@end itemize

@example

Line 468 y

Line 716 y

[2,11,(2-11*@@i)]

@end example

@node eval,,, $B?t$N1i;;(B

\JP @node eval deval ,,, $B?t$N1i;;(B

@subsection @code{eval}

\EG @node eval deval,,, Numbers

@subsection @code{eval}, @code{deval}

@findex eval

@findex deval

@cindex PARI

@table @t

@item eval(@var{obj}[,@var{prec}])

:: @var{obj} $B$NCM$NI>2A(B.

@item deval(@var{obj})

\JP :: @var{obj} $B$NCM$NI>2A(B.

\EG :: Evaluate @var{obj} numerically.

@end table

@table @var

@item return

$B?t$"$k$$$O<0(B

\JP $B?t$"$k$$$O<0(B

\EG number or expression

@item obj

$B0lHL$N<0(B

\JP $B0lHL$N<0(B

\EG general expression

@item prec

$B@0?t(B

\JP $B@0?t(B

\EG integer

@end table

@itemize @bullet

\BJP

@item

@var{obj} $B$K4^$^$l$kH!?t$NCM$r2DG=$J8B$jI>2A$9$k(B.

@item

$B7W;;$O(B @b{PARI} (@xref{pari}) $B$,9T$&(B.

@code{deval} $B$OG\@:EYIbF0>.?t$r7k2L$H$7$F(B

@code{eval} $B$N>l9g(B, $BM-M}?t$O$=$N$^$^;D$k(B.

@item

@code{eval} $B$K$*$$$F$O(B, $B7W;;$O(B @b{MPFR} $B%i%$%V%i%j$,9T$&(B.

@code{deval} $B$K$*$$$F$O(B, $B7W;;$O(B C $B?t3X%i%$%V%i%j$N4X?t$rMQ$$$F9T$&(B.

@item

@code{deval} $B$OJ#AG?t$O07$($J$$(B.

@item

@code{eval} $B$K$*$$$F$O(B,

@var{prec} $B$r;XDj$7$?>l9g(B, $B7W;;$O(B, 10 $B?J(B @var{prec} $B7eDxEY$G9T$o$l$k(B.

@var{prec} $B$N;XDj$,$J$$>l9g(B, $B8=:_@_Dj$5$l$F$$$k@:EY$G9T$o$l$k(B.

(@xref{setprec})

(@xref{setbprec setprec}.)

@item

@table @t

@item $B07$($kH!?t$O(B, $B<!$NDL$j(B.

Line 510 y

Line 773 y

@code{exp}, @code{log}, @code{pow(a,b) (a^b)}

@end table

@item

$B0J2<$N5-9f$r?t$H$7$FI>2A$G$-$k(B.

$B0J2<$N5-9f$r?t$H$7$FI>2A$G$-$k(B. $B$?$@$7(B @code{@@i} $B$r07$($k$N$O(B

@code{eval}, @code{deval} $B$N$_$G$"$k(B.

@table @t

@item @@i

$B5u?tC10L(B

Line 519 y

Line 783 y

@item @@e

$B<+A3BP?t$NDl(B

@end table

\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

@end itemize

@example

Line 530 y

Line 839 y

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);

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

@fref{ctrl}, @fref{setprec}, @fref{pari}.

\EG @item References

@fref{ctrl}, @fref{setbprec setprec}.

@end table

@node pari,,, $B?t$N1i;;(B

\JP @node pari,,, $B?t$N1i;;(B

\EG @node pari,,, Numbers

@subsection @code{pari}

@findex pari

@cindex PARI

@table @t

@item pari(@var{func},@var{arg},@var{prec})

:: @b{PARI} $B$NH!?t(B @var{func} $B$r8F$S=P$9(B.

\JP :: @b{PARI} $B$NH!?t(B @var{func} $B$r8F$S=P$9(B.

\EG :: Call @b{PARI} function @var{func}.

@end table

@table @var

@item return

@var{func} $BKh$K0[$J$k(B.

\JP @var{func} $BKh$K0[$J$k(B.

\EG Depends on @var{func}.

@item func

@b{PARI} $B$NH!?tL>(B

\JP @b{PARI} $B$NH!?tL>(B

\EG Function name of @b{PARI}.

@item arg

@var{func} $B$N0z?t(B

\JP @var{func} $B$N0z?t(B

\EG Arguments of @var{func}.

@item prec

$B@0?t(B

\JP $B@0?t(B

\EG integer

@end table

@itemize @bullet

\BJP

@item

@b{PARI} $B$NH!?t$r8F$S=P$9(B.

Line 570 y

Line 891 y

$BH!?tCM$NI>2A$r9bB.$K9T$&$3$H$,$G$-$k(B. @b{PARI} $B$OB>$N%W%m%0%i%`$+$i(B

$B%5%V%k!<%A%s%i%$%V%i%j$H$7$FMQ$$$k$3$H$,$G$-(B, $B$^$?(B, @samp{gp} $B$H$$$&(B

@b{PARI}$B%i%$%V%i%j$N%$%s%?%U%'!<%9$K$h$j(B UNIX $B$N%"%W%j%1!<%7%g%s$H$7$F(B

$BMxMQ$9$k$3$H$b$G$-$k(B. $B8=:_$N%P!<%8%g%s$O(B @b{1.39} $B$G$$$/$D$+$N(B ftp

$BMxMQ$9$k$3$H$b$G$-$k(B.

site ($B$?$H$($P(B @code{math.ucla.edu:/pub/pari})

$B$+$i(B anonymous ftp $B$G$-$k(B.

@item

$B:G8e$N0z?t(B @var{prec} $B$G7W;;@:EY$r;XDj$G$-$k(B.

@var{prec} $B$r>JN,$7$?>l9g(B @code{setprec()} $B$G;XDj$7$?@:EY$H$J$k(B.

@item

$B8=;~E@$G<B9T$G$-$k(B @b{PARI} $B$NH!?t$O<!$NDL$j$G$"$k(B. $B$$$:$l$b(B

$B8=;~E@$G<B9T$G$-$k(B @b{PARI} $B$N<g$JH!?t$O<!$NDL$j$G$"$k(B. $B$$$:$l$b(B

1 $B0z?t$G(B @b{Asir} $B$,BP1~$G$-$k7?$N0z?t$r$H$kH!?t$G$"$k(B.

$B<B9T$G$-$kH!?t$NA4%j%9%H$O(B asir-contrib $B$N(B ox_pari $B%^%K%e%"%k$r;2>H(B.

$B$J$*3F!9$N5!G=$K$D$$$F$O(B @b{PARI} $B$N%^%K%e%"%k$r;2>H$N$3$H(B.

\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.)

@code{abs},

@code{adj},

Line 657 site ($B$?$H$($P(B @code{math.ucla.edu:/pub/pari})

Line 1004 site ($B$?$H$($P(B @code{math.ucla.edu:/pub/pari})

@code{lngamma},

@code{logagm},

@code{mat},

@code{matinvr},

@code{matrixqz2},

@code{matrixqz3},

@code{matsize},

Line 714 site ($B$?$H$($P(B @code{math.ucla.edu:/pub/pari})

Line 1060 site ($B$?$H$($P(B @code{math.ucla.edu:/pub/pari})

@code{wf2},

@code{zeta}

\BJP

@item

@b{Asir} $B$GMQ$$$F$$$k$N$O(B @b{PARI} $B$N$[$s$N0lIt$N5!G=$G$"$k$,(B, $B:#8e(B

@b{Asir} $B$GMQ$$$F$$$k$N$O(B @b{PARI} $B$N$[$s$N0lIt$N5!G=$G$"$k(B.

$B$h$jB?$/$N5!G=$,MxMQ$G$-$k$h$&2~NI$9$kM=Dj$G$"$k(B.

\BEG

@item

@b{Asir} currently uses only a very small subset of @b{PARI}.

@end itemize

@example

/* $B9TNs$N8GM-%Y%/%H%k$r5a$a$k(B. */

\JP /* $B9TNs$N8GM-%Y%/%H%k$r5a$a$k(B. */

\EG /* Eigen vectors of a numerical matrix */

[0] pari(eigen,newmat(2,2,[[1,1],[1,2]]));

[ -1.61803398874989484819771921990 0.61803398874989484826 ]

[ 1 1 ]

/* 1 $BJQ?tB?9`<0$N:,$r5a$a$k(B. */

\JP /* 1 $BJQ?tB?9`<0$N:,$r5a$a$k(B. */

\EG /* Roots of a polynomial */

[1] pari(roots,t^2-2);

[ -1.41421356237309504876 1.41421356237309504876 ]

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

@fref{setprec}.

\EG @item References

@fref{setbprec setprec}.

@end table

@node setprec,,, $B?t$N1i;;(B

\JP @node setbprec setprec,,, $B?t$N1i;;(B

@subsection @code{setprec}

\EG @node setbprec setprec,,, Numbers

@subsection @code{setbprec}, @code{setprec}

@findex setbprec

@findex setprec

@cindex PARI

@table @t

@item setprec([@var{n}])

@item setbprec([@var{n}])

:: @b{bigfloat} $B$N7e?t$r(B @var{n} $B7e$K@_Dj$9$k(B.

@itemx setprec([@var{n}])

\JP :: @b{setbprec}, @b{setprec} $B$O(B @b{bigfloat} $B$N@:EY$r$=$l$>$l(B 2 $B?J(B, 10$B?J(B @var{n} $B7e$K@_Dj$9$k(B.

\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

$B@0?t(B

\JP $B@0?t(B

\EG integer

@item n

$B@0?t(B

\JP $B@0?t(B

\EG integer

@end table

@itemize @bullet

\BJP

@item

$B0z?t$,$"$k>l9g(B, @b{bigfloat} $B$N7e?t$r(B @var{n} $B7e$K@_Dj$9$k(B.

$B0z?t$N$"$k$J$7$K$+$+$o$i$:(B, $B0JA0$K@_Dj$5$l$F$$$?CM$rJV$9(B.

@item

@b{bigfloat} $B$N7W;;$O(B @b{PARI} (@xref{pari}) $B$K$h$C$F9T$o$l$k(B.

@b{bigfloat} $B$N7W;;$O(B @b{MPFR} $B%i%$%V%i%j$K$h$C$F9T$o$l$k(B.

@item

@b{bigfloat} $B$G$N7W;;$KBP$7M-8z$G$"$k(B.

@b{bigfloat} $B$N(B flag $B$r(B on $B$K$9$kJ}K!$O(B, @code{ctrl} $B$r;2>H(B.

@item

$B@_Dj$G$-$k7e?t$K>e8B$O$J$$$,(B, $B;XDj$7$?7e?t$K@_Dj$5$l$k$H$O(B

$B8B$i$J$$(B. $BBg$-$a$NCM$r@_Dj$9$k$N$,0BA4$G$"$k(B.

\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.

@end itemize

@example

[1] setprec();

[2] setprec(100);

[3] setprec(100);

[4] setbprec();

332

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

@fref{ctrl}, @fref{eval}, @fref{pari}.

@fref{ctrl}, @fref{eval deval}.

@end table

@node setmod,,, $B?t$N1i;;(B

\JP @node setround,,, $B?t$N1i;;(B

\EG @node setround,,, Numbers

@subsection @code{setround}

@findex setround

@table @t

@item setround([@var{mode}])

\JP :: @b{bigfloat} $B$N4]$a%b!<%I$r(B @var{mode} $B$K@_Dj$9$k(B.

\EG :: Sets the rounding mode @var{mode}.

@end table

@table @var

@item return

\JP $B@0?t(B

\EG integer

@item mode

\JP $B@0?t(B

\EG integer

@end table

@itemize @bullet

\BJP

@item

$B0z?t$,$"$k>l9g(B, @b{bigfloat} $B$N4]$a%b!<%I$r(B @var{mode} $B$K@_Dj$9$k(B.

$B0z?t$N$"$k$J$7$K$+$+$o$i$:(B, $B0JA0$K@_Dj$5$l$F$$$?CM$rJV$9(B.

$B4]$a%b!<%I$N0UL#$O<!$N$H$*$j(B.

@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} $B$G$N7W;;$KBP$7M-8z$G$"$k(B.

@b{bigfloat} $B$N(B flag $B$r(B on $B$K$9$kJ}K!$O(B, @code{ctrl} $B$r;2>H(B.

\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.'

@end itemize

@example

[1] setprec();

[2] setprec(100);

[3] setprec(100);

[4] setbprec();

332

@end example

@table @t

\JP @item $B;2>H(B

@fref{ctrl}, @fref{eval deval}.

@end table

\JP @node setmod,,, $B?t$N1i;;(B

\EG @node setmod,,, Numbers

@subsection @code{setmod}

@findex setmod

@table @t

@item setmod([@var{p}])

:: $BM-8BBN$r(B GF(@var{p}) $B$K@_Dj$9$k(B.

\JP :: $BM-8BBN$r(B GF(@var{p}) $B$K@_Dj$9$k(B.

\EG :: Sets the ground field to GF(@var{p}).

@end table

@table @var

@item return

$B@0?t(B

\JP $B@0?t(B

\EG integer

@item n

2^27 $BL$K~$NAG?t(B

\JP 2^27 $BL$K~$NAG?t(B

\EG prime less than 2^27

@end table

@itemize @bullet

\BJP

@item

$BM-8BBN$r(B GF(@var{p}) $B$K@_Dj$9$k(B. $B@_DjCM$rJV$9(B.

@item

$BM-8BBN$N85$N7?$r;}$D?t$O(B, $B$=$l<+?H$O$I$NM-8BBN$KB0$9$k$+$N>pJs$r;}$?$:(B,

$B8=:_@_Dj$5$l$F$$$kAG?t(B @var{p} $B$K$h$j(B GF(@var{p}) $B>e$G$N1i;;$,E,MQ$5$l$k(B.

@item

$B0L?t$NBg$-$JM-8BBN$K4X$7$F$O(B @pxref{$BM-8BBN$K4X$9$k1i;;(B}.

\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}.

@end itemize

@example

Line 816 return to toplevel

Line 1295 return to toplevel

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

@fref{dp_mod dp_rat}, @fref{$B?t$N7?(B}.

\EG @item References

\JP @fref{dp_mod dp_rat}, @fref{$B?t$N7?(B}.

\EG @fref{dp_mod dp_rat}, @fref{Types of numbers}.

@end table

\JP @node ntoint32 int32ton,,, $B?t$N1i;;(B

\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 :: $BHsIi@0?t$HId9f$J$7(B 32bit $B@0?t$N4V$N7?JQ49(B.

\EG :: Type-conversion between a non-negative integer and an unsigned 32bit integer.

@end table

@table @var

@item return

\JP $BId9f$J$7(B 32bit $B@0?t$^$?$OHsIi@0?t(B

\EG unsigned 32bit integer or non-negative integer

@item n

\JP 2^32 $BL$K~$NHsIi@0?t(B

\EG non-negative interger less than 2^32

@item int32

\JP $BId9f$J$7(B 32bit $B@0?t(B

\EG unsigned 32bit integer

@end table

@itemize @bullet

\BJP

@item $BHsIi@0?t(B ($B<1JL;R(B 1) $B$NId9f$J$7(B 32bit $B@0?t(B ($B<1JL;R(B 10) $B$X$NJQ49(B,

$B$^$?$O$=$N5UJQ49$r9T$&(B.

@item 32bit $B@0?t$O(B @b{OpenXM} $B$N4pK\9=@.MWAG$G$"$j(B, $B@0?t$r$=$N7?$GAw?.(B

$B$9$kI,MW$,$"$k>l9g$KMQ$$$k(B.

\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.

@end itemize

@table @t

\JP @item $B;2>H(B

\EG @item References

\JP @fref{$BJ,;67W;;(B}, @fref{$B?t$N7?(B}.

\EG @fref{Distributed computation}, @fref{Types of numbers}.

@end table

\JP @node inttorat,,, $B?t$N1i;;(B

\EG @node inttorat,,, Numbers

@subsection @code{inttorat}

@findex inttorat

@table @t

@item inttorat(@var{a},@var{m},@var{b})

\JP :: $B@0?t(B-$BM-M}?tJQ49$r9T$&(B.

\EG :: Perform the rational reconstruction.

@end table

@table @var

@item return

\JP $B%j%9%H$^$?$O(B 0

\EG list or 0

@item a

@itemx m

@itemx b

\JP $B@0?t(B

\EG integer

@end table

@itemize @bullet

\BJP

@item

$B@0?t(B @var{a} $B$KBP$7(B, @var{xa=y} mod @var{m} $B$rK~$?$9@5@0?t(B @var{x}, $B@0?t(B @var{y}

(@var{x}, @var{|y|} < @var{b}, @var{GCD(x,y)=1}) $B$r5a$a$k(B.

@item

$B$3$N$h$&$J(B @var{x}, @var{y} $B$,B8:_$9$k$J$i(B @var{[y,x]} $B$rJV$7(B, $BB8:_$7$J$$>l9g$K$O(B 0 $B$rJV$9(B.

@item

@var{b} $B$r(B @var{floor(sqrt(m/2))} $B$H<h$l$P(B, @var{x}, @var{y} $B$OB8:_$9$l$P0l0U$G$"$k(B.

@var{floor(sqrt(m/2))} $B$O(B @code{isqrt(floor(m/2))} $B$G7W;;$G$-$k(B.

@end itemize

\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

@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 $B;2>H(B

\EG @item References

@fref{floor}, @fref{isqrt}.

@end table

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