Annotation of OpenXM/src/asir-doc/parts/type.texi, Revision 1.4
1.4 ! noro 1: @comment $OpenXM: OpenXM/src/asir-doc/parts/type.texi,v 1.3 1999/12/21 02:47:32 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}
1.1 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:
1.3 noro 292: \JP $B9=B$BN$K4X$7$F$O(B, $B>O$r2~$a$F2r@b$9$kM=Dj$G$"$k(B.
293: \EG For type @b{structure}, we shall describe it in a later chapter.
294: (Not written yet.)
1.1 noro 295:
1.3 noro 296: \JP @item 9 @b{$BJ,;6I=8=B?9`<0(B}
297: \EG @item 9 @b{distributed polynomial}
1.1 noro 298:
299: @example
300: 2*<<0,1,2,3>>-3*<<1,2,3,4>>
301: @end example
302:
1.3 noro 303: \BJP
1.1 noro 304: $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
305: $B$J$k$3$H$O$^$:$J$$$,(B, $B%0%l%V%J4pDl7W;;%Q%C%1!<%8<+BN$,%f!<%68@8l(B
306: $B$G=q$+$l$F$$$k$?$a(B, $B%f!<%6$,A`:n$G$-$k$h$&FHN)$7$?7?$H$7$F(B @b{Asir}
307: $B$G;HMQ$G$-$k$h$&$K$7$F$"$k(B. $B$3$l$K$D$$$F$O(B @xref{$B%0%l%V%J4pDl$N7W;;(B}.
1.3 noro 308: \E
309: \BEG
310: This is the short for `Distributed representation of polynomials.'
311: This type is specially devised for computation of Groebner bases.
312: Though for ordinary users this type may never be needed, it is provided
313: as a distinguished type that user can operate by @code{Asir}.
314: This is because the Groebner basis package provided with
315: @code{Risa/Asir} is written in the @code{Asir} user language.
316: For details @xref{Groebner basis computation}.
317: \E
318:
319: \JP @item 10 @b{$BId9f$J$7%^%7%s(B 32bit $B@0?t(B}
320: \EG @item 10 @b{32bit unsigned integer}
321:
322: \JP @item 11 @b{$B%(%i!<%*%V%8%'%/%H(B}
323: \EG @item 11 @b{error object}
1.1 noro 324:
1.3 noro 325: \JP $B0J>eFs$D$O(B, Open XM $B$K$*$$$FMQ$$$i$l$kFC<l%*%V%8%'%/%H$G$"$k(B.
326: \EG These are special objects used for OpenXM.
1.1 noro 327:
1.3 noro 328: \JP @item 12 @b{GF(2) $B>e$N9TNs(B}
329: \EG @item 12 @b{matrix over GF(2)}
1.1 noro 330:
1.3 noro 331: \BJP
1.1 noro 332: $B8=:_(B, $BI8?t(B 2 $B$NM-8BBN$K$*$1$k4pDlJQ49$N$?$a$N%*%V%8%'%/%H$H$7$FMQ$$$i$l(B
333: $B$k(B.
1.3 noro 334: \E
335: \BEG
336: This is used for basis conversion in finite fields of characteristic 2.
337: \E
1.1 noro 338:
1.3 noro 339: \JP @item 13 @b{MATHCAP $B%*%V%8%'%/%H(B}
340: \EG @item 13 @b{MATHCAP object}
1.1 noro 341:
1.3 noro 342: \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.
343: \EG This object is used to express available funcionalities for Open XM.
1.1 noro 344:
1.2 noro 345: @item 14 @b{first order formula}
1.1 noro 346:
1.3 noro 347: \JP quantifier elimination $B$GMQ$$$i$l$k0l3,=R8lO@M}<0(B.
348: \EG This expresses a first order formula used in quantifier elimination.
1.2 noro 349:
1.3 noro 350: \JP @item -1 @b{VOID $B%*%V%8%'%/%H(B}
351: \EG @item -1 @b{VOID object}
1.2 noro 352:
1.3 noro 353: \JP $B7?<1JL;R(B -1 $B$r$b$D%*%V%8%'%/%H$O4X?t$NLa$jCM$J$I$,L58z$G$"$k$3$H$r<($9(B.
354: \BEG
355: The object with the object identifier -1 indicates that a return value
356: of a function is void.
357: \E
1.1 noro 358: @end table
359:
1.3 noro 360: \BJP
1.1 noro 361: @node $B?t$N7?(B,,, $B7?(B
362: @section $B?t$N7?(B
1.3 noro 363: \E
364: \BEG
365: @node Types of numbers,,, Data types
366: @section Types of numbers
367: \E
1.1 noro 368:
369: @table @code
370: @item 0
1.3 noro 371: \JP @b{$BM-M}?t(B}
372: \EG @b{rational number}
1.1 noro 373:
1.3 noro 374: \BJP
1.1 noro 375: $BM-M}?t$O(B, $BG$0UB?G\D9@0?t(B (@b{bignum}) $B$K$h$j<B8=$5$l$F$$$k(B. $BM-M}?t$O>o$K(B
376: $B4{LsJ,?t$GI=8=$5$l$k(B.
1.3 noro 377: \E
378: \BEG
379: Rational numbers are implemented by arbitrary precision integers
380: (@b{bignum}). A rational number is always expressed by a fraction of
381: lowest terms.
382: \E
1.1 noro 383:
384: @item 1
1.3 noro 385: \JP @b{$BG\@:EYIbF0>.?t(B}
386: \EG @b{double precision floating point number (double float)}
1.1 noro 387:
1.3 noro 388: \BJP
1.1 noro 389: $B%^%7%s$NDs6!$9$kG\@:EYIbF0>.?t$G$"$k(B. @b{Asir} $B$N5/F0;~$K$O(B,
390: $BDL>o$N7A<0$GF~NO$5$l$?IbF0>.?t$O$3$N7?$KJQ49$5$l$k(B. $B$?$@$7(B,
391: @code{ctrl()} $B$K$h$j(B @b{bigfloat} $B$,A*Br$5$l$F$$$k>l9g$K$O(B
392: @b{bigfloat} $B$KJQ49$5$l$k(B.
1.3 noro 393: \E
394: \BEG
395: The numbers of this type are numbers provided by the computer hardware.
396: By default, when @b{Asir} is started, floating point numbers in a
397: ordinary form are transformed into numbers of this type. However,
398: they will be transformed into @b{bigfloat} numbers
399: when the switch @b{bigfloat} is turned on (enabled) by @code{ctrl()}
400: command.
401: \E
1.1 noro 402:
403: @example
404: [0] 1.2;
405: 1.2
406: [1] 1.2e-1000;
407: 0
408: [2] ctrl("bigfloat",1);
409: 1
410: [3] 1.2e-1000;
411: 1.20000000000000000513 E-1000
412: @end example
413:
1.3 noro 414: \BJP
1.1 noro 415: $BG\@:EYIbF0>.?t$HM-M}?t$N1i;;$O(B, $BM-M}?t$,IbF0>.?t$KJQ49$5$l$F(B,
416: $BIbF0>.?t$H$7$F1i;;$5$l$k(B.
1.3 noro 417: \E
418: \BEG
419: A rational number shall be converted automatically into a double float
420: number before the operation with another double float number and the
421: result shall be computed as a double float number.
422: \E
1.1 noro 423:
424: @item 2
1.3 noro 425: \JP @b{$BBe?tE*?t(B}
426: \EG @b{algebraic number}
1.1 noro 427:
1.3 noro 428: \JP @xref{$BBe?tE*?t$K4X$9$k1i;;(B}.
429: \EG @xref{Algebraic numbers}.
1.1 noro 430:
431: @item 3
432: @b{bigfloat}
433:
1.3 noro 434: \BJP
1.1 noro 435: @b{bigfloat} $B$O(B, @b{Asir} $B$G$O(B @b{PARI} $B%i%$%V%i%j$K$h$j(B
436: $B<B8=$5$l$F$$$k(B. @b{PARI} $B$K$*$$$F$O(B, @b{bigfloat} $B$O(B, $B2>?tIt(B
437: $B$N$_G$0UB?G\D9$G(B, $B;X?tIt$O(B 1 $B%o!<%I0JFb$N@0?t$K8B$i$l$F$$$k(B.
438: @code{ctrl()} $B$G(B @b{bigfloat} $B$rA*Br$9$k$3$H$K$h$j(B, $B0J8e$NIbF0>.?t(B
439: $B$NF~NO$O(B @b{bigfloat} $B$H$7$F07$o$l$k(B. $B@:EY$O%G%U%)%k%H$G$O(B
440: 10 $B?J(B 9 $B7eDxEY$G$"$k$,(B, @code{setprec()} $B$K$h$j;XDj2DG=$G$"$k(B.
1.3 noro 441: \E
442: \BEG
443: The @b{bigfloat} numbers of @b{Asir} is realized by @b{PARI} library.
444: A @b{bigfloat} number of @b{PARI} has an arbitrary precision mantissa
445: part. However, its exponent part admits only an integer with a single
446: word precision.
447: Floating point operations will be performed all in @b{bigfloat} after
448: activating the @b{bigfloat} switch by @code{ctrl()} command.
449: The default precision is about 9 digits, which can be specified by
450: @code{setprec()} command.
451: \E
1.1 noro 452:
453: @example
454: [0] ctrl("bigfloat",1);
455: 1
456: [1] eval(2^(1/2));
457: 1.414213562373095048763788073031
458: [2] setprec(100);
459: 9
460: [3] eval(2^(1/2));
461: 1.41421356237309504880168872420969807856967187537694807317654396116148
462: @end example
463:
1.3 noro 464: \BJP
1.1 noro 465: @code{eval()} $B$O(B, $B0z?t$K4^$^$l$kH!?tCM$r2DG=$J8B$j?tCM2=$9$kH!?t$G$"$k(B.
466: @code{setprec()} $B$G;XDj$5$l$?7e?t$O(B, $B7k2L$N@:EY$rJ]>Z$9$k$b$N$G$O$J$/(B,
467: @b{PARI} $BFbIt$GMQ$$$i$l$kI=8=$N%5%$%:$r<($9$3$H$KCm0U$9$Y$-$G$"$k(B.
1.3 noro 468: \E
469: \BEG
470: Function @code{eval()} evaluates numerically its argument as far as
471: possible.
472: Notice that the integer given for the argument of @code{setprec()} does
473: not guarantee the accuracy of the result,
474: but it indicates the representation size of numbers with which internal
475: operations of @b{PARI} are performed.
476: \E
1.1 noro 477: (@ref{eval}, @xref{pari})
478:
479: @item 4
1.3 noro 480: \JP @b{$BJ#AG?t(B}
481: \EG @b{complex number}
1.1 noro 482:
1.3 noro 483: \BJP
1.1 noro 484: $BJ#AG?t$O(B, $BM-M}?t(B, $BG\@:EYIbF0>.?t(B, @b{bigfloat} $B$r<BIt(B, $B5uIt$H$7$F(B
485: @code{a+b*@@i} (@@i $B$O5u?tC10L(B) $B$H$7$FM?$($i$l$k?t$G$"$k(B. $B<BIt(B, $B5uIt$O(B
486: $B$=$l$>$l(B @code{real()}, @code{imag()} $B$G<h$j=P$;$k(B.
1.3 noro 487: \E
488: \BEG
489: A @b{complex} number of @b{Risa/Asir} is a number with the form
490: @code{a+b*@@i}, where @@i is the unit of imaginary number, and @code{a}
491: and @code{b}
492: are either a @b{rational} number, @b{double float} number or
493: @b{bigfloat} number, respectively.
494: The real part and the imaginary part of a @b{complex} number can be
495: taken out by @code{real()} and @code{imag()} respectively.
496: \E
1.1 noro 497:
498: @item 5
1.3 noro 499: \JP @b{$B>.I8?t$NM-8BAGBN$N85(B}
500: \EG @b{element of a small finite prime field}
1.1 noro 501:
1.3 noro 502: \BJP
1.1 noro 503: $B$3$3$G8@$&>.I8?t$H$O(B, $BI8?t$,(B 2^27 $BL$K~$N$b$N$N$3$H$G$"$k(B. $B$3$N$h$&$JM-8B(B
504: $BBN$O(B, $B8=:_$N$H$3$m%0%l%V%J4pDl7W;;$K$*$$$FFbItE*$KMQ$$$i$l(B, $BM-8BBN78?t$N(B
505: $BJ,;6I=8=B?9`<0$N78?t$r<h$j=P$9$3$H$GF@$i$l$k(B. $B$=$l<+?H$OB0$9$kM-8BBN$K4X(B
506: $B$9$k>pJs$O;}$?$:(B, @code{setmod()} $B$G@_Dj$5$l$F$$$kAG?t(B @var{p} $B$rMQ$$$F(B
507: GF(@var{p}) $B>e$G$N1i;;$,E,MQ$5$l$k(B.
1.3 noro 508: \E
509: \BEG
510: Here a small finite fieid means that its characteristic is less than
511: 2^27.
512: At present small finite fields are used mainly
513: for groebner basis computation, and elements in such finite fields
514: can be extracted by taking coefficients of distributed polynomials
515: whose coefficients are in finite fields. Such an element itself does not
516: have any information about the field to which the element belongs, and
517: field operations are executed by using a prime @var{p} which is set by
518: @code{setmod()}.
519: \E
1.1 noro 520:
521: @item 6
1.3 noro 522: \JP @b{$BBgI8?t$NM-8BAGBN$N85(B}
523: \EG @b{element of large finite prime field}
1.1 noro 524:
1.3 noro 525: \BJP
1.1 noro 526: $BI8?t$H$7$FG$0U$NAG?t$,$H$l$k(B.
527: $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 528: \E
529: \BEG
530: This type expresses an element of a finite prime field whose characteristic
531: is an arbitrary prime. An object of this type is obtained by applying
532: @code{simp_ff} to an integer.
533: \E
1.1 noro 534:
535: @item 7
1.3 noro 536: \JP @b{$BI8?t(B 2 $B$NM-8BBN$N85(B}
537: \EG @b{element of a finite field of characteristic 2}
1.1 noro 538:
1.3 noro 539: \BJP
1.1 noro 540: $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
541: [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
542: 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 543: $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 544: 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 545: \E
546: \BEG
547: This type expresses an element of a finite field of characteristic 2.
548: Let @var{F} be a finite field of characteristic 2. If @var{[F:GF(2)]}
549: is equal to @var{n}, then @var{F} is expressed as @var{F=GF(2)[t]/(f(t))},
550: where @var{f(t)} is an irreducible polynomial over @var{GF(2)}
551: of degree @var{n}.
552: As an element @var{g} of @var{GF(2)[t]} can be expressed by a bit string,
553: An element @var{g mod f} in @var{F} can be expressed by two bit strings
554: representing @var{g} and @var{f} respectively.
555: \E
1.1 noro 556:
1.3 noro 557: \JP F $B$N85$rF~NO$9$k$$$/$D$+$NJ}K!$,MQ0U$5$l$F$$$k(B.
558: \EG Several methods to input an element of @var{F} are provided.
1.1 noro 559:
560: @itemize @bullet
561: @item
562: @code{@@}
563:
1.3 noro 564: \BJP
1.1 noro 565: @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,
566: $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.
567: $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 568: \E
569: \BEG
570: @code{@@} represents @var{t mod f} in @var{F=GF(2)[t](f(t))}.
571: By using @code{@@} one can input an element of @var{F}. For example
572: @code{@@^10+@@+1} represents an element of @var{F}.
573: \E
1.1 noro 574:
575: @item
576: @code{ptogf2n}
577:
1.3 noro 578: \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.
579: \BEG
580: @code{ptogf2n} converts a univariate polynomial into an element of @var{F}.
581: \E
1.1 noro 582:
583: @item
584: @code{ntogf2n}
585:
1.3 noro 586: \BJP
1.1 noro 587: $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,
588: 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 589: \E
590: \BEG
591: As a bit string, a non-negative integer can be regarded as an element
592: of @var{F}. Note that one can input a non-negative integer in decimal,
593: hexadecimal (@code{0x} prefix) and binary (@code{0b} prefix) formats.
594: \E
1.1 noro 595:
596: @item
1.3 noro 597: \JP @code{$B$=$NB>(B}
598: \EG @code{micellaneous}
1.1 noro 599:
1.3 noro 600: \BJP
1.1 noro 601: $BB?9`<0$N78?t$r4]$4$H(B F $B$N85$KJQ49$9$k$h$&$J>l9g(B, @code{simp_ff}
602: $B$K$h$jJQ49$G$-$k(B.
1.3 noro 603: \E
604: \BEG
605: @code{simp_ff} is available if one wants to convert the whole
606: coefficients of a polynomial.
607: \E
1.1 noro 608:
609: @end itemize
610: @end table
611:
1.3 noro 612: \BJP
1.1 noro 613: $BBgI8?tAGBN$NI8?t(B, $BI8?t(B 2 $B$NM-8BBN$NDj5AB?9`<0$O(B, @code{setmod_ff}
614: $B$G@_Dj$9$k(B.
615: $BM-8BBN$N85$I$&$7$N1i;;$G$O(B, @code{setmod_ff} $B$K$h$j@_Dj$5$l$F$$$k(B
616: modulus $B$G(B, $BB0$9$kBN$,J,$+$j(B, $B$=$NCf$G1i;;$,9T$o$l$k(B.
617: $B0lJ}$,M-M}?t$N>l9g$K$O(B, $B$=$NM-M}?t$O<+F0E*$K8=:_@_Dj$5$l$F$$$k(B
618: $BM-8BBN$N85$KJQ49$5$l(B, $B1i;;$,9T$o$l$k(B.
1.3 noro 619: \E
620: \BEG
621: The characteristic of a large finite prime field and the defining
622: polynomial of a finite field of characteristic 2 are set by @code{setmod_ff}.
623: Elements of finite fields do not have informations about the modulus.
624: Upon an arithmetic operation, the modulus set by @code{setmod_ff} is
625: used. If one of the operands is a rational number, it is automatically
626: converted into an element of the finite field currently set and
627: the operation is done in the finite field.
628: \E
1.1 noro 629:
1.3 noro 630: \BJP
1.1 noro 631: @node $BITDj85$N7?(B,,, $B7?(B
632: @section $BITDj85$N7?(B
1.3 noro 633: \E
634: \BEG
635: @node Types of indeterminates,,, Data types
636: @section Types of indeterminates
637: \E
1.1 noro 638:
639: @noindent
1.3 noro 640: \BJP
1.1 noro 641: $BB?9`<0$NJQ?t$H$J$jF@$kBP>]$r(B@b{$BITDj85(B}$B$H$h$V(B. @b{Asir} $B$G$O(B,
642: $B1Q>.J8;z$G;O$^$j(B, $BG$0U8D$N%"%k%U%!%Y%C%H(B, $B?t;z(B, @samp{_} $B$+$i$J$kJ8;zNs(B
643: $B$rITDj85$H$7$F07$&$,(B, $B$=$NB>$K$b%7%9%F%`$K$h$jITDj85$H$7$F07$o$l$k$b$N(B
644: $B$,$$$/$D$+$"$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
645: $B7?$r;}$D$,(B, $B?t$HF1MM(B, $BITDj85$N7?$K$h$j6hJL$5$l$k(B.
1.3 noro 646: \E
647: \BEG
648: An algebraic object is recognized as an indeterminate when it can be
649: a (so-called) variable in polynomials.
650: An ordinary indeterminate is usually denoted by a string that start with
651: a small alphabetical letter followed by an arbitrary number of
652: alphabetical letters, digits or @samp{_}.
653: In addition to such ordinary indeterminates,
654: there are other kinds of indeterminates in a wider sense in @b{Asir}.
655: Such indeterminates in the wider sense have type @b{polynomial},
656: and further are classified into sub-types of the type @b{indeterminate}.
657: \E
1.1 noro 658:
659: @table @code
660: @item 0
1.3 noro 661: \JP @b{$B0lHLITDj85(B}
662: \EG @b{ordinary indeterminate}
1.1 noro 663:
1.3 noro 664: \JP $B1Q>.J8;z$G;O$^$kJ8;zNs(B. $BB?9`<0$NJQ?t$H$7$F:G$bIaDL$KMQ$$$i$l$k(B.
665: \BEG
666: An object of this sub-type is denoted by a string that start with
667: a small alphabetical letter followed by an arbitrary number of
668: alphabetical letters, digits or @samp{_}.
669: This kind of indeterminates are most commonly used for variables of
670: polynomials.
671: \E
1.1 noro 672:
673: @example
674: [0] [vtype(a),vtype(aA_12)];
675: [0,0]
676: @end example
677:
678: @item 1
1.3 noro 679: \JP @b{$BL$Dj78?t(B}
680: \EG @b{undetermined coefficient}
1.1 noro 681:
1.3 noro 682: \BJP
1.1 noro 683: @code{uc()} $B$O(B, @samp{_} $B$G;O$^$kJ8;zNs$rL>A0$H$9$kITDj85$r@8@.$9$k(B.
684: $B$3$l$i$O(B, $B%f!<%6$,F~NO$G$-$J$$$H$$$&$@$1$G(B, $B0lHLITDj85$HJQ$o$i$J$$$,(B,
685: $B%f!<%6$,F~NO$7$?ITDj85$H>WFM$7$J$$$H$$$&@-<A$rMxMQ$7$FL$Dj78?t$N(B
686: $B<+F0@8@.$J$I$KMQ$$$k$3$H$,$G$-$k(B.
1.3 noro 687: \E
688: \BEG
689: The function @code{uc()} creates an indeterminate which is denoted by
690: a string that begins with @samp{_}. Such an indeterminate cannot be
691: directly input by its name. Other properties are the same as those of
692: @b{ordinary indeterminate}. Therefore, it has a property that it cannot
693: cause collision with the name of ordinary indeterminates input by the
694: user. And this property is conveniently used to create undetermined
695: coefficients dynamically by programs.
696: \E
1.1 noro 697:
698: @example
699: [1] U=uc();
700: _0
701: [2] vtype(U);
702: 1
703: @end example
704:
705: @item 2
1.3 noro 706: \JP @b{$BH!?t7A<0(B}
707: \EG @b{function form}
1.1 noro 708:
1.3 noro 709: \BJP
1.1 noro 710: $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
711: $BFbIt7A<0$KJQ49$5$l$k$,(B, @code{sin(x)}, @code{cos(x+1)} $B$J$I$O(B, $BI>2A8e(B
712: $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
713: $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
714: $B<+A3BP?t$NDl(B @code{@@e} $B$bH!?t7A<0$H$7$F07$o$l$k(B.
1.3 noro 715: \E
716: \BEG
717: A function call to a built-in function or to an user defined function
718: is usually evaluated by @b{Asir} and retained in a proper internal form.
719: Some expressions, however, will remain in the same form after evaluation.
720: For example, @code{sin(x)} and @code{cos(x+1)} will remain as if they
721: were not evaluated. These (unevaluated) forms are called
722: `function forms' and are treated as if they are indeterminates in a
723: wider sense. Also, special forms such as @code{@@pi} the ratio of
724: circumference and diameter, and @code{@@e} Napier's number, will be
725: treated as `function forms.'
726: \E
1.1 noro 727:
728: @example
729: [3] V=sin(x);
730: sin(x)
731: [4] vtype(V);
732: 2
733: [5] vars(V^2+V+1);
734: [sin(x)]
735: @end example
736:
737: @item 3
1.3 noro 738: \JP @b{$BH!?t;R(B}
739: \EG @b{functor}
1.1 noro 740:
1.3 noro 741: \BJP
1.1 noro 742: $BH!?t8F$S=P$7$O(B, @var{fname(args)} $B$H$$$&7A$G9T$J$o$l$k$,(B, @var{fname} $B$N(B
743: $BItJ,$rH!?t;R$H8F$V(B. $BH!?t;R$K$O(B, $BH!?t$N<oN`$K$h$jAH$_9~$_H!?t;R(B,
744: $B%f!<%6Dj5AH!?t;R(B, $B=iEyH!?t;R$J$I$,$"$k$,(B, $BH!?t;R$OC1FH$GITDj85$H$7$F(B
745: $B5!G=$9$k(B.
1.3 noro 746: \E
747: \BEG
748: A function call (or a function form) has a form @var{fname(args)}.
749: Here, @var{fname} alone is called a @b{functor}.
750: There are several kinds of @b{functor}s: built-in functor, user defined
751: functor and functor for the elementary functions. A functor alone is
752: treated as an indeterminate in a wider sense.
753: \E
1.1 noro 754:
755: @example
756: [6] vtype(sin);
757: 3
758: @end example
759: @end table
760:
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