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version 1.3, 1999/12/21 02:47:32

version 1.15, 2019/09/13 09:31:00

Line 1

@comment $OpenXM$

@comment $OpenXM: OpenXM/src/asir-doc/parts/type.texi,v 1.14 2016/03/22 07:25:14 noro Exp $

\BJP

@node $B7?(B,,, Top

@chapter $B7?(B

Line 52 Each example shows possible forms of inputs for @b{Asi

@table @code

@item 0 @b{0}

\BJP

$B<B:]$K$O(B 0 $B$r<1JL;R$K$b$DBP>]$OB8:_$7$J$$(B. 0 $B$O(B, C $B$K$*$1$k(B 0 $B%]%$%s%?$K(B

$B$h$jI=8=$5$l$F$$$k(B. $B$7$+$7(B, $BJX59>e(B @b{Asir} $B$N(B @code{type(0)} $B$O(B

Line 83 x afo (2.3*x+y)^10

\BJP

$BB?9`<0$O(B, $BA4$FE83+$5$l(B, $B$=$N;~E@$K$*$1$kJQ?t=g=x$K=>$C$F(B, $B:F5"E*$K(B

1 $BJQ?tB?9`<0$H$7$F9_QQ$N=g$K@0M}$5$l$k(B (@xref{$BJ,;6I=8=B?9`<0(B}).

1 $BJQ?tB?9`<0$H$7$F9_QQ$N=g$K@0M}$5$l$k(B. (@xref{$BJ,;6I=8=B?9`<0(B}.)

$B$3$N;~(B, $B$=$NB?9`<0$K8=$l$k=g=x:GBg$NJQ?t$r(B @b{$B<gJQ?t(B} $B$H8F$V(B.

\BEG

Every polynomial is maintained internally in its full expanded form,

represented as a nested univariate polynomial, according to the current

variable ordering, arranged by the descending order of exponents.

(@xref{Distributed polynomial}).

(@xref{Distributed polynomial}.)

In the representation, the indeterminate (or variable), appearing in

the polynomial, with maximum ordering is called the @b{main variable}.

Moreover, we call the coefficient of the maximum degree term of

Line 207 on the whole value of that vector.

[1] for (I=0;I<3;I++)A3[I] = newvect(3);

[2] for (I=0;I<3;I++)for(J=0;J<3;J++)A3[I][J]=newvect(3);

[3] A3;

[ [ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ] [ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ] [ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ] ]

[ [ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ]

[ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ]

[ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ] ]

[4] A3[0];

[ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ]

[5] A3[0][0];

Line 288 afotake

Line 290 afotake

newstruct(afo)

@end example

\JP $B9=B$BN$K4X$7$F$O(B, $B>O$r2~$a$F2r@b$9$kM=Dj$G$"$k(B.

\BJP

\EG For type @b{structure}, we shall describe it in a later chapter.

Asir $B$K$*$1$k9=B$BN$O(B, C $B$K$*$1$k9=B$BN$r4J0W2=$7$?$b$N$G$"$k(B.

(Not written yet.)

$B8GDjD9G[Ns$N3F@.J,$rL>A0$G%"%/%;%9$G$-$k%*%V%8%'%/%H$G(B,

$B9=B$BNDj5AKh$KL>A0$r$D$1$k(B.

\BEG

The type @b{structure} is a simplified version of that in C language.

It is defined as a fixed length array and each entry of the array

is accessed by its name. A name is associated with each structure.

\JP @item 9 @b{$BJ,;6I=8=B?9`<0(B}

\EG @item 9 @b{distributed polynomial}

Line 320 For details @xref{Groebner basis computation}.

Line 330 For details @xref{Groebner basis computation}.

\JP @item 11 @b{$B%(%i!<%*%V%8%'%/%H(B}

\EG @item 11 @b{error object}

\JP $B0J>eFs$D$O(B, Open XM $B$K$*$$$FMQ$$$i$l$kFC<l%*%V%8%'%/%H$G$"$k(B.

\EG These are special objects used for OpenXM.

\JP @item 12 @b{GF(2) $B>e$N9TNs(B}

\EG @item 12 @b{matrix over GF(2)}

\BJP

$B8=:_(B, $BI8?t(B 2 $B$NM-8BBN$K$*$1$k4pDlJQ49$N$?$a$N%*%V%8%'%/%H$H$7$FMQ$$$i$l(B

$B$k(B.

Line 337 This is used for basis conversion in finite fields of

Line 347 This is used for basis conversion in finite fields of

\JP @item 13 @b{MATHCAP $B%*%V%8%'%/%H(B}

\EG @item 13 @b{MATHCAP object}

\JP Open XM $B$K$*$$$F(B, $B<BAu$5$l$F$$$k5!G=$rAw<u?.$9$k$?$a$N%*%V%8%'%/%H$G$"$k(B.

\EG This object is used to express available funcionalities for Open XM.

@item 14 @b{first order formula}

\JP quantifier elimination $B$GMQ$$$i$l$k0l3,=R8lO@M}<0(B.

\EG This expresses a first order formula used in quantifier elimination.

@item 15 @b{matrix over GF(@var{p})}

\JP $B>.I8?tM-8BBN>e$N9TNs(B.

\EG A matrix over a small finite field.

@item 16 @b{byte array}

\JP $BId9f$J$7(B byte $B$NG[Ns(B

\EG An array of unsigned bytes.

\JP @item 26 @b{$BJ,;6I=8=2C72B?9`<0(B}

\EG @item 26 @b{distributed module polynomial}

@example

2*<<0,1,2,3:1>>-3*<<1,2,3,4:2>>

@end example

\BJP

$B$3$l$O(B, $BB?9`<04D>e$N<+M32C72$N85$r(B, $B2C72C19`<0$NOB$H$7$FFbItI=8=$7$?$b$N$G$"$k(B.

$B$3$3$G(B, $B2C72C19`<0$H$OC19`<0$H2C72$NI8=`4pDl$N@Q$G$"$k(B.

$B$3$l$K$D$$$F$O(B @xref{$B%0%l%V%J4pDl$N7W;;(B}.

\BEG

This represents an element in a free module over a polynomial ring

as a linear sum of module monomials, where a module monomial is

the product of a monomial in the polynomial ring and a standard base of the free module.

For details @xref{Groebner basis computation}.

\JP @item -1 @b{VOID $B%*%V%8%'%/%H(B}

\EG @item -1 @b{VOID object}

\JP $B7?<1JL;R(B -1 $B$r$b$D%*%V%8%'%/%H$O4X?t$NLa$jCM$J$I$,L58z$G$"$k$3$H$r<($9(B.

\BEG

The object with the object identifier -1 indicates that a return value

Line 369 of a function is void.

Line 407 of a function is void.

@item 0

\JP @b{$BM-M}?t(B}

\EG @b{rational number}

\BJP

$BM-M}?t$O(B, $BG$0UB?G\D9@0?t(B (@b{bignum}) $B$K$h$j<B8=$5$l$F$$$k(B. $BM-M}?t$O>o$K(B

$B4{LsJ,?t$GI=8=$5$l$k(B.

Line 383 lowest terms.

Line 421 lowest terms.

@item 1

\JP @b{$BG\@:EYIbF0>.?t(B}

\EG @b{double precision floating point number (double float)}

\BJP

$B%^%7%s$NDs6!$9$kG\@:EYIbF0>.?t$G$"$k(B. @b{Asir} $B$N5/F0;~$K$O(B,

$BDL>o$N7A<0$GF~NO$5$l$?IbF0>.?t$O$3$N7?$KJQ49$5$l$k(B. $B$?$@$7(B,

Line 423 result shall be computed as a double float number.

Line 461 result shall be computed as a double float number.

@item 2

\JP @b{$BBe?tE*?t(B}

\EG @b{algebraic number}

\JP @xref{$BBe?tE*?t$K4X$9$k1i;;(B}.

\EG @xref{Algebraic numbers}.

@item 3

@b{bigfloat}

\BJP

@b{bigfloat} $B$O(B, @b{Asir} $B$G$O(B @b{PARI} $B%i%$%V%i%j$K$h$j(B

@b{bigfloat} $B$O(B, @b{Asir} $B$G$O(B @b{MPFR} $B%i%$%V%i%j$K$h$j(B

$B<B8=$5$l$F$$$k(B. @b{PARI} $B$K$*$$$F$O(B, @b{bigfloat} $B$O(B, $B2>?tIt(B

$B<B8=$5$l$F$$$k(B. @b{MPFR} $B$K$*$$$F$O(B, @b{bigfloat} $B$O(B, $B2>?tIt(B

$B$N$_G$0UB?G\D9$G(B, $B;X?tIt$O(B 1 $B%o!<%I0JFb$N@0?t$K8B$i$l$F$$$k(B.

$B$N$_G$0UB?G\D9$G(B, $B;X?tIt$O(B 64bit $B@0?t$G$"$k(B.

@code{ctrl()} $B$G(B @b{bigfloat} $B$rA*Br$9$k$3$H$K$h$j(B, $B0J8e$NIbF0>.?t(B

$B$NF~NO$O(B @b{bigfloat} $B$H$7$F07$o$l$k(B. $B@:EY$O%G%U%)%k%H$G$O(B

10 $B?J(B 9 $B7eDxEY$G$"$k$,(B, @code{setprec()} $B$K$h$j;XDj2DG=$G$"$k(B.

10 $B?J(B 9 $B7eDxEY$G$"$k$,(B, @code{setprec()}, @code{setbprec()} $B$K$h$j;XDj2DG=$G$"$k(B.

\BEG

The @b{bigfloat} numbers of @b{Asir} is realized by @b{PARI} library.

The @b{bigfloat} numbers of @b{Asir} is realized by @b{MPFR} library.

A @b{bigfloat} number of @b{PARI} has an arbitrary precision mantissa

A @b{bigfloat} number of @b{MPFR} has an arbitrary precision mantissa

part. However, its exponent part admits only an integer with a single

part. However, its exponent part admits only a 64bit integer.

word precision.

Floating point operations will be performed all in @b{bigfloat} after

activating the @b{bigfloat} switch by @code{ctrl()} command.

The default precision is about 9 digits, which can be specified by

The default precision is 53 bits (about 15 digits), which can be specified by

@code{setprec()} command.

@code{setbprec()} and @code{setprec()} command.

@example

[0] ctrl("bigfloat",1);

[1] eval(2^(1/2));

1.414213562373095048763788073031

1.4142135623731

[2] setprec(100);

[3] eval(2^(1/2));

1.41421356237309504880168872420969807856967187537694807317654396116148

1.41421356237309504880168872420969807856967187537694...764157

[4] setbprec(100);

332

[5] 1.41421356237309504880168872421

@end example

\BJP

@code{eval()} $B$O(B, $B0z?t$K4^$^$l$kH!?tCM$r2DG=$J8B$j?tCM2=$9$kH!?t$G$"$k(B.

@code{setprec()} $B$G;XDj$5$l$?7e?t$O(B, $B7k2L$N@:EY$rJ]>Z$9$k$b$N$G$O$J$/(B,

@code{setbprec()} $B$G;XDj$5$l$?(B2 $B?J7e?t$O(B, $B4]$a%b!<%I$K1~$8$?7k2L$N@:EY$rJ]>Z$9$k(B. @code{setprec()} $B$G;XDj$5$l$k(B10$B?J7e?t$O(B 2 $B?J7e?t$KJQ49$5$l$F@_Dj$5$l$k(B.

@b{PARI} $BFbIt$GMQ$$$i$l$kI=8=$N%5%$%:$r<($9$3$H$KCm0U$9$Y$-$G$"$k(B.

\BEG

Function @code{eval()} evaluates numerically its argument as far as

possible.

Notice that the integer given for the argument of @code{setprec()} does

Notice that the integer given for the argument of @code{setbprec()}

not guarantee the accuracy of the result,

guarantees the accuracy of the result according to the current rounding mode.

but it indicates the representation size of numbers with which internal

The argument of @code{setbprec()} is converted to the corresonding bit length

operations of @b{PARI} are performed.

and set.

(@ref{eval}, @xref{pari})

(@xref{eval deval}.)

@item 4

\JP @b{$BJ#AG?t(B}

\EG @b{complex number}

\BJP

$BJ#AG?t$O(B, $BM-M}?t(B, $BG\@:EYIbF0>.?t(B, @b{bigfloat} $B$r<BIt(B, $B5uIt$H$7$F(B

@code{a+b*@@i} (@@i $B$O5u?tC10L(B) $B$H$7$FM?$($i$l$k?t$G$"$k(B. $B<BIt(B, $B5uIt$O(B

Line 497 taken out by @code{real()} and @code{imag()} respectiv

Line 537 taken out by @code{real()} and @code{imag()} respectiv

@item 5

\JP @b{$B>.I8?t$NM-8BAGBN$N85(B}

\EG @b{element of a small finite prime field}

\BJP

$B$3$3$G8@$&>.I8?t$H$O(B, $BI8?t$,(B 2^27 $BL$K~$N$b$N$N$3$H$G$"$k(B. $B$3$N$h$&$JM-8B(B

$BBN$O(B, $B8=:_$N$H$3$m%0%l%V%J4pDl7W;;$K$*$$$FFbItE*$KMQ$$$i$l(B, $BM-8BBN78?t$N(B

Line 520 field operations are executed by using a prime @var{p}

Line 560 field operations are executed by using a prime @var{p}

@item 6

\JP @b{$BBgI8?t$NM-8BAGBN$N85(B}

\EG @b{element of large finite prime field}

\BJP

$BI8?t$H$7$FG$0U$NAG?t$,$H$l$k(B.

$B$3$N7?$N?t$O(B, $B@0?t$KBP$7(B@code{simp_ff} $B$rE,MQ$9$k$3$H$K$h$jF@$i$l$k(B.

Line 534 is an arbitrary prime. An object of this type is obtai

Line 574 is an arbitrary prime. An object of this type is obtai

@item 7

\JP @b{$BI8?t(B 2 $B$NM-8BBN$N85(B}

\EG @b{element of a finite field of characteristic 2}

\BJP

$BI8?t(B 2 $B$NG$0U$NM-8BBN$N85$rI=8=$9$k(B. $BI8?t(B 2 $B$NM-8BBN(B F $B$O(B, $B3HBg<!?t(B

[F:GF(2)] $B$r(B n $B$H$9$l$P(B, GF(2) $B>e4{Ls$J(B n $B<!B?9`<0(B f(t) $B$K$h$j(B

Line 544 g mod f $B$O(B, g, f $B$r$"$i$o$9(B 2 $B$D$N%S%C

Line 584 g mod f $B$O(B, g, f $B$r$"$i$o$9(B 2 $B$D$N%S%C

\BEG

This type expresses an element of a finite field of characteristic 2.

Let @var{F} be a finite field of characteristic 2. If @var{[F:GF(2)]}

Let @var{F} be a finite field of characteristic 2. If [F:GF(2)]

is equal to @var{n}, then @var{F} is expressed as @var{F=GF(2)[t]/(f(t))},

is equal to @var{n}, then @var{F} is expressed as F=GF(2)[t]/(f(t)),

where @var{f(t)} is an irreducible polynomial over @var{GF(2)}

where f(t) is an irreducible polynomial over GF(2)

of degree @var{n}.

As an element @var{g} of @var{GF(2)[t]} can be expressed by a bit string,

As an element @var{g} of GF(2)[t] can be expressed by a bit string,

An element @var{g mod f} in @var{F} can be expressed by two bit strings

representing @var{g} and @var{f} respectively.

Line 559 representing @var{g} and @var{f} respectively.

Line 599 representing @var{g} and @var{f} respectively.

@itemize @bullet

@item

@code{@@}

\BJP

@code{@@} $B$O$=$N8e$m$K?t;z(B, $BJ8;z$rH<$C$F(B, $B%R%9%H%j$dFC<l$J?t$r$"$i$o$9$,(B,

$BC1FH$G8=$l$?>l9g$K$O(B, F=GF(2)[t]/(f(t)) $B$K$*$1$k(B t mod f $B$r$"$i$o$9(B.

$B$h$C$F(B, @@ $B$NB?9`<0$H$7$F(B F $B$N85$rF~NO$G$-$k(B. (@@^10+@@+1 $B$J$I(B)

\BEG

@code{@@} represents @var{t mod f} in @var{F=GF(2)[t](f(t))}.

@code{@@} represents @var{t mod f} in F=GF(2)[t](f(t)).

By using @code{@@} one can input an element of @var{F}. For example

@code{@@^10+@@+1} represents an element of @var{F}.

@item

@code{ptogf2n}

\JP $BG$0UJQ?t$N(B 1 $BJQ?tB?9`<0$r(B, @code{ptogf2n} $B$K$h$jBP1~$9$k(B F $B$N85$KJQ49$9$k(B.

\BEG

@code{ptogf2n} converts a univariate polynomial into an element of @var{F}.

Line 581 } one can input an element of @var{F}

Line 621 } one can input an element of @var{F}

@item

@code{ntogf2n}

\BJP

$BG$0U$N<+A3?t$r(B, $B<+A3$J;EJ}$G(B F $B$N85$H$_$J$9(B. $B<+A3?t$H$7$F$O(B, 10 $B?J(B,

16 $B?J(B (0x $B$G;O$^$k(B), 2 $B?J(B (0b $B$G;O$^$k(B) $B$GF~NO$,2DG=$G$"$k(B.

Line 595 hexadecimal (@code{0x} prefix) and binary (@code{0b} p

Line 635 hexadecimal (@code{0x} prefix) and binary (@code{0b} p

@item

\JP @code{$B$=$NB>(B}

\EG @code{micellaneous}

\BJP

$BB?9`<0$N78?t$r4]$4$H(B F $B$N85$KJQ49$9$k$h$&$J>l9g(B, @code{simp_ff}

$B$K$h$jJQ49$G$-$k(B.

Line 606 coefficients of a polynomial.

Line 646 coefficients of a polynomial.

@end itemize

@item 8

\JP @b{$B0L?t(B @var{p^n} $B$NM-8BBN$N85(B}

\EG @b{element of a finite field of characteristic @var{p^n}}

\BJP

$B0L?t$,(B @var{p^n} (@var{p} $B$OG$0U$NAG?t(B, @var{n} $B$O@5@0?t(B) $B$O(B,

$BI8?t(B @var{p} $B$*$h$S(B GF(@var{p}) $B>e4{Ls$J(B @var{n} $B<!B?9`<0(B m(x)

$B$r(B @code{setmod_ff} $B$K$h$j;XDj$9$k$3$H$K$h$j@_Dj$9$k(B.

$B$3$NBN$N85$O(B m(x) $B$rK!$H$9$k(B GF(@var{p}) $B>e$NB?9`<0$H$7$F(B

$BI=8=$5$l$k(B.

\BEG

A finite field of order @var{p^n}, where @var{p} is an arbitrary prime

and @var{n} is a positive integer, is set by @code{setmod_ff}

by specifying its characteristic @var{p} and an irreducible polynomial

of degree @var{n} over GF(@var{p}). An element of this field

is represented by a polynomial over GF(@var{p}) modulo m(x).

@item 9

\JP @b{$B0L?t(B @var{p^n} $B$NM-8BBN$N85(B ($B>.0L?t(B)}

\EG @b{element of a finite field of characteristic @var{p^n} (small order)}

\BJP

$B0L?t$,(B @var{p^n} $B$NM-8BBN(B (@var{p^n} $B$,(B @var{2^29} $B0J2<(B, @var{p} $B$,(B @var{2^14} $B0J>e(B

$B$J$i(B @var{n} $B$O(B 1) $B$O(B,

$BI8?t(B @var{p} $B$*$h$S3HBg<!?t(B @var{n}

$B$r(B @code{setmod_ff} $B$K$h$j;XDj$9$k$3$H$K$h$j@_Dj$9$k(B.

$B$3$NBN$N(B 0 $B$G$J$$85$O(B, @var{p} $B$,(B @var{2^14} $BL$K~$N>l9g(B,

GF(@var{p^n}) $B$N>hK!72$N@8@.85$r8GDj$9$k$3$H(B

$B$K$h$j(B, $B$3$N85$N$Y$-$H$7$FI=$5$l$k(B. $B$3$l$K$h$j(B, $B$3$NBN$N(B 0 $B$G$J$$85(B

$B$O(B, $B$3$N$Y$-;X?t$H$7$FI=8=$5$l$k(B. @var{p} $B$,(B @var{2^14} $B0J>e(B

$B$N>l9g$ODL>o$N>jM>$K$h$kI=8=$H$J$k$,(B, $B6&DL$N%W%m%0%i%`$G(B

$BAPJ}$N>l9g$r07$($k$h$&$K$3$N$h$&$J;EMM$H$J$C$F$$$k(B.

\BEG

A finite field of order @var{p^n}, where @var{p^n} must be less than

@var{2^29} and @var{n} must be equal to 1 if @var{p} is greater or

equal to @var{2^14},

is set by @code{setmod_ff}

by specifying its characteristic @var{p} the extension degree

@var{n}. If @var{p} is less than @var{2^14}, each non-zero element

of this field

is a power of a fixed element, which is a generator of the multiplicative

group of the field, and it is represented by its exponent.

Otherwise, each element is represented by the redue modulo @var{p}.

This specification is useful for treating both cases in a single

program.

@item 10

\JP @b{$B0L?t(B @var{p^n} $B$N>.0L?tM-8BBN$NBe?t3HBg$N85(B}

\EG @b{element of a finite field which is an algebraic extension of a small finite field of characteristic @var{p^n}}

\BJP

$BA09`$N(B, $B0L?t$,(B @var{p^n} $B$N>.0L?tM-8BBN$N(B @var{m} $B<!3HBg$N85$G$"$k(B.

$BI8?t(B @var{p} $B$*$h$S3HBg<!?t(B @var{n}, @var{m}

$B$r(B @code{setmod_ff} $B$K$h$j;XDj$9$k$3$H$K$h$j@_Dj$9$k(B. $B4pACBN>e$N(B @var{m}

$B<!4{LsB?9`<0$,<+F0@8@.$5$l(B, $B$=$NBe?t3HBg$N@8@.85$NDj5AB?9`<0$H$7$FMQ$$$i$l$k(B.

$B@8@.85$O(B @code{@@s} $B$G$"$k(B.

\BEG

An extension field @var{K} of the small finite field @var{F} of order @var{p^n}

is set by @code{setmod_ff}

by specifying its characteristic @var{p} the extension degree

@var{n} and @var{m}=[@var{K}:@var{F}]. An irreducible polynomial of degree @var{m}

over @var{K} is automatically generated and used as the defining polynomial of

the generator of the extension @var{K/F}. The generator is denoted by @code{@@s}.

@item 11

\JP @b{$BJ,;6I=8=$NBe?tE*?t(B}

\EG @b{algebraic number represented by a distributed polynomial}

\JP @xref{$BBe?tE*?t$K4X$9$k1i;;(B}.

\EG @xref{Algebraic numbers}.

\BJP

\BEG

@end table

\BJP

$BBgI8?tAGBN$NI8?t(B, $BI8?t(B 2 $B$NM-8BBN$NDj5AB?9`<0$O(B, @code{setmod_ff}

$B>.I8?tM-8BAGBN0J30$NM-8BBN$O(B @code{setmod_ff} $B$G@_Dj$9$k(B.

$B$G@_Dj$9$k(B.

$BM-8BBN$N85$I$&$7$N1i;;$G$O(B,

$BM-8BBN$N85$I$&$7$N1i;;$G$O(B, @code{setmod_ff} $B$K$h$j@_Dj$5$l$F$$$k(B

modulus $B$G(B, $BB0$9$kBN$,J,$+$j(B, $B$=$NCf$G1i;;$,9T$o$l$k(B.

$B0lJ}$,M-M}?t$N>l9g$K$O(B, $B$=$NM-M}?t$O<+F0E*$K8=:_@_Dj$5$l$F$$$k(B

$BM-8BBN$N85$KJQ49$5$l(B, $B1i;;$,9T$o$l$k(B.

\BEG

The characteristic of a large finite prime field and the defining

Finite fields other than small finite prime fields are

polynomial of a finite field of characteristic 2 are set by @code{setmod_ff}.

set by @code{setmod_ff}.

Elements of finite fields do not have informations about the modulus.

Upon an arithmetic operation, the modulus set by @code{setmod_ff} is

Upon an arithmetic operation, i

used. If one of the operands is a rational number, it is automatically

f one of the operands is a rational number, it is automatically

converted into an element of the finite field currently set and

the operation is done in the finite field.

Line 659 and further are classified into sub-types of the type

Line 783 and further are classified into sub-types of the type

@item 0

\JP @b{$B0lHLITDj85(B}

\EG @b{ordinary indeterminate}

\JP $B1Q>.J8;z$G;O$^$kJ8;zNs(B. $BB?9`<0$NJQ?t$H$7$F:G$bIaDL$KMQ$$$i$l$k(B.

\BEG

An object of this sub-type is denoted by a string that start with

Line 677 polynomials.

Line 801 polynomials.

@item 1

\JP @b{$BL$Dj78?t(B}

\EG @b{undetermined coefficient}

\BJP

@code{uc()} $B$O(B, @samp{_} $B$G;O$^$kJ8;zNs$rL>A0$H$9$kITDj85$r@8@.$9$k(B.

$B$3$l$i$O(B, $B%f!<%6$,F~NO$G$-$J$$$H$$$&$@$1$G(B, $B0lHLITDj85$HJQ$o$i$J$$$,(B,

Line 704 _0

Line 828 _0

@item 2

\JP @b{$BH!?t7A<0(B}

\EG @b{function form}

\BJP

$BAH$_9~$_H!?t(B, $B%f!<%6H!?t$N8F$S=P$7$O(B, $BI>2A$5$l$F2?$i$+$N(B @b{Asir} $B$N(B

$BFbIt7A<0$KJQ49$5$l$k$,(B, @code{sin(x)}, @code{cos(x+1)} $B$J$I$O(B, $BI>2A8e(B

Line 736 sin(x)

Line 860 sin(x)

@item 3

\JP @b{$BH!?t;R(B}

\EG @b{functor}

\BJP

$BH!?t8F$S=P$7$O(B, @var{fname(args)} $B$H$$$&7A$G9T$J$o$l$k$,(B, @var{fname} $B$N(B

$BH!?t8F$S=P$7$O(B, @var{fname}(@var{args}) $B$H$$$&7A$G9T$J$o$l$k$,(B, @var{fname} $B$N(B

$BItJ,$rH!?t;R$H8F$V(B. $BH!?t;R$K$O(B, $BH!?t$N<oN`$K$h$jAH$_9~$_H!?t;R(B,

$B%f!<%6Dj5AH!?t;R(B, $B=iEyH!?t;R$J$I$,$"$k$,(B, $BH!?t;R$OC1FH$GITDj85$H$7$F(B

$B5!G=$9$k(B.

\BEG

A function call (or a function form) has a form @var{fname(args)}.

A function call (or a function form) has a form @var{fname}(@var{args}).

Here, @var{fname} alone is called a @b{functor}.

There are several kinds of @b{functor}s: built-in functor, user defined

functor and functor for the elementary functions. A functor alone is

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