===================================================================
RCS file: /home/cvs/OpenXM/src/asir-doc/parts/type.texi,v
retrieving revision 1.7
retrieving revision 1.14
diff -u -p -r1.7 -r1.14
--- OpenXM/src/asir-doc/parts/type.texi	2000/11/13 00:16:35	1.7
+++ OpenXM/src/asir-doc/parts/type.texi	2016/03/22 07:25:14	1.14
@@ -1,4 +1,4 @@
-@comment $OpenXM: OpenXM/src/asir-doc/parts/type.texi,v 1.6 2000/09/23 07:53:25 noro Exp $
+@comment $OpenXM: OpenXM/src/asir-doc/parts/type.texi,v 1.13 2007/02/15 02:41:38 noro Exp $
 \BJP
 @node $B7?(B,,, Top
 @chapter $B7?(B
@@ -83,14 +83,14 @@ 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. 
 \E
 \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
@@ -207,7 +207,8 @@ 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 ] ] ]
 [4] A3[0];
 [ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ]
@@ -355,7 +356,7 @@ This is used for basis conversion in finite fields of 
 \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(p)}
+@item 15 @b{matrix over GF(@var{p})}
 @*
 \JP $B>.I8?tM-8BBN>e$N9TNs(B.
 \EG A matrix over a small finite field.
@@ -450,49 +451,51 @@ result shall be computed as a double float number.
 @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<B8=$5$l$F$$$k(B. @b{PARI} $B$K$*$$$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$$$k(B. 
+@b{bigfloat} $B$O(B, @b{Asir} $B$G$O(B @b{MPFR} $B%i%$%V%i%j$K$h$j(B
+$B<B8=$5$l$F$$$k(B. @b{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 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. 
 \E
 \BEG
-The @b{bigfloat} numbers of @b{Asir} is realized by @b{PARI} library.
-A @b{bigfloat} number of @b{PARI} has an arbitrary precision mantissa
-part.  However, its exponent part admits only an integer with a single
-word precision.
+The @b{bigfloat} numbers of @b{Asir} is realized by @b{MPFR} library.
+A @b{bigfloat} number of @b{MPFR} has an arbitrary precision mantissa
+part.  However, its exponent part admits only a 64bit integer.
 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
-@code{setprec()} command.
+The default precision is 53 bits (about 15 digits), which can be specified by
+@code{setbprec()} and @code{setprec()} command.
 \E
 
 @example
 [0] ctrl("bigfloat",1);
 1
 [1] eval(2^(1/2));
-1.414213562373095048763788073031
+1.4142135623731
 [2] setprec(100);      
-9
+15
 [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, 
-@b{PARI} $BFbIt$GMQ$$$i$l$kI=8=$N%5%$%:$r<($9$3$H$KCm0U$9$Y$-$G$"$k(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.
+
 \E
 \BEG
 Function @code{eval()} evaluates numerically its argument as far as
 possible.
-Notice that the integer given for the argument of @code{setprec()} does
-not guarantee the accuracy of the result,
-but it indicates the representation size of numbers with which internal
-operations of @b{PARI} are performed.
+Notice that the integer given for the argument of @code{setbprec()}
+guarantees the accuracy of the result according to the current rounding mode.
+The argument of @code{setbprec()} is converted to the corresonding bit length
+and set.
 \E
-(@ref{eval}, @xref{pari})
+(@xref{eval deval}.)
 
 @item 4
 \JP @b{$BJ#AG?t(B}
@@ -563,11 +566,11 @@ g mod f $B$O(B, g, f  $B$r$"$i$o$9(B 2 $B$D$N%S%C
 \E
 \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)]}
-is equal to @var{n}, then @var{F} is expressed as @var{F=GF(2)[t]/(f(t))},
-where @var{f(t)} is an irreducible polynomial over @var{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 F=GF(2)[t]/(f(t)),
+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.
 \E
@@ -585,7 +588,7 @@ representing @var{g} and @var{f} respectively.
 $B$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)
 \E
 \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}.
 \E
@@ -625,22 +628,106 @@ coefficients of a polynomial.
 \E
 
 @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$k(B. 
+\E
+\BEG
+A finite field of order @var{p^n}, where @var{p} is an arbitrary prime
+and @var{n} is a positive integer, is set by @code{setmod_ff}
+by specifying its characteristic @var{p} and an irreducible polynomial
+of degree @var{n} over GF(@var{p}). An element of this field
+is represented by a polynomial over GF(@var{p}) modulo m(x).
+\E
+
+@item 9
+\JP @b{$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$$$k(B. 
+
+\E
+\BEG
+A finite field of order @var{p^n}, where @var{p^n} must be less than
+@var{2^29} and @var{n} must be equal to 1 if @var{p} is greater or
+equal to @var{2^14}, 
+is set by @code{setmod_ff}
+by specifying its characteristic @var{p} the extension degree
+@var{n}. If @var{p} is less than @var{2^14}, each non-zero element 
+of this field
+is a power of a fixed element, which is a generator of the multiplicative
+group of the field, and it is represented by its exponent.
+Otherwise, each element is represented by the redue modulo @var{p}.
+This specification is useful for treating both cases in a single 
+program.
+\E
+
+@item 10
+\JP @b{$B0L?t(B @var{p^n} $B$N>.0L?tM-8BBN$NBe?t3HBg$N85(B}
+\EG @b{element of a finite field which is an algebraic extension of a small finite field of characteristic @var{p^n}}
+
+\BJP
+$BA09`$N(B, $B0L?t$,(B @var{p^n} $B$N>.0L?tM-8BBN$N(B @var{m} $B<!3HBg$N85$G$"$k(B.
+$BI8?t(B @var{p} $B$*$h$S3HBg<!?t(B @var{n}, @var{m} 
+$B$r(B @code{setmod_ff} $B$K$h$j;XDj$9$k$3$H$K$h$j@_Dj$9$k(B. $B4pACBN>e$N(B @var{m}
+$B<!4{LsB?9`<0$,<+F0@8@.$5$l(B, $B$=$NBe?t3HBg$N@8@.85$NDj5AB?9`<0$H$7$FMQ$$$i$l$k(B.
+$B@8@.85$O(B @code{@@s} $B$G$"$k(B.
+
+\E
+\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}.
+\E
+
+@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
+
+\E
+\BEG
+\E
 @end table
 
 \BJP
-$BBgI8?tAGBN$NI8?t(B, $BI8?t(B 2 $B$NM-8BBN$NDj5AB?9`<0$O(B, @code{setmod_ff}
-$B$G@_Dj$9$k(B. 
-$BM-8BBN$N85$I$&$7$N1i;;$G$O(B, @code{setmod_ff} $B$K$h$j@_Dj$5$l$F$$$k(B
-modulus $B$G(B, $BB0$9$kBN$,J,$+$j(B, $B$=$NCf$G1i;;$,9T$o$l$k(B. 
+$B>.I8?tM-8BAGBN0J30$NM-8BBN$O(B @code{setmod_ff} $B$G@_Dj$9$k(B. 
+$BM-8BBN$N85$I$&$7$N1i;;$G$O(B,
 $B0lJ}$,M-M}?t$N>l9g$K$O(B, $B$=$NM-M}?t$O<+F0E*$K8=:_@_Dj$5$l$F$$$k(B
 $BM-8BBN$N85$KJQ49$5$l(B, $B1i;;$,9T$o$l$k(B. 
 \E
 \BEG
-The characteristic of a large finite prime field and the defining
-polynomial of a finite field of characteristic 2 are set by @code{setmod_ff}.
+Finite fields other than small finite prime fields are
+set by @code{setmod_ff}.
 Elements of finite fields do not have informations about the modulus.
-Upon an arithmetic operation, the modulus set by @code{setmod_ff} is
-used. If one of the operands is a rational number, it is automatically
+Upon an arithmetic operation, i
+f one of the operands is a rational number, it is automatically
 converted into an element of the finite field currently set and
 the operation is done in the finite field.
 \E
@@ -757,13 +844,13 @@ sin(x)
 \EG @b{functor}
 @*
 \BJP
-$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. 
 \E
 \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