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Annotation of OpenXM/src/asir-doc/parts/type.texi, Revision 1.7

1.7     ! noro        1: @comment $OpenXM: OpenXM/src/asir-doc/parts/type.texi,v 1.6 2000/09/23 07:53:25 noro Exp $
1.3       noro        2: \BJP
1.1       noro        3: @node $B7?(B,,, Top
                      4: @chapter $B7?(B
1.3       noro        5: \E
                      6: \BEG
                      7: @node Data types,,, Top
                      8: @chapter Data types
                      9: \E
1.1       noro       10:
                     11: @menu
1.3       noro       12: \BJP
1.1       noro       13: * Asir $B$G;HMQ2DG=$J7?(B::
                     14: * $B?t$N7?(B::
                     15: * $BITDj85$N7?(B::
1.3       noro       16: \E
                     17: \BEG
                     18: * Types in Asir::
                     19: * Types of numbers::
                     20: * Types of indeterminates::
                     21: \E
1.1       noro       22: @end menu
                     23:
1.3       noro       24: \BJP
1.1       noro       25: @node Asir $B$G;HMQ2DG=$J7?(B,,, $B7?(B
                     26: @section @b{Asir} $B$G;HMQ2DG=$J7?(B
1.3       noro       27: \E
                     28: \BEG
                     29: @node Types in Asir,,, Data types
                     30: @section Types in @b{Asir}
                     31: \E
1.1       noro       32:
                     33: @noindent
1.3       noro       34: \BJP
1.1       noro       35: @b{Asir} $B$K$*$$$F$O(B, $B2DFI$J7A<0$GF~NO$5$l$?$5$^$6$^$JBP>]$O(B, $B%Q!<%6$K$h$j(B
                     36: $BCf4V8@8l$KJQ49$5$l(B, $B%$%s%?%W%j%?$K$h$j(B @b{Risa} $B$N7W;;%(%s%8%s$r8F$S=P$7(B
                     37: $B$J$,$iFbIt7A<0$KJQ49$5$l$k(B. $BJQ49$5$l$?BP>]$O(B, $B<!$N$$$:$l$+$N7?$r;}$D(B.
                     38: $B3FHV9f$O(B, $BAH$_9~$_H!?t(B @code{type()} $B$K$h$jJV$5$l$kCM$KBP1~$7$F$$$k(B.
                     39: $B3FNc$O(B, @b{Asir} $B$N%W%m%s%W%H$KBP$9$kF~NO$,2DG=$J7A<0$N$$$/$D$+$r(B
                     40: $B<($9(B.
1.3       noro       41: \E
                     42: \BEG
                     43: In @b{Asir}, various objects described according to the syntax of
                     44: @b{Asir} are translated to intermediate forms and by @b{Asir}
                     45: interpreter further translated into internal forms with the help of
                     46: basic algebraic engine.  Such an object in an internal form has one of
                     47: the following types listed below.
                     48: In the list, the number coincides with the value returned by the
                     49: built-in function @code{type()}.
                     50: Each example shows possible forms of inputs for @b{Asir}'s prompt.
                     51: \E
1.1       noro       52:
                     53: @table @code
1.2       noro       54: @item 0 @b{0}
1.5       noro       55: @*
1.3       noro       56: \BJP
1.1       noro       57: $B<B:]$K$O(B 0 $B$r<1JL;R$K$b$DBP>]$OB8:_$7$J$$(B. 0 $B$O(B, C $B$K$*$1$k(B 0 $B%]%$%s%?$K(B
                     58: $B$h$jI=8=$5$l$F$$$k(B. $B$7$+$7(B, $BJX59>e(B @b{Asir} $B$N(B @code{type(0)} $B$O(B
                     59: $BCM(B 0 $B$rJV$9(B.
1.3       noro       60: \E
                     61: \BEG
                     62: As a matter of fact, no object exists that has 0 as its identification
                     63: number.  The number 0 is implemented as a null (0) pointer of C language.
                     64: For convenience's sake,  a 0 is returned for the input @code{type(0)}.
                     65: \E
1.1       noro       66:
1.3       noro       67: \JP @item 1 @b{$B?t(B}
                     68: \EG @item 1 @b{number}
1.1       noro       69:
                     70: @example
                     71: 1 2/3  14.5  3+2*@@i
                     72: @end example
                     73:
1.3       noro       74: \JP $B?t$O(B, $B$5$i$K$$$/$D$+$N7?$KJ,$1$i$l$k(B. $B$3$l$K$D$$$F$O2<$G=R$Y$k(B.
                     75: \EG Numbers have sub-types. @xref{Types of numbers}.
1.1       noro       76:
1.3       noro       77: \JP @item 2 @b{$BB?9`<0(B} ($B?t$G$J$$(B)
                     78: \EG @item 2 @b{polynomial} (but not a number)
1.1       noro       79:
                     80: @example
                     81: x  afo  (2.3*x+y)^10
                     82: @end example
                     83:
1.3       noro       84: \BJP
1.1       noro       85: $BB?9`<0$O(B, $BA4$FE83+$5$l(B, $B$=$N;~E@$K$*$1$kJQ?t=g=x$K=>$C$F(B, $B:F5"E*$K(B
                     86: 1 $BJQ?tB?9`<0$H$7$F9_QQ$N=g$K@0M}$5$l$k(B (@xref{$BJ,;6I=8=B?9`<0(B}).
                     87: $B$3$N;~(B, $B$=$NB?9`<0$K8=$l$k=g=x:GBg$NJQ?t$r(B @b{$B<gJQ?t(B} $B$H8F$V(B.
1.3       noro       88: \E
                     89: \BEG
                     90: Every polynomial is maintained internally in its full expanded form,
                     91: represented as a nested univariate polynomial, according to the current
                     92: variable ordering, arranged by the descending order of exponents.
                     93: (@xref{Distributed polynomial}).
                     94: In the representation, the indeterminate (or variable), appearing in
                     95: the polynomial, with maximum ordering is called the @b{main variable}.
                     96: Moreover, we call the coefficient of the maximum degree term of
                     97: the polynomial with respect to the main variable the @b{leading coefficient}.
                     98: \E
1.1       noro       99:
1.3       noro      100: \JP @item 3 @b{$BM-M}<0(B} ($BB?9`<0$G$J$$(B)
                    101: \EG @item 3 @b{rational expression} (not a polynomial)
1.1       noro      102:
                    103: @example
                    104: (x+1)/(y^2-y-x)  x/x
                    105: @end example
                    106:
1.3       noro      107: \BJP
1.1       noro      108: $BM-M}<0$O(B, $BJ,JlJ,;R$,LsJ,2DG=$G$b(B, $BL@<(E*$K(B @code{red()} $B$,8F$P$l$J$$(B
                    109: $B8B$jLsJ,$O9T$o$l$J$$(B. $B$3$l$O(B, $BB?9`<0$N(B GCD $B1i;;$,6K$a$F=E$$1i;;$G$"$k(B
                    110: $B$?$a$G(B, $BM-M}<0$N1i;;$OCm0U$,I,MW$G$"$k(B.
1.3       noro      111: \E
                    112: \BEG
                    113: Note that in @b{Risa/Asir} a rational expression is not simplified
                    114: by reducing the common divisors unless @code{red()} is called
                    115: explicitly, even if it is possible.  This is because the GCD computation
                    116: of polynomials is a considerably heavy operation.  You have to be careful
                    117: enough in operating rational expressions.
                    118: \E
1.1       noro      119:
1.3       noro      120: \JP @item 4 @b{$B%j%9%H(B}
                    121: \EG @item 4 @b{list}
1.1       noro      122:
                    123: @example
                    124: []  [1,2,[3,4],[x,y]]
                    125: @end example
                    126:
1.3       noro      127: \BJP
1.1       noro      128: $B%j%9%H$OFI$_=P$7@lMQ$G$"$k(B. @code{[]} $B$O6u%j%9%H$r0UL#$9$k(B. $B%j%9%H$KBP$9$k(B
                    129: $BA`:n$H$7$F$O(B, @code{car()}, @code{cdr()}, @code{cons()} $B$J$I$K$h$kA`:n$NB>$K(B,
                    130: $BFI$_=P$7@lMQ$NG[Ns$H$_$J$7$F(B, @code{[@var{index}]} $B$rI,MW$J$@$1$D$1$k$3$H$K$h$j(B
                    131: $BMWAG$N<h$j=P$7$r9T$&$3$H$,$G$-$k(B. $BNc$($P(B
1.3       noro      132: \E
                    133: \BEG
                    134: Lists are all read-only object. A null list is specified by @code{[]}.
                    135: There are operations for lists: @code{car()}, @code{cdr()},
                    136: @code{cons()} etc.  And further more, element referencing by indexing is
                    137: available.  Indexing is done by putting @code{[@var{index}]}'s after a
                    138: program variable as many as are required.
                    139: For example,
                    140: \E
1.1       noro      141:
                    142: @example
                    143: [0] L = [[1,2,3],[4,[5,6]],7]$
                    144: [1] L[1][1];
                    145: [5,6]
                    146: @end example
                    147:
1.3       noro      148: \BJP
1.1       noro      149: $BCm0U$9$Y$-$3$H$O(B, $B%j%9%H(B, $BG[Ns(B ($B9TNs(B, $B%Y%/%H%k(B) $B6&$K(B, $B%$%s%G%C%/%9$O(B
                    150: 0 $B$+$i;O$^$k$3$H$H(B, $B%j%9%H$NMWAG$N<h$j=P$7$r%$%s%G%C%/%9$G9T$&$3$H$O(B,
                    151: $B7k6I$O@hF,$+$i%]%$%s%?$r$?$I$k$3$H$KAjEv$9$k$?$a(B, $BG[Ns$KBP$9$kA`:n$K(B
                    152: $BHf3S$7$FBg$-$J%j%9%H$G$O;~4V$,$+$+$k>l9g$,$"$k$H$$$&$3$H$G$"$k(B.
1.3       noro      153: \E
                    154: \BEG
                    155: Notice that for lists, matrices and vectors, the index begins with
                    156: number 0.  Also notice that referencing list elements is done by
                    157: following pointers from the first element.  Therefore, it sometimes takes
                    158: much more time to perform referencing operations on a large list than
                    159: on a vectors or a matrices with the same size.
                    160: \E
1.1       noro      161:
1.3       noro      162: \JP @item 5 @b{$B%Y%/%H%k(B}
                    163: \EG @item 5 @b{vector}
1.1       noro      164:
                    165: @example
                    166: newvect(3)  newvect(2,[a,1])
                    167: @end example
                    168:
1.3       noro      169: \BJP
1.1       noro      170: $B%Y%/%H%k$O(B, @code{newvect()} $B$GL@<(E*$K@8@.$9$kI,MW$,$"$k(B. $BA0<T$NNc$G(B
                    171: $B$O(B2 $B@.J,$N(B 0 $B%Y%/%H%k$,@8@.$5$l(B, $B8e<T$G$O(B, $BBh(B 0 $B@.J,$,(B @code{a}, $BBh(B 1
                    172: $B@.J,$,(B @code{1} $B$N%Y%/%H%k$,@8@.$5$l$k(B. $B=i4|2=$N$?$a$N(B $BBh(B 2 $B0z?t$O(B, $BBh(B
                    173: 1 $B0z?t0J2<$ND9$5$N%j%9%H$r<u$1IU$1$k(B. $B%j%9%H$NMWAG$O:8$+$iMQ$$$i$l(B, $BB-(B
                    174: $B$j$J$$J,$O(B 0 $B$,Jd$o$l$k(B. $B@.J,$O(B @code{[@var{index}]} $B$K$h$j<h$j=P$;$k(B. $B<B:](B
                    175: $B$K$O(B, $B3F@.J,$K(B, $B%Y%/%H%k(B, $B9TNs(B, $B%j%9%H$r4^$`G$0U$N7?$NBP>]$rBeF~$G$-$k(B
                    176: $B$N$G(B, $BB?<!85G[Ns$r%Y%/%H%k$GI=8=$9$k$3$H$,$G$-$k(B.
1.3       noro      177: \E
                    178: \BEG
                    179: Vector objects are created only by explicit execution of @code{newvect()}
                    180: command.  The first example above creates a null vector object with
                    181: 3 elements.  The other example creates a vector object
                    182: with 2 elements which is initialized such that its 0-th element
                    183: is @code{a} and 1st element is @code{1}.
                    184: The second argument for @code{newvect} is used to initialize
                    185: elements of the newly created vector.  A list with size smaller or equal
                    186: to the first argument will be accepted.  Elements of the initializing
                    187: list is used from the left to the right.  If the list is too short to
                    188: specify all the vector elements,
                    189: the unspecified elements are filled with as many 0's as are required.
                    190: Any vector element is designated by indexing, e.g.,
                    191: @code{[@var{index}]}.
                    192: @code{Asir} allows any type, including vector, matrix and list,
                    193: for each respective element of a vector.
                    194: As a matter of course, arrays with arbitrary dimensions can be
                    195: represented by vectors, because each element of a vector can be a vector
                    196: or matrix itself.
                    197: An element designator of a vector can be a left value of assignment
                    198: statement.  This implies that an element designator is treated just like
                    199: a simple program variable.
                    200: Note that an assignment to the element designator of a vector has effect
                    201: on the whole value of that vector.
                    202: \E
1.1       noro      203:
                    204: @example
                    205: [0] A3 = newvect(3);
                    206: [ 0 0 0 ]
                    207: [1] for (I=0;I<3;I++)A3[I] = newvect(3);
                    208: [2] for (I=0;I<3;I++)for(J=0;J<3;J++)A3[I][J]=newvect(3);
                    209: [3] A3;
1.4       noro      210: [ [ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ] [ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ]
                    211: [ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ] ]
1.1       noro      212: [4] A3[0];
                    213: [ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ]
                    214: [5] A3[0][0];
                    215: [ 0 0 0 ]
                    216: @end example
                    217:
1.3       noro      218: \JP @item 6 @b{$B9TNs(B}
                    219: \EG @item 6 @b{matrix}
1.1       noro      220:
                    221: @example
                    222: newmat(2,2)  newmat(2,3,[[x,y],[z]])
                    223: @end example
                    224:
1.3       noro      225: \BJP
1.1       noro      226: $B9TNs$N@8@.$b(B @code{newmat()} $B$K$h$jL@<(E*$K9T$o$l$k(B. $B=i4|2=$b(B, $B0z?t(B
                    227: $B$,%j%9%H$N%j%9%H$H$J$k$3$H$r=|$$$F$O%Y%/%H%k$HF1MM$G(B, $B%j%9%H$N3FMWAG(B
                    228: ($B$3$l$O$^$?%j%9%H$G$"$k(B) $B$O(B, $B3F9T$N=i4|2=$K;H$o$l(B, $BB-$j$J$$ItJ,$K$O(B
                    229: 0 $B$,Kd$a$i$l$k(B. $B9TNs$b(B, $B3FMWAG$K$OG$0U$NBP>]$rBeF~$G$-$k(B. $B9TNs$N3F(B
                    230: $B9T$O(B, $B%Y%/%H%k$H$7$F<h$j=P$9$3$H$,$G$-$k(B.
1.3       noro      231: \E
                    232: \BEG
                    233: Like vector objects, matrix objects are also created only by explicit
                    234: execution of @code{newmat()} command. Initialization of the matrix
                    235: elements are done in a similar manner with that of the vector elements
                    236: except that the elements are specified by a list of lists.  Each element,
                    237: again a list, is used to initialize each row; if the list is too short
                    238: to specify all the row elements, unspecified elements are filled with
                    239: as many 0's as are required.
                    240: Like vectors, any matrix element is designated by indexing, e.g.,
                    241: @code{[@var{index}][@var{index}]}.
                    242: @code{Asir} also allows any type, including vector, matrix and list,
                    243: for each respective element of a matrix.
                    244: An element designator of a matrix can also be a left value of assignment
                    245: statement.  This implies that an element designator is treated just like
                    246: a simple program variable.
                    247: Note that an assignment to the element designator of a matrix has effect
                    248: on the whole value of that matrix.
                    249: Note also that every row, (not column,) of a matrix can be extracted
                    250: and referred to as a vector.
                    251: \E
1.1       noro      252:
                    253: @example
                    254: [0] M=newmat(2,3);
                    255: [ 0 0 0 ]
                    256: [ 0 0 0 ]
                    257: [1] M[1];
                    258: [ 0 0 0 ]
                    259: [2] type(@@@@);
                    260: 5
                    261: @end example
                    262:
1.3       noro      263: \JP @item 7 @b{$BJ8;zNs(B}
                    264: \EG @item 7 @b{string}
1.1       noro      265:
                    266: @example
                    267: ""  "afo"
                    268: @end example
                    269:
1.3       noro      270: \BJP
1.1       noro      271: $BJ8;zNs$O(B, $B<g$K%U%!%$%kL>$J$I$KMQ$$$i$l$k(B. $BJ8;zNs$KBP$7$F$O2C;;$N$_$,(B
                    272: $BDj5A$5$l$F$$$F(B, $B7k2L$O(B 2 $B$D$NJ8;zNs$N7k9g$G$"$k(B.
1.3       noro      273: \E
                    274: \BEG
                    275: Strings are used mainly for naming files.  It is also used for giving
                    276: comments of the results.  Operator symbol @code{+} denote the
                    277: concatenation  operation of two strings.
                    278: \E
1.1       noro      279:
                    280: @example
                    281: [0] "afo"+"take";
                    282: afotake
                    283: @end example
1.2       noro      284:
1.3       noro      285: \JP @item 8 @b{$B9=B$BN(B}
                    286: \EG @item 8 @b{structure}
1.1       noro      287:
                    288: @example
                    289: newstruct(afo)
                    290: @end example
                    291:
1.6       noro      292: \BJP
                    293: Asir $B$K$*$1$k9=B$BN$O(B, C $B$K$*$1$k9=B$BN$r4J0W2=$7$?$b$N$G$"$k(B.
                    294: $B8GDjD9G[Ns$N3F@.J,$rL>A0$G%"%/%;%9$G$-$k%*%V%8%'%/%H$G(B,
                    295: $B9=B$BNDj5AKh$KL>A0$r$D$1$k(B.
                    296: \E
                    297:
                    298: \BEG
                    299: The type @b{structure} is a simplified version of that in C language.
                    300: It is defined as a fixed length array and each entry of the array
                    301: is accessed by its name. A name is associated with each structure.
                    302: \E
1.1       noro      303:
1.3       noro      304: \JP @item 9 @b{$BJ,;6I=8=B?9`<0(B}
                    305: \EG @item 9 @b{distributed polynomial}
1.1       noro      306:
                    307: @example
                    308: 2*<<0,1,2,3>>-3*<<1,2,3,4>>
                    309: @end example
                    310:
1.3       noro      311: \BJP
1.1       noro      312: $B$3$l$O(B, $B$[$H$s$I%0%l%V%J4pDl@lMQ$N7?$G(B, $BDL>o$N7W;;$G$3$N7?$,I,MW$H(B
                    313: $B$J$k$3$H$O$^$:$J$$$,(B, $B%0%l%V%J4pDl7W;;%Q%C%1!<%8<+BN$,%f!<%68@8l(B
                    314: $B$G=q$+$l$F$$$k$?$a(B, $B%f!<%6$,A`:n$G$-$k$h$&FHN)$7$?7?$H$7$F(B  @b{Asir}
                    315: $B$G;HMQ$G$-$k$h$&$K$7$F$"$k(B. $B$3$l$K$D$$$F$O(B @xref{$B%0%l%V%J4pDl$N7W;;(B}.
1.3       noro      316: \E
                    317: \BEG
                    318: This is the short for `Distributed representation of polynomials.'
                    319: This type is specially devised for computation of Groebner bases.
                    320: Though for ordinary users this type may never be needed, it is provided
                    321: as a distinguished type that user can operate by @code{Asir}.
                    322: This is because the Groebner basis package provided with
                    323: @code{Risa/Asir} is written in the @code{Asir} user language.
                    324: For details @xref{Groebner basis computation}.
                    325: \E
                    326:
                    327: \JP @item 10 @b{$BId9f$J$7%^%7%s(B 32bit $B@0?t(B}
                    328: \EG @item 10 @b{32bit unsigned integer}
                    329:
                    330: \JP @item 11 @b{$B%(%i!<%*%V%8%'%/%H(B}
                    331: \EG @item 11 @b{error object}
1.5       noro      332: @*
1.3       noro      333: \JP $B0J>eFs$D$O(B, Open XM $B$K$*$$$FMQ$$$i$l$kFC<l%*%V%8%'%/%H$G$"$k(B.
                    334: \EG These are special objects used for OpenXM.
1.1       noro      335:
1.3       noro      336: \JP @item 12 @b{GF(2) $B>e$N9TNs(B}
                    337: \EG @item 12 @b{matrix over GF(2)}
1.5       noro      338: @*
1.3       noro      339: \BJP
1.1       noro      340: $B8=:_(B, $BI8?t(B 2 $B$NM-8BBN$K$*$1$k4pDlJQ49$N$?$a$N%*%V%8%'%/%H$H$7$FMQ$$$i$l(B
                    341: $B$k(B.
1.3       noro      342: \E
                    343: \BEG
                    344: This is used for basis conversion in finite fields of characteristic 2.
                    345: \E
1.1       noro      346:
1.3       noro      347: \JP @item 13 @b{MATHCAP $B%*%V%8%'%/%H(B}
                    348: \EG @item 13 @b{MATHCAP object}
1.5       noro      349: @*
1.3       noro      350: \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.
                    351: \EG This object is used to express available funcionalities for Open XM.
1.1       noro      352:
1.2       noro      353: @item 14 @b{first order formula}
1.5       noro      354: @*
1.3       noro      355: \JP quantifier elimination $B$GMQ$$$i$l$k0l3,=R8lO@M}<0(B.
                    356: \EG This expresses a first order formula used in quantifier elimination.
1.7     ! noro      357:
        !           358: @item 15 @b{matrix over GF(p)}
        !           359: @*
        !           360: \JP $B>.I8?tM-8BBN>e$N9TNs(B.
        !           361: \EG A matrix over a small finite field.
        !           362:
        !           363: @item 16 @b{byte array}
        !           364: @*
        !           365: \JP $BId9f$J$7(B byte $B$NG[Ns(B
        !           366: \EG An array of unsigned bytes.
1.2       noro      367:
1.3       noro      368: \JP @item -1 @b{VOID $B%*%V%8%'%/%H(B}
                    369: \EG @item -1 @b{VOID object}
1.5       noro      370: @*
1.3       noro      371: \JP $B7?<1JL;R(B -1 $B$r$b$D%*%V%8%'%/%H$O4X?t$NLa$jCM$J$I$,L58z$G$"$k$3$H$r<($9(B.
                    372: \BEG
                    373: The object with the object identifier -1 indicates that a return value
                    374: of a function is void.
                    375: \E
1.1       noro      376: @end table
                    377:
1.3       noro      378: \BJP
1.1       noro      379: @node $B?t$N7?(B,,, $B7?(B
                    380: @section $B?t$N7?(B
1.3       noro      381: \E
                    382: \BEG
                    383: @node Types of numbers,,, Data types
                    384: @section Types of numbers
                    385: \E
1.1       noro      386:
                    387: @table @code
                    388: @item 0
1.3       noro      389: \JP @b{$BM-M}?t(B}
                    390: \EG @b{rational number}
1.5       noro      391: @*
1.3       noro      392: \BJP
1.1       noro      393: $BM-M}?t$O(B, $BG$0UB?G\D9@0?t(B (@b{bignum}) $B$K$h$j<B8=$5$l$F$$$k(B. $BM-M}?t$O>o$K(B
                    394: $B4{LsJ,?t$GI=8=$5$l$k(B.
1.3       noro      395: \E
                    396: \BEG
                    397: Rational numbers are implemented by arbitrary precision integers
                    398: (@b{bignum}).  A rational number is always expressed by a fraction of
                    399: lowest terms.
                    400: \E
1.1       noro      401:
                    402: @item 1
1.3       noro      403: \JP @b{$BG\@:EYIbF0>.?t(B}
                    404: \EG @b{double precision floating point number (double float)}
1.5       noro      405: @*
1.3       noro      406: \BJP
1.1       noro      407: $B%^%7%s$NDs6!$9$kG\@:EYIbF0>.?t$G$"$k(B. @b{Asir} $B$N5/F0;~$K$O(B,
                    408: $BDL>o$N7A<0$GF~NO$5$l$?IbF0>.?t$O$3$N7?$KJQ49$5$l$k(B. $B$?$@$7(B,
                    409: @code{ctrl()} $B$K$h$j(B @b{bigfloat} $B$,A*Br$5$l$F$$$k>l9g$K$O(B
                    410: @b{bigfloat} $B$KJQ49$5$l$k(B.
1.3       noro      411: \E
                    412: \BEG
                    413: The numbers of this type are numbers provided by the computer hardware.
                    414: By default, when @b{Asir} is started, floating point numbers in a
                    415: ordinary form are transformed into numbers of this type.  However,
                    416: they will be transformed into @b{bigfloat} numbers
                    417: when the switch @b{bigfloat} is turned on (enabled) by @code{ctrl()}
                    418: command.
                    419: \E
1.1       noro      420:
                    421: @example
                    422: [0] 1.2;
                    423: 1.2
                    424: [1] 1.2e-1000;
                    425: 0
                    426: [2] ctrl("bigfloat",1);
                    427: 1
                    428: [3] 1.2e-1000;
                    429: 1.20000000000000000513 E-1000
                    430: @end example
                    431:
1.3       noro      432: \BJP
1.1       noro      433: $BG\@:EYIbF0>.?t$HM-M}?t$N1i;;$O(B, $BM-M}?t$,IbF0>.?t$KJQ49$5$l$F(B,
                    434: $BIbF0>.?t$H$7$F1i;;$5$l$k(B.
1.3       noro      435: \E
                    436: \BEG
                    437: A rational number shall be converted automatically into a double float
                    438: number before the operation with another double float number and the
                    439: result shall be computed as a double float number.
                    440: \E
1.1       noro      441:
                    442: @item 2
1.3       noro      443: \JP @b{$BBe?tE*?t(B}
                    444: \EG @b{algebraic number}
1.5       noro      445: @*
1.3       noro      446: \JP @xref{$BBe?tE*?t$K4X$9$k1i;;(B}.
                    447: \EG @xref{Algebraic numbers}.
1.1       noro      448:
                    449: @item 3
                    450: @b{bigfloat}
1.5       noro      451: @*
1.3       noro      452: \BJP
1.1       noro      453: @b{bigfloat} $B$O(B, @b{Asir} $B$G$O(B @b{PARI} $B%i%$%V%i%j$K$h$j(B
                    454: $B<B8=$5$l$F$$$k(B. @b{PARI} $B$K$*$$$F$O(B, @b{bigfloat} $B$O(B, $B2>?tIt(B
                    455: $B$N$_G$0UB?G\D9$G(B, $B;X?tIt$O(B 1 $B%o!<%I0JFb$N@0?t$K8B$i$l$F$$$k(B.
                    456: @code{ctrl()} $B$G(B @b{bigfloat} $B$rA*Br$9$k$3$H$K$h$j(B, $B0J8e$NIbF0>.?t(B
                    457: $B$NF~NO$O(B @b{bigfloat} $B$H$7$F07$o$l$k(B. $B@:EY$O%G%U%)%k%H$G$O(B
                    458: 10 $B?J(B 9 $B7eDxEY$G$"$k$,(B, @code{setprec()} $B$K$h$j;XDj2DG=$G$"$k(B.
1.3       noro      459: \E
                    460: \BEG
                    461: The @b{bigfloat} numbers of @b{Asir} is realized by @b{PARI} library.
                    462: A @b{bigfloat} number of @b{PARI} has an arbitrary precision mantissa
                    463: part.  However, its exponent part admits only an integer with a single
                    464: word precision.
                    465: Floating point operations will be performed all in @b{bigfloat} after
                    466: activating the @b{bigfloat} switch by @code{ctrl()} command.
                    467: The default precision is about 9 digits, which can be specified by
                    468: @code{setprec()} command.
                    469: \E
1.1       noro      470:
                    471: @example
                    472: [0] ctrl("bigfloat",1);
                    473: 1
                    474: [1] eval(2^(1/2));
                    475: 1.414213562373095048763788073031
                    476: [2] setprec(100);
                    477: 9
                    478: [3] eval(2^(1/2));
                    479: 1.41421356237309504880168872420969807856967187537694807317654396116148
                    480: @end example
                    481:
1.3       noro      482: \BJP
1.1       noro      483: @code{eval()} $B$O(B, $B0z?t$K4^$^$l$kH!?tCM$r2DG=$J8B$j?tCM2=$9$kH!?t$G$"$k(B.
                    484: @code{setprec()} $B$G;XDj$5$l$?7e?t$O(B, $B7k2L$N@:EY$rJ]>Z$9$k$b$N$G$O$J$/(B,
                    485: @b{PARI} $BFbIt$GMQ$$$i$l$kI=8=$N%5%$%:$r<($9$3$H$KCm0U$9$Y$-$G$"$k(B.
1.3       noro      486: \E
                    487: \BEG
                    488: Function @code{eval()} evaluates numerically its argument as far as
                    489: possible.
                    490: Notice that the integer given for the argument of @code{setprec()} does
                    491: not guarantee the accuracy of the result,
                    492: but it indicates the representation size of numbers with which internal
                    493: operations of @b{PARI} are performed.
                    494: \E
1.1       noro      495: (@ref{eval}, @xref{pari})
                    496:
                    497: @item 4
1.3       noro      498: \JP @b{$BJ#AG?t(B}
                    499: \EG @b{complex number}
1.5       noro      500: @*
1.3       noro      501: \BJP
1.1       noro      502: $BJ#AG?t$O(B, $BM-M}?t(B, $BG\@:EYIbF0>.?t(B, @b{bigfloat} $B$r<BIt(B, $B5uIt$H$7$F(B
                    503: @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
                    504: $B$=$l$>$l(B @code{real()}, @code{imag()} $B$G<h$j=P$;$k(B.
1.3       noro      505: \E
                    506: \BEG
                    507: A @b{complex} number of @b{Risa/Asir} is a number with the form
                    508: @code{a+b*@@i}, where @@i is the unit of imaginary number, and @code{a}
                    509: and @code{b}
                    510: are either a @b{rational} number, @b{double float} number or
                    511: @b{bigfloat} number, respectively.
                    512: The real part and the imaginary part of a @b{complex} number can be
                    513: taken out by @code{real()} and @code{imag()} respectively.
                    514: \E
1.1       noro      515:
                    516: @item 5
1.3       noro      517: \JP @b{$B>.I8?t$NM-8BAGBN$N85(B}
                    518: \EG @b{element of a small finite prime field}
1.5       noro      519: @*
1.3       noro      520: \BJP
1.1       noro      521: $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
                    522: $BBN$O(B, $B8=:_$N$H$3$m%0%l%V%J4pDl7W;;$K$*$$$FFbItE*$KMQ$$$i$l(B, $BM-8BBN78?t$N(B
                    523: $BJ,;6I=8=B?9`<0$N78?t$r<h$j=P$9$3$H$GF@$i$l$k(B. $B$=$l<+?H$OB0$9$kM-8BBN$K4X(B
                    524: $B$9$k>pJs$O;}$?$:(B, @code{setmod()} $B$G@_Dj$5$l$F$$$kAG?t(B @var{p} $B$rMQ$$$F(B
                    525: GF(@var{p}) $B>e$G$N1i;;$,E,MQ$5$l$k(B.
1.3       noro      526: \E
                    527: \BEG
                    528: Here a small finite fieid means that its characteristic is less than
                    529: 2^27.
                    530: At present small finite fields are used mainly
                    531: for groebner basis computation, and elements in such finite fields
                    532: can be extracted by taking coefficients of distributed polynomials
                    533: whose coefficients are in finite fields. Such an element itself does not
                    534: have any information about the field to which the element belongs, and
                    535: field operations are executed by using a prime @var{p} which is set by
                    536: @code{setmod()}.
                    537: \E
1.1       noro      538:
                    539: @item 6
1.3       noro      540: \JP @b{$BBgI8?t$NM-8BAGBN$N85(B}
                    541: \EG @b{element of large finite prime field}
1.5       noro      542: @*
1.3       noro      543: \BJP
1.1       noro      544: $BI8?t$H$7$FG$0U$NAG?t$,$H$l$k(B.
                    545: $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.
1.3       noro      546: \E
                    547: \BEG
                    548: This type expresses an element of a finite prime field whose characteristic
                    549: is an arbitrary prime. An object of this type is obtained by applying
                    550: @code{simp_ff} to an integer.
                    551: \E
1.1       noro      552:
                    553: @item 7
1.3       noro      554: \JP @b{$BI8?t(B 2 $B$NM-8BBN$N85(B}
                    555: \EG @b{element of a finite field of characteristic 2}
1.5       noro      556: @*
1.3       noro      557: \BJP
1.1       noro      558: $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
                    559: [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
                    560: F=GF(2)[t]/(f(t)) $B$H$"$i$o$5$l$k(B. $B$5$i$K(B, GF(2)[t] $B$N85(B g $B$O(B, f(t)
1.3       noro      561: $B$b4^$a$F<+A3$J;EJ}$G%S%C%HNs$H$_$J$5$l$k$?$a(B, $B7A<0>e$O(B, F $B$N85(B
1.1       noro      562: g mod f $B$O(B, g, f  $B$r$"$i$o$9(B 2 $B$D$N%S%C%HNs$GI=8=$9$k$3$H$,$G$-$k(B.
1.3       noro      563: \E
                    564: \BEG
                    565: This type expresses an element of a finite field of characteristic 2.
                    566: Let @var{F} be a finite field of characteristic 2. If @var{[F:GF(2)]}
                    567: is equal to @var{n}, then @var{F} is expressed as @var{F=GF(2)[t]/(f(t))},
                    568: where @var{f(t)} is an irreducible polynomial over @var{GF(2)}
                    569: of degree @var{n}.
                    570: As an element @var{g} of @var{GF(2)[t]} can be expressed by a bit string,
                    571: An element @var{g mod f} in @var{F} can be expressed by two bit strings
                    572: representing @var{g} and @var{f} respectively.
                    573: \E
1.1       noro      574:
1.3       noro      575: \JP F $B$N85$rF~NO$9$k$$$/$D$+$NJ}K!$,MQ0U$5$l$F$$$k(B.
                    576: \EG Several methods to input an element of @var{F} are provided.
1.1       noro      577:
                    578: @itemize @bullet
                    579: @item
                    580: @code{@@}
1.5       noro      581: @*
1.3       noro      582: \BJP
1.1       noro      583: @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,
                    584: $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.
                    585: $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)
1.3       noro      586: \E
                    587: \BEG
                    588: @code{@@} represents @var{t mod f} in @var{F=GF(2)[t](f(t))}.
                    589: By using @code{@@} one can input an element of @var{F}. For example
                    590: @code{@@^10+@@+1} represents an element of @var{F}.
                    591: \E
1.1       noro      592:
                    593: @item
                    594: @code{ptogf2n}
1.5       noro      595: @*
1.3       noro      596: \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.
                    597: \BEG
                    598: @code{ptogf2n} converts a univariate polynomial into an element of @var{F}.
                    599: \E
1.1       noro      600:
                    601: @item
                    602: @code{ntogf2n}
1.5       noro      603: @*
1.3       noro      604: \BJP
1.1       noro      605: $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,
                    606: 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.
1.3       noro      607: \E
                    608: \BEG
                    609: As a bit string, a non-negative integer can be regarded as an element
                    610: of @var{F}. Note that one can input a non-negative integer in decimal,
                    611: hexadecimal (@code{0x} prefix) and binary (@code{0b} prefix) formats.
                    612: \E
1.1       noro      613:
                    614: @item
1.3       noro      615: \JP @code{$B$=$NB>(B}
                    616: \EG @code{micellaneous}
1.5       noro      617: @*
1.3       noro      618: \BJP
1.1       noro      619: $BB?9`<0$N78?t$r4]$4$H(B F $B$N85$KJQ49$9$k$h$&$J>l9g(B, @code{simp_ff}
                    620: $B$K$h$jJQ49$G$-$k(B.
1.3       noro      621: \E
                    622: \BEG
                    623: @code{simp_ff} is available if one wants to convert the whole
                    624: coefficients of a polynomial.
                    625: \E
1.1       noro      626:
                    627: @end itemize
                    628: @end table
                    629:
1.3       noro      630: \BJP
1.1       noro      631: $BBgI8?tAGBN$NI8?t(B, $BI8?t(B 2 $B$NM-8BBN$NDj5AB?9`<0$O(B, @code{setmod_ff}
                    632: $B$G@_Dj$9$k(B.
                    633: $BM-8BBN$N85$I$&$7$N1i;;$G$O(B, @code{setmod_ff} $B$K$h$j@_Dj$5$l$F$$$k(B
                    634: modulus $B$G(B, $BB0$9$kBN$,J,$+$j(B, $B$=$NCf$G1i;;$,9T$o$l$k(B.
                    635: $B0lJ}$,M-M}?t$N>l9g$K$O(B, $B$=$NM-M}?t$O<+F0E*$K8=:_@_Dj$5$l$F$$$k(B
                    636: $BM-8BBN$N85$KJQ49$5$l(B, $B1i;;$,9T$o$l$k(B.
1.3       noro      637: \E
                    638: \BEG
                    639: The characteristic of a large finite prime field and the defining
                    640: polynomial of a finite field of characteristic 2 are set by @code{setmod_ff}.
                    641: Elements of finite fields do not have informations about the modulus.
                    642: Upon an arithmetic operation, the modulus set by @code{setmod_ff} is
                    643: used. If one of the operands is a rational number, it is automatically
                    644: converted into an element of the finite field currently set and
                    645: the operation is done in the finite field.
                    646: \E
1.1       noro      647:
1.3       noro      648: \BJP
1.1       noro      649: @node $BITDj85$N7?(B,,, $B7?(B
                    650: @section $BITDj85$N7?(B
1.3       noro      651: \E
                    652: \BEG
                    653: @node Types of indeterminates,,, Data types
                    654: @section Types of indeterminates
                    655: \E
1.1       noro      656:
                    657: @noindent
1.3       noro      658: \BJP
1.1       noro      659: $BB?9`<0$NJQ?t$H$J$jF@$kBP>]$r(B@b{$BITDj85(B}$B$H$h$V(B. @b{Asir} $B$G$O(B,
                    660: $B1Q>.J8;z$G;O$^$j(B, $BG$0U8D$N%"%k%U%!%Y%C%H(B, $B?t;z(B, @samp{_} $B$+$i$J$kJ8;zNs(B
                    661: $B$rITDj85$H$7$F07$&$,(B, $B$=$NB>$K$b%7%9%F%`$K$h$jITDj85$H$7$F07$o$l$k$b$N(B
                    662: $B$,$$$/$D$+$"$k(B. @b{Asir} $B$NFbIt7A<0$H$7$F$O(B, $B$3$l$i$OA4$FB?9`<0$H$7$F$N(B
                    663: $B7?$r;}$D$,(B, $B?t$HF1MM(B, $BITDj85$N7?$K$h$j6hJL$5$l$k(B.
1.3       noro      664: \E
                    665: \BEG
                    666: An algebraic object is recognized as an indeterminate when it can be
                    667: a (so-called) variable in polynomials.
                    668: An ordinary indeterminate is usually denoted by a string that start with
                    669: a small alphabetical letter followed by an arbitrary number of
                    670: alphabetical letters, digits or @samp{_}.
                    671: In addition to such ordinary indeterminates,
                    672: there are other kinds of indeterminates in a wider sense in @b{Asir}.
                    673: Such indeterminates in the wider sense have type @b{polynomial},
                    674: and further are classified into sub-types of the type @b{indeterminate}.
                    675: \E
1.1       noro      676:
                    677: @table @code
                    678: @item 0
1.3       noro      679: \JP @b{$B0lHLITDj85(B}
                    680: \EG @b{ordinary indeterminate}
1.5       noro      681: @*
1.3       noro      682: \JP $B1Q>.J8;z$G;O$^$kJ8;zNs(B. $BB?9`<0$NJQ?t$H$7$F:G$bIaDL$KMQ$$$i$l$k(B.
                    683: \BEG
                    684: An object of this sub-type is denoted by a string that start with
                    685: a small alphabetical letter followed by an arbitrary number of
                    686: alphabetical letters, digits or @samp{_}.
                    687: This kind of indeterminates are most commonly used for variables of
                    688: polynomials.
                    689: \E
1.1       noro      690:
                    691: @example
                    692: [0] [vtype(a),vtype(aA_12)];
                    693: [0,0]
                    694: @end example
                    695:
                    696: @item 1
1.3       noro      697: \JP @b{$BL$Dj78?t(B}
                    698: \EG @b{undetermined coefficient}
1.5       noro      699: @*
1.3       noro      700: \BJP
1.1       noro      701: @code{uc()} $B$O(B, @samp{_} $B$G;O$^$kJ8;zNs$rL>A0$H$9$kITDj85$r@8@.$9$k(B.
                    702: $B$3$l$i$O(B, $B%f!<%6$,F~NO$G$-$J$$$H$$$&$@$1$G(B, $B0lHLITDj85$HJQ$o$i$J$$$,(B,
                    703: $B%f!<%6$,F~NO$7$?ITDj85$H>WFM$7$J$$$H$$$&@-<A$rMxMQ$7$FL$Dj78?t$N(B
                    704: $B<+F0@8@.$J$I$KMQ$$$k$3$H$,$G$-$k(B.
1.3       noro      705: \E
                    706: \BEG
                    707: The function @code{uc()} creates an indeterminate which is denoted by
                    708: a string that begins with @samp{_}.  Such an indeterminate cannot be
                    709: directly input by its name.  Other properties are the same as those of
                    710: @b{ordinary indeterminate}.  Therefore, it has a property that it cannot
                    711: cause collision with the name of ordinary indeterminates input by the
                    712: user.  And this property is conveniently used to create undetermined
                    713: coefficients dynamically by programs.
                    714: \E
1.1       noro      715:
                    716: @example
                    717: [1] U=uc();
                    718: _0
                    719: [2] vtype(U);
                    720: 1
                    721: @end example
                    722:
                    723: @item 2
1.3       noro      724: \JP @b{$BH!?t7A<0(B}
                    725: \EG @b{function form}
1.5       noro      726: @*
1.3       noro      727: \BJP
1.1       noro      728: $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
                    729: $BFbIt7A<0$KJQ49$5$l$k$,(B, @code{sin(x)}, @code{cos(x+1)} $B$J$I$O(B, $BI>2A8e(B
                    730: $B$b$=$N$^$^$N7A$GB8:_$9$k(B. $B$3$l$OH!?t7A<0$H8F$P$l(B, $B$=$l<+?H$,(B 1 $B$D$N(B
                    731: $BITDj85$H$7$F07$o$l$k(B. $B$^$?$d$dFC<l$JNc$H$7$F(B, $B1_<~N((B @code{@@pi} $B$d(B
                    732: $B<+A3BP?t$NDl(B @code{@@e} $B$bH!?t7A<0$H$7$F07$o$l$k(B.
1.3       noro      733: \E
                    734: \BEG
                    735: A function call to a built-in function or to an user defined function
                    736: is usually evaluated by @b{Asir} and retained in a proper internal form.
                    737: Some expressions, however, will remain in the same form after evaluation.
                    738: For example, @code{sin(x)} and @code{cos(x+1)} will remain as if they
                    739: were not evaluated.  These (unevaluated) forms are called
                    740: `function forms' and are treated as if they are indeterminates in a
                    741: wider sense.  Also, special forms such as @code{@@pi} the ratio of
                    742: circumference and diameter, and @code{@@e} Napier's number, will be
                    743: treated as `function forms.'
                    744: \E
1.1       noro      745:
                    746: @example
                    747: [3] V=sin(x);
                    748: sin(x)
                    749: [4] vtype(V);
                    750: 2
                    751: [5] vars(V^2+V+1);
                    752: [sin(x)]
                    753: @end example
                    754:
                    755: @item 3
1.3       noro      756: \JP @b{$BH!?t;R(B}
                    757: \EG @b{functor}
1.5       noro      758: @*
1.3       noro      759: \BJP
1.1       noro      760: $BH!?t8F$S=P$7$O(B, @var{fname(args)} $B$H$$$&7A$G9T$J$o$l$k$,(B, @var{fname} $B$N(B
                    761: $BItJ,$rH!?t;R$H8F$V(B. $BH!?t;R$K$O(B, $BH!?t$N<oN`$K$h$jAH$_9~$_H!?t;R(B,
                    762: $B%f!<%6Dj5AH!?t;R(B, $B=iEyH!?t;R$J$I$,$"$k$,(B, $BH!?t;R$OC1FH$GITDj85$H$7$F(B
                    763: $B5!G=$9$k(B.
1.3       noro      764: \E
                    765: \BEG
                    766: A function call (or a function form) has a form @var{fname(args)}.
                    767: Here, @var{fname} alone is called a @b{functor}.
                    768: There are several kinds of @b{functor}s: built-in functor, user defined
                    769: functor and functor for the elementary functions.  A functor alone is
                    770: treated as an indeterminate in a wider sense.
                    771: \E
1.1       noro      772:
                    773: @example
                    774: [6] vtype(sin);
                    775: 3
                    776: @end example
                    777: @end table
                    778: