Annotation of OpenXM/src/asir-doc/parts/type.texi, Revision 1.11
1.11 ! noro 1: @comment $OpenXM: OpenXM/src/asir-doc/parts/type.texi,v 1.10 2003/04/19 10:36:30 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$$(B)
78: \EG @item 2 @b{polynomial} (but not a number)
1.1 noro 79:
80: @example
81: x afo (2.3*x+y)^10
82: @end example
83:
1.3 noro 84: \BJP
1.1 noro 85: $BB?9`<0$O(B, $BA4$FE83+$5$l(B, $B$=$N;~E@$K$*$1$kJQ?t=g=x$K=>$C$F(B, $B:F5"E*$K(B
1.9 noro 86: 1 $BJQ?tB?9`<0$H$7$F9_QQ$N=g$K@0M}$5$l$k(B. (@xref{$BJ,;6I=8=B?9`<0(B}.)
1.1 noro 87: $B$3$N;~(B, $B$=$NB?9`<0$K8=$l$k=g=x:GBg$NJQ?t$r(B @b{$B<gJQ?t(B} $B$H8F$V(B.
1.3 noro 88: \E
89: \BEG
90: Every polynomial is maintained internally in its full expanded form,
91: represented as a nested univariate polynomial, according to the current
92: variable ordering, arranged by the descending order of exponents.
1.9 noro 93: (@xref{Distributed polynomial}.)
1.3 noro 94: In the representation, the indeterminate (or variable), appearing in
95: the polynomial, with maximum ordering is called the @b{main variable}.
96: Moreover, we call the coefficient of the maximum degree term of
97: the polynomial with respect to the main variable the @b{leading coefficient}.
98: \E
1.1 noro 99:
1.3 noro 100: \JP @item 3 @b{$BM-M}<0(B} ($BB?9`<0$G$J$$(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$$$i$l$k0l3,=R8lO@M}<0(B.
356: \EG This expresses a first order formula used in quantifier elimination.
1.7 noro 357:
1.11 ! noro 358: @item 15 @b{matrix over GF(@var{p})}
1.7 noro 359: @*
360: \JP $B>.I8?tM-8BBN>e$N9TNs(B.
361: \EG A matrix over a small finite field.
362:
363: @item 16 @b{byte array}
364: @*
365: \JP $BId9f$J$7(B byte $B$NG[Ns(B
366: \EG An array of unsigned bytes.
1.2 noro 367:
1.3 noro 368: \JP @item -1 @b{VOID $B%*%V%8%'%/%H(B}
369: \EG @item -1 @b{VOID object}
1.5 noro 370: @*
1.3 noro 371: \JP $B7?<1JL;R(B -1 $B$r$b$D%*%V%8%'%/%H$O4X?t$NLa$jCM$J$I$,L58z$G$"$k$3$H$r<($9(B.
372: \BEG
373: The object with the object identifier -1 indicates that a return value
374: of a function is void.
375: \E
1.1 noro 376: @end table
377:
1.3 noro 378: \BJP
1.1 noro 379: @node $B?t$N7?(B,,, $B7?(B
380: @section $B?t$N7?(B
1.3 noro 381: \E
382: \BEG
383: @node Types of numbers,,, Data types
384: @section Types of numbers
385: \E
1.1 noro 386:
387: @table @code
388: @item 0
1.3 noro 389: \JP @b{$BM-M}?t(B}
390: \EG @b{rational number}
1.5 noro 391: @*
1.3 noro 392: \BJP
1.1 noro 393: $BM-M}?t$O(B, $BG$0UB?G\D9@0?t(B (@b{bignum}) $B$K$h$j<B8=$5$l$F$$$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$$$i$l$kI=8=$N%5%$%:$r<($9$3$H$KCm0U$9$Y$-$G$"$k(B.
1.3 noro 486: \E
487: \BEG
488: Function @code{eval()} evaluates numerically its argument as far as
489: possible.
490: Notice that the integer given for the argument of @code{setprec()} does
491: not guarantee the accuracy of the result,
492: but it indicates the representation size of numbers with which internal
493: operations of @b{PARI} are performed.
494: \E
1.9 noro 495: (@xref{eval deval}, @ref{pari}.)
1.1 noro 496:
497: @item 4
1.3 noro 498: \JP @b{$BJ#AG?t(B}
499: \EG @b{complex number}
1.5 noro 500: @*
1.3 noro 501: \BJP
1.1 noro 502: $BJ#AG?t$O(B, $BM-M}?t(B, $BG\@:EYIbF0>.?t(B, @b{bigfloat} $B$r<BIt(B, $B5uIt$H$7$F(B
503: @code{a+b*@@i} (@@i $B$O5u?tC10L(B) $B$H$7$FM?$($i$l$k?t$G$"$k(B. $B<BIt(B, $B5uIt$O(B
504: $B$=$l$>$l(B @code{real()}, @code{imag()} $B$G<h$j=P$;$k(B.
1.3 noro 505: \E
506: \BEG
507: A @b{complex} number of @b{Risa/Asir} is a number with the form
508: @code{a+b*@@i}, where @@i is the unit of imaginary number, and @code{a}
509: and @code{b}
510: are either a @b{rational} number, @b{double float} number or
511: @b{bigfloat} number, respectively.
512: The real part and the imaginary part of a @b{complex} number can be
513: taken out by @code{real()} and @code{imag()} respectively.
514: \E
1.1 noro 515:
516: @item 5
1.3 noro 517: \JP @b{$B>.I8?t$NM-8BAGBN$N85(B}
518: \EG @b{element of a small finite prime field}
1.5 noro 519: @*
1.3 noro 520: \BJP
1.1 noro 521: $B$3$3$G8@$&>.I8?t$H$O(B, $BI8?t$,(B 2^27 $BL$K~$N$b$N$N$3$H$G$"$k(B. $B$3$N$h$&$JM-8B(B
522: $BBN$O(B, $B8=:_$N$H$3$m%0%l%V%J4pDl7W;;$K$*$$$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.
1.11 ! noro 566: Let @var{F} be a finite field of characteristic 2. If [F:GF(2)]
! 567: is equal to @var{n}, then @var{F} is expressed as F=GF(2)[t]/(f(t)),
! 568: where f(t) is an irreducible polynomial over GF(2)
1.3 noro 569: of degree @var{n}.
1.11 ! noro 570: As an element @var{g} of GF(2)[t] can be expressed by a bit string,
1.3 noro 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
1.11 ! noro 588: @code{@@} represents @var{t mod f} in F=GF(2)[t](f(t)).
1.3 noro 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
1.10 noro 628:
629:
630: @item 8
631: \JP @b{$B0L?t(B @var{p^n} $B$NM-8BBN$N85(B}
632: \EG @b{element of a finite field of characteristic @var{p^n}}
633:
634: \BJP
635: $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,
1.11 ! noro 636: $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)
1.10 noro 637: $B$r(B @code{setmod_ff} $B$K$h$j;XDj$9$k$3$H$K$h$j@_Dj$9$k(B.
1.11 ! noro 638: $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
1.10 noro 639: $BI=8=$5$l$k(B.
640: \E
641: \BEG
642: A finite field of order @var{p^n}, where @var{p} is an arbitrary prime
643: and @var{n} is a positive integer, is set by @code{setmod_ff}
644: by specifying its characteristic @var{p} and an irreducible polynomial
1.11 ! noro 645: of degree @var{n} over GF(@var{p}). An element of this field
! 646: is represented by a polynomial over GF(@var{p}) modulo m(x).
1.10 noro 647: \E
648:
649: @item 9
650: \JP @b{$B0L?t(B @var{p^n} $B$NM-8BBN$N85(B ($B>.0L?t(B)}
651: \EG @b{element of a finite field of characteristic @var{p^n} (small order)}
652:
653: \BJP
654: $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
655: $B$J$i(B @var{n} $B$O(B 1) $B$O(B,
656: $BI8?t(B @var{p} $B$*$h$S3HBg<!?t(B @var{n}
657: $B$r(B @code{setmod_ff} $B$K$h$j;XDj$9$k$3$H$K$h$j@_Dj$9$k(B.
658: $B$3$NBN$N(B 0 $B$G$J$$85$O(B, @var{p} $B$,(B @var{2^14} $BL$K~$N>l9g(B,
1.11 ! noro 659: GF(@var{p^n}) $B$N>hK!72$N@8@.85$r8GDj$9$k$3$H(B
1.10 noro 660: $B$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$$85(B
661: $B$O(B, $B$3$N$Y$-;X?t$H$7$FI=8=$5$l$k(B. @var{p} $B$,(B @var{2^14} $B0J>e(B
662: $B$N>l9g$ODL>o$N>jM>$K$h$kI=8=$H$J$k$,(B, $B6&DL$N%W%m%0%i%`$G(B
663: $BAPJ}$N>l9g$r07$($k$h$&$K$3$N$h$&$J;EMM$H$J$C$F$$$k(B.
664:
665: \E
666: \BEG
667: A finite field of order @var{p^n}, where @var{p^n} must be less than
668: @var{2^29} and @var{n} must be equal to 1 if @var{p} is greater or
669: equal to @var{2^14}@,
670: is set by @code{setmod_ff}
671: by specifying its characteristic @var{p} the extension degree
672: @var{n}. If @var{p} is less than @var{2^14}, each non-zero element
673: of this field
674: is a power of a fixed element, which is a generator of the multiplicative
675: group of the field, and it is represented by its exponent.
676: Otherwise, each element is represented by the redue modulo @var{p}.
677: This specification is useful for treating both cases in a single
678: program.
679: \E
680:
1.1 noro 681: @end table
682:
1.3 noro 683: \BJP
1.10 noro 684: $B>.I8?tM-8BAGBN0J30$NM-8BBN$O(B @code{setmod_ff} $B$G@_Dj$9$k(B.
685: $BM-8BBN$N85$I$&$7$N1i;;$G$O(B,
1.1 noro 686: $B0lJ}$,M-M}?t$N>l9g$K$O(B, $B$=$NM-M}?t$O<+F0E*$K8=:_@_Dj$5$l$F$$$k(B
687: $BM-8BBN$N85$KJQ49$5$l(B, $B1i;;$,9T$o$l$k(B.
1.3 noro 688: \E
689: \BEG
1.10 noro 690: Finite fields other than small finite prime fields are
691: set by @code{setmod_ff}.
1.3 noro 692: Elements of finite fields do not have informations about the modulus.
1.10 noro 693: Upon an arithmetic operation, i
694: f one of the operands is a rational number, it is automatically
1.3 noro 695: converted into an element of the finite field currently set and
696: the operation is done in the finite field.
697: \E
1.1 noro 698:
1.3 noro 699: \BJP
1.1 noro 700: @node $BITDj85$N7?(B,,, $B7?(B
701: @section $BITDj85$N7?(B
1.3 noro 702: \E
703: \BEG
704: @node Types of indeterminates,,, Data types
705: @section Types of indeterminates
706: \E
1.1 noro 707:
708: @noindent
1.3 noro 709: \BJP
1.1 noro 710: $BB?9`<0$NJQ?t$H$J$jF@$kBP>]$r(B@b{$BITDj85(B}$B$H$h$V(B. @b{Asir} $B$G$O(B,
711: $B1Q>.J8;z$G;O$^$j(B, $BG$0U8D$N%"%k%U%!%Y%C%H(B, $B?t;z(B, @samp{_} $B$+$i$J$kJ8;zNs(B
712: $B$rITDj85$H$7$F07$&$,(B, $B$=$NB>$K$b%7%9%F%`$K$h$jITDj85$H$7$F07$o$l$k$b$N(B
713: $B$,$$$/$D$+$"$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
714: $B7?$r;}$D$,(B, $B?t$HF1MM(B, $BITDj85$N7?$K$h$j6hJL$5$l$k(B.
1.3 noro 715: \E
716: \BEG
717: An algebraic object is recognized as an indeterminate when it can be
718: a (so-called) variable in polynomials.
719: An ordinary indeterminate is usually denoted by a string that start with
720: a small alphabetical letter followed by an arbitrary number of
721: alphabetical letters, digits or @samp{_}.
722: In addition to such ordinary indeterminates,
723: there are other kinds of indeterminates in a wider sense in @b{Asir}.
724: Such indeterminates in the wider sense have type @b{polynomial},
725: and further are classified into sub-types of the type @b{indeterminate}.
726: \E
1.1 noro 727:
728: @table @code
729: @item 0
1.3 noro 730: \JP @b{$B0lHLITDj85(B}
731: \EG @b{ordinary indeterminate}
1.5 noro 732: @*
1.3 noro 733: \JP $B1Q>.J8;z$G;O$^$kJ8;zNs(B. $BB?9`<0$NJQ?t$H$7$F:G$bIaDL$KMQ$$$i$l$k(B.
734: \BEG
735: An object of this sub-type is denoted by a string that start with
736: a small alphabetical letter followed by an arbitrary number of
737: alphabetical letters, digits or @samp{_}.
738: This kind of indeterminates are most commonly used for variables of
739: polynomials.
740: \E
1.1 noro 741:
742: @example
743: [0] [vtype(a),vtype(aA_12)];
744: [0,0]
745: @end example
746:
747: @item 1
1.3 noro 748: \JP @b{$BL$Dj78?t(B}
749: \EG @b{undetermined coefficient}
1.5 noro 750: @*
1.3 noro 751: \BJP
1.1 noro 752: @code{uc()} $B$O(B, @samp{_} $B$G;O$^$kJ8;zNs$rL>A0$H$9$kITDj85$r@8@.$9$k(B.
753: $B$3$l$i$O(B, $B%f!<%6$,F~NO$G$-$J$$$H$$$&$@$1$G(B, $B0lHLITDj85$HJQ$o$i$J$$$,(B,
754: $B%f!<%6$,F~NO$7$?ITDj85$H>WFM$7$J$$$H$$$&@-<A$rMxMQ$7$FL$Dj78?t$N(B
755: $B<+F0@8@.$J$I$KMQ$$$k$3$H$,$G$-$k(B.
1.3 noro 756: \E
757: \BEG
758: The function @code{uc()} creates an indeterminate which is denoted by
759: a string that begins with @samp{_}. Such an indeterminate cannot be
760: directly input by its name. Other properties are the same as those of
761: @b{ordinary indeterminate}. Therefore, it has a property that it cannot
762: cause collision with the name of ordinary indeterminates input by the
763: user. And this property is conveniently used to create undetermined
764: coefficients dynamically by programs.
765: \E
1.1 noro 766:
767: @example
768: [1] U=uc();
769: _0
770: [2] vtype(U);
771: 1
772: @end example
773:
774: @item 2
1.3 noro 775: \JP @b{$BH!?t7A<0(B}
776: \EG @b{function form}
1.5 noro 777: @*
1.3 noro 778: \BJP
1.1 noro 779: $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
780: $BFbIt7A<0$KJQ49$5$l$k$,(B, @code{sin(x)}, @code{cos(x+1)} $B$J$I$O(B, $BI>2A8e(B
781: $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
782: $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
783: $B<+A3BP?t$NDl(B @code{@@e} $B$bH!?t7A<0$H$7$F07$o$l$k(B.
1.3 noro 784: \E
785: \BEG
786: A function call to a built-in function or to an user defined function
787: is usually evaluated by @b{Asir} and retained in a proper internal form.
788: Some expressions, however, will remain in the same form after evaluation.
789: For example, @code{sin(x)} and @code{cos(x+1)} will remain as if they
790: were not evaluated. These (unevaluated) forms are called
791: `function forms' and are treated as if they are indeterminates in a
792: wider sense. Also, special forms such as @code{@@pi} the ratio of
793: circumference and diameter, and @code{@@e} Napier's number, will be
794: treated as `function forms.'
795: \E
1.1 noro 796:
797: @example
798: [3] V=sin(x);
799: sin(x)
800: [4] vtype(V);
801: 2
802: [5] vars(V^2+V+1);
803: [sin(x)]
804: @end example
805:
806: @item 3
1.3 noro 807: \JP @b{$BH!?t;R(B}
808: \EG @b{functor}
1.5 noro 809: @*
1.3 noro 810: \BJP
1.11 ! noro 811: $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
1.1 noro 812: $BItJ,$rH!?t;R$H8F$V(B. $BH!?t;R$K$O(B, $BH!?t$N<oN`$K$h$jAH$_9~$_H!?t;R(B,
813: $B%f!<%6Dj5AH!?t;R(B, $B=iEyH!?t;R$J$I$,$"$k$,(B, $BH!?t;R$OC1FH$GITDj85$H$7$F(B
814: $B5!G=$9$k(B.
1.3 noro 815: \E
816: \BEG
1.11 ! noro 817: A function call (or a function form) has a form @var{fname}(@var{args}).
1.3 noro 818: Here, @var{fname} alone is called a @b{functor}.
819: There are several kinds of @b{functor}s: built-in functor, user defined
820: functor and functor for the elementary functions. A functor alone is
821: treated as an indeterminate in a wider sense.
822: \E
1.1 noro 823:
824: @example
825: [6] vtype(sin);
826: 3
827: @end example
828: @end table
829:
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