Annotation of OpenXM/doc/OpenXM-specs/cmo-basic1.tex, Revision 1.10
1.10 ! takayama 1: %% $OpenXM: OpenXM/doc/OpenXM-specs/cmo-basic1.tex,v 1.9 2000/09/12 23:09:18 noro Exp $
1.1 noro 2: //&jp \section{ 数, 多項式 の CMO 表現 }
3: //&eg \section{ CMOexpressions for numbers and polynomials }
1.4 noro 4: \label{sec:basic1}
1.1 noro 5: /*&C
6: @../SSkan/plugin/cmotag.h
7: \begin{verbatim}
8: #define CMO_MONOMIAL32 19
9: #define CMO_ZZ 20
10: #define CMO_QQ 21
11: #define CMO_ZERO 22
12: #define CMO_DMS_GENERIC 24
13: #define CMO_DMS_OF_N_VARIABLES 25
14: #define CMO_RING_BY_NAME 26
15: #define CMO_DISTRIBUTED_POLYNOMIAL 31
16: #define CMO_RATIONAL 34
17:
18:
19: #define CMO_INDETERMINATE 60
20: #define CMO_TREE 61
21: #define CMO_LAMBDA 62 /* for function definition */
22: \end{verbatim}
23:
24: */
25:
26: /*&jp
1.4 noro 27: 以下, グループ CMObject/Basic, CMObject/Tree
1.1 noro 28: および CMObject/DistributedPolynomial
29: に属する CMObject の形式を説明する.
30:
1.5 noro 31: \noindent
32: {\tt OpenXM/src/ox\_toolkit} にある {\tt bconv} をもちいると
33: CMO expression を binary format に変換できるので,
34: これを参考にするといい.
1.1 noro 35: */
36: /*&eg
37: In the sequel, we will explain on the groups
1.4 noro 38: CMObject/Basic, CMObject/Tree
1.1 noro 39: and CMObject/DistributedPolynomial.
1.5 noro 40:
41: \noindent
42: The program {\tt bconv} at {\tt OpenXM/src/ox\_toolkit}
43: translates
44: CMO expressions into binary formats.
45: It is convinient to understand the binary formats explained in
46: this section.
1.1 noro 47: */
48:
1.5 noro 49: /*&C
50: \noindent Example:
51: \begin{verbatim}
52: bash$ ./bconv
53: > (CMO_ZZ,123123);
54: 00 00 00 14 00 00 00 01 00 01 e0 f3
55: \end{verbatim}
56: */
1.1 noro 57: /*&jp
58:
59: \bigbreak
60: \noindent
1.4 noro 61: Group CMObject/Basic requires CMObject/Primitive. \\
62: ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/Basic. \\
1.1 noro 63: \begin{eqnarray*}
64: \mbox{Zero} &:& ({\tt CMO\_ZERO}) \\
65: & & \mbox{ --- ユニバーサルな ゼロを表す. } \\
66: \mbox{ZZ} &:& ({\tt CMO\_ZZ},{\sl int32}\, {\rm f}, {\sl byte}\, \mbox{a[1]}, \ldots
1.9 noro 67: {\sl byte}\, \mbox{a[$|$f$|$]} ) \\
1.1 noro 68: &:& \mbox{ --- bignum をあらわす. a[i] についてはあとで説明}\\
1.9 noro 69: \mbox{QQ} &:& ({\tt CMO\_QQ},
70: {\sl int32}\, {\rm m}, {\sl byte}\, \mbox{a[1]}, \ldots, {\sl byte}\, \mbox{a[$|$m$|$]},
71: {\sl int32}\, {\rm n}, {\sl byte}\, \mbox{b[1]}, \ldots, {\sl byte}\, \mbox{b[$|$n$|$]})\\
1.1 noro 72: & & \mbox{ --- 有理数 $a/b$ を表す. } \\
73: \mbox{Rational} &:& ({\tt CMO\_RATIONAL}, {\sl CMObject}\, {\rm a}, {\sl CMObject}\, {\rm b}) \\
74: & & \mbox{ --- $a/b$ を表す. } \\
75: \mbox{Indeterminate} &:& ({\tt CMO\_INDETERMINATE}, {\sl Cstring}\, {\rm v}) \\
76: & & \mbox{ --- 変数名 $v$ . } \\
77: \end{eqnarray*}
78: */
79: /*&eg
80:
81: \bigbreak
82: \noindent
1.4 noro 83: Group CMObject/Basic requires CMObject/Primitive. \\
84: ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/Basic. \\
1.1 noro 85: \begin{eqnarray*}
86: \mbox{Zero} &:& ({\tt CMO\_ZERO}) \\
87: & & \mbox{ --- Universal zero } \\
88: \mbox{ZZ} &:& ({\tt CMO\_ZZ},{\sl int32}\, {\rm f}, {\sl byte}\, \mbox{a[1]}, \ldots
1.9 noro 89: {\sl byte}\, \mbox{a[$|$m$|$]} ) \\
1.1 noro 90: &:& \mbox{ --- bignum. The meaning of a[i] will be explained later.}\\
1.9 noro 91: \mbox{QQ} &:& ({\tt CMO\_QQ},
92: {\sl int32}\, {\rm m}, {\sl byte}\, \mbox{a[1]}, \ldots, {\sl byte}\, \mbox{a[$|$m$|$]},
93: {\sl int32}\, {\rm n}, {\sl byte}\, \mbox{b[1]}, \ldots, {\sl byte}\, \mbox{b[$|$n$|$]})\\
1.1 noro 94: & & \mbox{ --- Rational number $a/b$. } \\
95: \mbox{Rational} &:& ({\tt CMO\_RATIONAL}, {\sl CMObject}\, {\rm a}, {\sl CMObject}\, {\rm b}) \\
96: & & \mbox{ --- Rational expression $a/b$. } \\
97: \mbox{Indeterminate} &:& ({\tt CMO\_INDETERMINATE}, {\sl Cstring}\, {\rm v}) \\
98: & & \mbox{ --- Variable name $v$ . } \\
99: \end{eqnarray*}
100: */
101: /*&C
102:
103: */
104:
105: /*&jp
106: Indeterminate は変数名をあらわす.
107: v はバイト列であればなにを用いてもよいが,
108: システム毎に変数名として用いられるバイト列は制限がある.
109: 各システム xxx は任意の文字列を各システム固有の変数名へ1対1に変換できるように
110: 実装しないといけない.
111: (これを
112: {\tt Dx} は {\tt \#dx} と変換するなどの
113: escape sequence を用いて実現するのは, 無理があるようである.
114: テーブルを作成する必要があるであろう.)
115: */
116: /*&eg
117: Indeterminate is a name of a variable.
118: v may be any sequence of bytes, but each system has its own
119: restrictions on the names of variables.
120: Indeterminates of CMO and internal variable names must be translated
121: in one to one correspondence.
122: */
123:
124: /*&jp
125:
126: \noindent
1.4 noro 127: Group CMObject/Tree requires CMObject/Basic. \\
128: Tree, Lambda $\in$ CMObject/Basic. \\
1.1 noro 129: \begin{eqnarray*}
130: \mbox{Tree} &:& ({\tt CMO\_TREE}, {\sl Cstring}\, {\rm name},
131: {\sl Cstring}\, {\rm cdname}, {\sl List}\, {\rm leaves}) \\
132: & & \mbox{ --- 名前 name の定数または関数. 関数の評価はおこなわない. } \\
133: & & \mbox{ --- cdname は空文字列でなければ name の意味が説明されている }\\
134: & & \mbox{ --- OpenMath CD (content dictionary) の名前. } \\
135: \mbox{Lambda} &:& ({\tt CMO\_LAMBDA}, {\sl List}\, {\rm args},
136: {\sl Tree} {\rm body}) \\
137: & & \mbox{ --- body を args を引数とする関数とする. } \\
138: & & \mbox{ --- optional な引数が必要なときは, leaves の後へつづける.} \\
139: \end{eqnarray*}
140: */
141: /*&eg
142:
143: \noindent
1.4 noro 144: Group CMObject/Tree requires CMObject/Basic. \\
145: Tree, Lambda $\in$ CMObject/Basic. \\
1.1 noro 146: \begin{eqnarray*}
147: \mbox{Tree} &:& ({\tt CMO\_TREE}, {\sl Cstring}\, {\rm name},
148: {\sl Cstring}\, {\rm cdname}, {\sl List}\, {\rm leaves}) \\
149: & & \mbox{ --- A function or a constant of name. Functions are not evaluated. } \\
150: & & \mbox{ --- cdname may be a null. If it is not null, it is the name of}\\
151: & & \mbox{ --- the OpenMath CD (content dictionary). } \\
152: \mbox{Lambda} &:& ({\tt CMO\_LAMBDA}, {\sl List}\, {\rm args},
153: {\sl Tree} {\rm body}) \\
154: & & \mbox{ --- a function with the arguments body. } \\
155: & & \mbox{ --- optional arguments come after leaves.} \\
156: \end{eqnarray*}
157: */
158:
159: /*&jp
160: 数式を処理するシステムでは, Tree 構造が一般にもちいられる.
161: たとえば, $\sin(x+e)$ は,
162: {\tt (sin, (plus, x, e))}
163: なる Tree であらわすのが一般的である.
164: Tree の表現を スタックマシンのレベルでおこなうとすると,
165: {\tt ox\_BEGIN\_BLOCK}, {\tt ox\_END\_BLOCK} で評価を抑制するのが
166: 一つの方法である (cf. Postscript の {\tt \{ }, {\tt \} }).
167: たとえば上の方法では
168: {\tt x, e, plus, sin } を begin block, end block でかこめばよろしい.
169: われわれはスタックマシンの実装をなるべく簡単にするという立場をとりたい,
170: また数学オブジェクトを OX スタックマシンと CMObject を混在して表現したく
171: ない.
172: したがって,
173: Tree 構造は Open Math 風の表現をもちいた CMO を導入することにした.
174: またこのほうが, われわれの想定するシステム xxx において, Open XM 対応が
175: はるかに容易である.
176: なお, Tree は, Open Math では, Symbol, Application のメカニズムに相当する.
177: */
178: /*&eg
179: In many computer algebra systems, mathematical expressions are usually
180: expressed in terms of a tree structure.
181: For example,
182: $\sin(x+e)$ is expressed as
183: {\tt (sin, (plus, x, e))}
184: as a tree.
1.4 noro 185: Tree may be expressed by putting the expression between
1.3 noro 186: {\tt SM\_beginBlock} and {\tt SM\_endBlock}, which are
187: stack machine commands for delayed evaluation.
188: (cf. {\tt \{ }, {\tt \} } in PostScript).
189: However it makes the implementation of stack machines complicated.
190: It is desirable that CMObject is independent of OX stack machine.
191: Therefore we introduce an OpenMath like tree representation for CMO
1.7 noro 192: Tree object.
1.3 noro 193: This method allows us to implement tree structure easily
194: on individual OpenXM systems.
195: Note that CMO Tree corresponds to Symbol and Application in OpenMath.
1.1 noro 196: */
197:
198:
199: /*&C
200:
201: */
202: /*&jp
203: Lambda は関数を定義するための関数である.
204: Lisp の Lambda 表現と同じ.
1.3 noro 205: */
206: /*&eg
207: Lambda is used to define functions.
208: It is the same as the Lambda expression in Lisp.
209: */
1.1 noro 210:
211: \noindent
1.3 noro 212: //&jp 例: $sin(x+e)$ の表現.
213: //&eg Example: the expression of $sin(x+e)$.
1.1 noro 214: \begin{verbatim}
215: (CMO_TREE, (CMO_STRING, "sin"), (CMO_STRING, "basic"),
216: (CMO_LIST,[size=]1,
217: (CMO_TREE, (CMO_STRING, "plus"), (CMO_STRING, "basic"),
218: (CMO_LIST,[size=]2, (CMO_INDETERMINATE,"x"),
1.6 noro 219: //&jp (CMO_TREE,(CMO_STRING, "e"), 自然対数の底
220: //&eg (CMO_TREE,(CMO_STRING, "e"), the base of natural logarithms
221: (CMO_STRING, "basic"))
1.1 noro 222: ))
223: )
224: )
225: \end{verbatim}
1.10 ! takayama 226: //&jp Leave の成分には, 多項式を含む任意のオブジェクトがきてよい.
! 227: //&eg Elements of the leave may be any objects including polynomials.
1.1 noro 228:
229: \noindent
230: Example:
231: \begin{verbatim}
232: sm1> [(plus) (Basic) [(123).. (345)..]] [(class) (tree)] dc ::
1.8 takayama 233: Class.tree [ $plus$ , $basic$ , [ 123 , 345 ] ]
1.1 noro 234: \end{verbatim}
235:
236:
237:
238: \bigbreak
1.3 noro 239: //&jp 次に, 分散表現多項式に関係するグループを定義しよう.
1.4 noro 240: /*&eg
241: Let us define a group for distributed polynomials. In the following
242: DMS stands for Distributed Monomial System.
243: */
1.1 noro 244:
245: \medbreak
246: \noindent
1.4 noro 247: Group CMObject/DistributedPolynomials requires CMObject/Primitive,
248: CMObject/Basic. \\
1.1 noro 249: Monomial, Monomial32, Coefficient, Dpolynomial, DringDefinition,
250: Generic DMS ring, RingByName, DMS of N variables $\in$
251: CMObject/DistributedPolynomials. \\
1.3 noro 252: /*&jp
1.1 noro 253: \begin{eqnarray*}
254: \mbox{Monomial} &:& \mbox{Monomial32}\, |\, \mbox{Zero} \\
255: \mbox{Monomial32}&:& ({\tt CMO\_MONOMIAL32}, {\sl int32}\, n,
256: {\sl int32}\, \mbox{e[1]}, \ldots,
257: {\sl int32}\, \mbox{e[n]}, \\
258: & & \ \mbox{Coefficient}) \\
259: & & \mbox{ --- e[i] で, $n$ 変数 monomial
260: $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$
261: をあらわす.} \\
262: \mbox{Coefficient}&:& \mbox{ZZ} | \mbox{Integer32} \\
263: \mbox{Dpolynomial}&:& \mbox{Zero} \\
264: & & |\ ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},{\sl int32} m, \\
265: & & \ \ \mbox{DringDefinition},
266: [\mbox{Monomial32}|\mbox{Zero}], \\
267: & &\ \
268: \{\mbox{Monomial32}\}) \\
269: & &\mbox{--- m はモノミアルの個数である.}\\
270: \mbox{DringDefinition}
271: &:& \mbox{DMS of N variables} \\
272: & & |\ \mbox{RingByName} \\
273: & & |\ \mbox{Generic DMS ring} \\
274: & & \mbox{ --- 分散表現多項式環の定義. } \\
275: \mbox{Generic DMS ring}
1.2 noro 276: &:& \mbox{({\tt CMO\_DMS\_GENERIC}) --- 新版はこちら}\\
1.1 noro 277: \mbox{RingByName}&:& ({\tt CMO\_RING\_BY\_NAME}, {\sl Cstring}\ {\rm s}) \\
278: & & \mbox{ --- 名前 s で, 格納された ring 定義.} \\
279: \mbox{DMS of N variables}
280: &:& ({\tt CMO\_DMS\_OF\_N\_VARIABLES}, \\
281: & & \ ({\tt CMO\_LIST}, {\sl int32}\, \mbox{m},
282: {\sl Integer32}\, \mbox{n}, {\sl Integer32}\,\mbox{p} \\
283: & & \ \ [,{\sl object}\,\mbox{s}, {\sl Cstring}\,\mbox{c},
284: {\sl List}\, \mbox{vlist},
285: {\sl List}\, \mbox{wvec}, {\sl List}\, \mbox{outord}]) \\
286: & & \mbox{ --- m はあとに続く要素の数} \\
287: & & \mbox{ --- n は変数の数, p は 標数} \\
288: & & \mbox{ --- s は ring の名前} \\
289: & & \mbox{ --- c は係数環, QQ, ZZ の場合は文字列で QQ, ZZ と書く.} \\
290: & & \mbox{ --- vlist は Indeterminate のリスト(新版). 多項式環の変数リスト} \\
291: & & \mbox{ --- wvec は order をきめる weight vector,} \\
292: & & \mbox{ --- outord は出力するときの変数順序.} \\
293: \end{eqnarray*}
1.3 noro 294: */
295: /*&eg
296: \begin{eqnarray*}
297: \mbox{Monomial} &:& \mbox{Monomial32}\, |\, \mbox{Zero} \\
298: \mbox{Monomial32}&:& ({\tt CMO\_MONOMIAL32}, {\sl int32}\, n,
299: {\sl int32}\, \mbox{e[1]}, \ldots,
300: {\sl int32}\, \mbox{e[n]}, \\
301: & & \ \mbox{Coefficient}) \\
302: & & \mbox{ --- e[i] is the exponent $e_i$ of the monomial
303: $x^e = x_1^{e_1} \cdots x_n^{e_n}$. } \\
304: \mbox{Coefficient}&:& \mbox{ZZ} | \mbox{Integer32} \\
305: \mbox{Dpolynomial}&:& \mbox{Zero} \\
306: & & |\ ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},{\sl int32} m, \\
307: & & \ \ \mbox{DringDefinition}, [\mbox{Monomial32}|\mbox{Zero}], \\
308: & &\ \
309: \{\mbox{Monomial32}\}) \\
310: & &\mbox{--- m is equal to the number of monomials.}\\
311: \mbox{DringDefinition}
312: &:& \mbox{DMS of N variables} \\
313: & & |\ \mbox{RingByName} \\
314: & & |\ \mbox{Generic DMS ring} \\
315: & & \mbox{ --- definition of the ring of distributed polynomials. } \\
316: \mbox{Generic DMS ring}
317: &:& ({\tt CMO\_DMS\_GENERIC}) \\
318: \mbox{RingByName}&:& ({\tt CMO\_RING\_BY\_NAME}, {\sl Cstring} s) \\
1.4 noro 319: & & \mbox{ --- The ring definition referred by the name ``s''.} \\
1.3 noro 320: \mbox{DMS of N variables}
321: &:& ({\tt CMO\_DMS\_OF\_N\_VARIABLES}, \\
322: & & \ ({\tt CMO\_LIST}, {\sl int32}\, \mbox{m},
323: {\sl Integer32}\, \mbox{n}, {\sl Integer32}\, \mbox{p} \\
324: & & \ \ [,{\sl Cstring}\,\mbox{s}, {\sl List}\, \mbox{vlist},
325: {\sl List}\, \mbox{wvec}, {\sl List}\, \mbox{outord}]) \\
326: & & \mbox{ --- m is the number of elements.} \\
327: & & \mbox{ --- n is the number of variables, p is the characteristic} \\
328: & & \mbox{ --- s is the name of the ring, vlist is the list of variables.} \\
329: & & \mbox{ --- wvec is the weight vector.} \\
330: & & \mbox{ --- outord is the order of variables to output.} \\
331: \end{eqnarray*}
332: */
1.1 noro 333:
1.3 noro 334: /*&jp
1.1 noro 335: RingByName や DMS of N variables はなくても, DMS を定義できる.
336: したがって, これらを実装してないシステムで DMS を扱うものが
337: あってもかまわない.
338:
339: 以下, 以上の CMObject にたいする,
340: xxx = asir, kan の振舞いを記述する.
1.3 noro 341: */
342: /*&eg
343: Note that it is possible to define DMS without RingByName and
344: DMS of N variables.
345:
346: In the following we describe how the above CMObjects
347: are implemented on Asir and Kan.
348: */
1.1 noro 349:
350: \subsection{ Zero}
1.3 noro 351: /*&jp
352: CMO では ゼロの表現法がなんとおりもあるが,
1.1 noro 353: どのようなゼロをうけとっても,
354: システムのゼロに変換できるべきである.
1.3 noro 355: */
356: /*&eg
357: Though CMO has various representations of zero,
358: each representation should be translated into zero
359: in the system.
360: */
1.1 noro 361:
362:
1.3 noro 363: //&jp \subsection{ 整数 ZZ }
364: //&eg \subsection{ Integer ZZ }
1.1 noro 365:
366: \begin{verbatim}
367: #define CMO_ZZ 20
368: \end{verbatim}
369:
1.3 noro 370: /*&jp
371: この節ではOpen xxx 規約における任意の大きさの整数(bignum)の扱いについ
372: て説明する. Open XM 規約における多重精度整数を表すデータ型 CMO\_ZZ は
373: GNU MPライブラリなどを参考にして設計されていて, 符号付き絶対値表現を用
374: いている. (cf. {\tt kan/sm1} の配布ディレクトリのなかの {\tt
375: plugin/cmo-gmp.c}) CMO\_ZZ は次の形式をとる.
376: */
377: /*&eg
378: We describe the bignum (multi-precision integer) representation in OpenXM.
379: In OpenXM {\tt CMO\_ZZ} is used to represent bignum. Its design is similar
380: to that in GNU MP. (cf. {\tt plugin/cmo-gmp.c} in the {\tt kan/sm1}
381: distribution). CMO\_ZZ is defined as follows.
382: */
1.1 noro 383:
384: \begin{tabular}{|c|c|c|c|c|}
385: \hline
386: {\tt int32 CMO\_ZZ} & {\tt int32 $f$} & {\tt int32 $b_0$} & $\cdots$ &
387: {\tt int32 $b_{n}$} \\
388: \hline
1.3 noro 389: \end{tabular}
390:
391: /*&jp
392: $f$ は32bit整数である. $b_0, \ldots, b_n$ は unsigned int32 である.
393: $|f|$ は $n+1$ である. この CMO の符号は $f$ の符号で定める. 前述し
394: たように, 32bit整数の負数は 2 の補数表現で表される.
395:
396: Open xxx 規約では上の CMO は以下の整数を意味する. ($R = 2^{32}$)
397: */
398: /*&eg
399: $f$ is a 32bit integer. $b_0, \ldots, b_n$ are unsigned 32bit integers.
400: $|f|$ is equal to $n+1$.
401: The sign of $f$ represents that of the above CMO. As stated in Section
402: \ref{sec:basic0}, a negative 32bit integer is represented by
403: two's complement.
404:
405: In OpenXM the above CMO represents the following integer. ($R = 2^{32}$.)
406: */
1.1 noro 407:
408: \[
409: \mbox{sgn}(f)\times (b_0 R^{0}+ b_1 R^{1} + \cdots + b_{n-1}R^{n-1} + b_n R^n).
410: \]
1.3 noro 411:
412: /*&jp
413: {\tt int32} を network byte order で表現
414: しているとすると,例えば, 整数 $14$ は CMO\_ZZ で表わすと,
415: */
416: /*&eg
417: If we express {\tt int32} by the network byte order,
418: a CMO\_ZZ $14$ is expressed by
419: */
1.1 noro 420: \[
421: \mbox{(CMO\_ZZ, 1, 0, 0, 0, e)},
422: \]
1.3 noro 423: //&jp と表わす. これはバイト列では
1.4 noro 424: //&eg The corresponding byte sequence is
1.1 noro 425: \[
426: \mbox{\tt 00 00 00 14 00 00 00 01 00 00 00 0e}
427: \]
1.3 noro 428: //&jp となる.
1.1 noro 429:
430:
1.3 noro 431: //&jp なお ZZ の 0 ( (ZZ) 0 と書く ) は, {\tt (CMO\_ZZ, 00,00,00,00)} と表現する.
432: //&eg Note that CMO\_ZZ 0 is expressed by {\tt (CMO\_ZZ, 00,00,00,00)}.
1.1 noro 433:
434:
1.3 noro 435: //&jp \subsection{ 分散表現多項式 Dpolynomial }
436: //&eg \subsection{ Distributed polynomial Dpolynomial }
1.1 noro 437:
1.3 noro 438: /*&jp
1.1 noro 439: 環とそれに属する多項式は次のような考えかたであつかう.
440:
441: Generic DMS ring に属する元は,
442: 変数を $n$ 個持つ 適当な係数集合 $K$ を持つ多項式環 $K[x_1, \ldots, x_n]$
443: の元である.
444: 係数集合 $K$ がなにかは, 実際データを読み込み, Coefficient を見た段階で
445: わかる.
446: この環に属する多項式を CMO 形式でうけとった場合, 各サーバはその
447: サーバの対応する Object に変換しないといけない.
448: この変換の仕方は, 各サーバ毎にきめる.
449:
450: Asir の場合は, $K[x_1, \ldots, x_n]$ の元は分散表現多項式に変換される.
451: \noroa{ でも, order はどうなるの? }
452:
453: {\tt kan/sm1} の場合は事情は複雑である.
454: {\tt kan/sm1} は, Generic DMS ring にあたる クラスをもたない.
455: つまり, Default で存在する, $n$ 変数の分散表現多項式環は存在しないわけである.
456: したがって, {\tt kan/sm1} では, DMS of N variables が来た場合,
457: これを CurrentRing の元として読み込む. CurrentRing の変数の数が $n'$
458: で, $n' < n$ だと新しい多項式環を生成してデータを読み込む.
459: Order その他の optional 情報はすべて無視する.
460:
461: DMS の 2 番目のフィールドで,
462: Ring by Name を用いた場合, 現在の名前空間で変数 yyy に格納された ring object
463: の元として, この多項式を変換しなさいという意味になる.
464: {\tt kan/sm1} の場合, 環の定義は ring object として格納されており,
465: この ring object を 変数 yyy で参照することにより CMO としてうけとった
466: 多項式をこの ring の元として格納できる.
1.3 noro 467: */
1.1 noro 468:
1.3 noro 469: /*&eg
470: We treat polynomial rings and their elements as follows.
1.1 noro 471:
1.4 noro 472: Generic DMS ring is an $n$-variate polynomial ring $K[x_1, \ldots, x_n]$,
1.3 noro 473: where $K$ is some coefficient set. $K$ is unknown in advance
1.4 noro 474: and it is determined when coefficients of an element are received.
475: When a server has received an element in Generic DMS ring,
1.3 noro 476: the server has to translate it into the corresponding local object
477: on the server. Each server has its own translation scheme.
478: In Asir such an element are translated into a distributed polynomial.
479: In {\tt kan/sm1} things are complicated.
1.4 noro 480: {\tt kan/sm1} does not have any class corresponding to Generic DMS ring.
1.3 noro 481: {\tt kan/sm1} translates a DMS of N variables into an element of
482: the CurrentRing.
483: If the CurrentRing is $n'$-variate and $n' < n$, then
1.4 noro 484: an $n$-variate polynomial ring is newly created. Optional informations such as
1.3 noro 485: the term order are all ignored.
486:
1.4 noro 487: If RingByName ({\tt CMO\_RING\_BY\_NAME}, yyy)
1.3 noro 488: is specified as the second field of DMS,
489: it requests a sever to use a ring object whose name is yyy
490: as the destination ring for the translation.
491: This is done in {\tt kan/sm1}.
492: */
1.1 noro 493:
494: \medbreak \noindent
1.3 noro 495: //&jp {\bf Example}: (すべての数の表記は 16 進表記)
496: //&eg {\bf Example}: (all numbers are represented in hexadecimal notation)
1.1 noro 497: {\footnotesize \begin{verbatim}
498: Z/11Z [6 variables]
499: (kxx/cmotest.sm1) run
500: [(x,y) ring_of_polynomials ( ) elimination_order 11 ] define_ring ;
501: (3x^2 y). cmo /ff set ;
502: [(cmoLispLike) 1] extension ;
503: ff ::
504: Class.CMO CMO StandardEncoding: size = 52, size/sizeof(int) = 13,
505: tag=CMO_DISTRIBUTED_POLYNOMIAL
506:
507: 0 0 0 1f 0 0 0 1 0 0 0 18 0 0 0 13 0 0 0 6
508: 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 1
509: 0 0 0 0 0 0 0 2 0 0 0 3
510:
511: ff omc ::
512: (CMO_DISTRIBUTED_POLYNOMIAL[1f],[size=]1,(CMO_DMS_GENERIC[18],),
513: (CMO_MONOMIAL32[13],3*x^2*y),),
514: \end{verbatim} }
1.3 noro 515: /*&jp
516: $ 3 x^2 y$ は 6 変数の多項式環の 元としてみなされている.
517: */
518: /*&eg
519: $3 x^2 y$ is regarded as an element of a six-variate polynomial ring.
520: */
1.1 noro 521:
522:
1.3 noro 523: //&jp \subsection{再帰表現多項式の定義}
524: //&eg \subsection{Recursive polynomials}
1.1 noro 525:
526: \begin{verbatim}
527: #define CMO_RECURSIVE_POLYNOMIAL 27
528: #define CMO_POLYNOMIAL_IN_ONE_VARIABLE 33
529: \end{verbatim}
530:
1.4 noro 531: Group CMObject/RecursivePolynomial requires CMObject/Primitive, CMObject/Basic.\\
1.1 noro 532: Polynomial in 1 variable, Coefficient, Name of the main variable,
533: Recursive Polynomial, Ring definition for recursive polynomials
534: $\in$ CMObject/RecursivePolynomial \\
535:
1.3 noro 536: /*&jp
1.1 noro 537: \begin{eqnarray*}
538: \mbox{Polynomial in 1 variable} &:&
539: \mbox{({\tt CMO\_POLYNOMIAL\_IN\_ONE\_VARIABLE},\, {\sl int32}\, m, } \\
540: & & \quad \mbox{ Name of the main variable }, \\
541: & & \quad \mbox{ \{ {\sl int32} e, Coefficient \}} \\
542: & & \mbox{ --- m はモノミアルの個数. } \\
543: & & \mbox{ --- e, Coefficieint はモノミアルを表現している. } \\
544: & & \mbox{ --- 順序の高い順にならべる. 普通は巾の高い順.} \\
545: & & \mbox{ --- e は 1変数多項式の巾をあらわす. } \\
546: \mbox{Coefficient} &:& \mbox{ ZZ} \,|\, \mbox{ QQ } \,|\,
547: \mbox{ integer32 } \,|\,
548: \mbox{ Polynomial in 1 variable } \\
549: & & \quad \,|\, \mbox{Tree} \,|\, \mbox{Zero} \,|\,\mbox{Dpolynomial}\\
550: \mbox{Name of the main variable } &:&
551: \mbox{ {\sl int32} v } \\
552: & & \mbox{ --- v は 変数番号 (0 からはじまる) を表す. } \\
553: \mbox{Recursive Polynomial} &:&
554: \mbox{ ( {\tt CMO\_RECURSIVE\_POLYNOMIAL}, } \\
1.3 noro 555: & & \quad \mbox{ RringDefinition, } \\
1.1 noro 556: & & \quad
557: \mbox{ Polynomial in 1 variable}\, | \, \mbox{Coefficient} \\
1.3 noro 558: \mbox{RringDefinition}
1.1 noro 559: & : & \mbox{ {\sl List} v } \\
1.10 ! takayama 560: & & \quad \mbox{ --- v は, 変数名(indeterminate) または Tree のリスト. } \\
1.1 noro 561: & & \quad \mbox{ --- 順序の高い順. } \\
562: \end{eqnarray*}
1.3 noro 563: */
564: /*&eg
565: \begin{eqnarray*}
566: \mbox{Polynomial in 1 variable} &:&
567: \mbox{({\tt CMO\_POLYNOMIAL\_IN\_ONE\_VARIABLE},\, {\sl int32}\, m, } \\
568: & & \quad \mbox{ Name of the main variable }, \\
569: & & \quad \mbox{ \{ {\sl int32} e, Coefficient \}} \\
1.4 noro 570: & & \mbox{ --- m is the number of monomials. } \\
571: & & \mbox{ --- A pair of e and Coefficient represents a monomial. } \\
1.3 noro 572: & & \mbox{ --- The pairs of e and Coefficient are sorted in the } \\
573: & & \mbox{ \quad decreasing order, usually with respect to e.} \\
574: & & \mbox{ --- e denotes an exponent of a monomial with respect to } \\
575: & & \mbox{ \quad the main variable. } \\
576: \mbox{Coefficient} &:& \mbox{ ZZ} \,|\, \mbox{ QQ } \,|\,
577: \mbox{ integer32 } \,|\,
578: \mbox{ Polynomial in 1 variable } \\
579: & & \quad \,|\, \mbox{Tree} \,|\, \mbox{Zero} \,|\,\mbox{Dpolynomial}\\
580: \mbox{Name of the main variable } &:&
581: \mbox{ {\sl int32} v } \\
582: & & \mbox{ --- v denotes a variable number. } \\
583: \mbox{Recursive Polynomial} &:&
584: \mbox{ ( {\tt CMO\_RECURSIVE\_POLYNOMIAL}, } \\
585: & & \quad \mbox{ RringDefinition, } \\
586: & & \quad
587: \mbox{ Polynomial in 1 variable}\, | \, \mbox{Coefficient} \\
588: \mbox{RringDefinition}
589: & : & \mbox{ {\sl List} v } \\
1.10 ! takayama 590: & & \quad \mbox{ --- v is a list of names of indeterminates or trees. } \\
1.3 noro 591: & & \quad \mbox{ --- It is sorted in the decreasing order. } \\
592: \end{eqnarray*}
593: */
1.1 noro 594: \bigbreak
595: \noindent
1.3 noro 596: Example:
1.1 noro 597: \begin{verbatim}
598: (CMO_RECURSIEVE_POLYNOMIAL, ("x","y"),
599: (CMO_POLYNOMIAL_IN_ONE_VARIABLE, 2, 0, <--- "x"
600: 3, (CMO_POLYNOMIAL_IN_ONE_VARIABLE, 2, 1, <--- "y"
601: 5, 1234,
602: 0, 17),
603: 1, (CMO_POLYNOMIAL_IN_ONE_VARIABLE, 2, 1, <--- "y"
604: 10, 1,
605: 5, 31)))
606: \end{verbatim}
1.3 noro 607: //&jp これは,
608: //&eg This represents
609: $$ x^3 (1234 y^5 + 17 ) + x^1 (y^{10} + 31 y^5) $$
610: /*&jp
1.1 noro 611: をあらわす.
612: 非可換多項式もこの形式であらわしたいので, 積の順序を上のように
613: すること. つまり, 主変数かける係数の順番.
1.3 noro 614: */
615: /*&eg
616: We intend to represent non-commutative polynomials with the
617: same form. In such a case, the order of products are defined
1.4 noro 618: as above, that is a power of the main variable $\times$ a coeffcient.
1.3 noro 619: */
1.1 noro 620:
621: \noindent
622: \begin{verbatim}
623: sm1
624: sm1>(x^2-h). [(class) (recursivePolynomial)] dc /ff set ;
625: sm1>ff ::
626: Class.recursivePolynomial h * ((-1)) + (x^2 * (1))
627: \end{verbatim}
628:
1.3 noro 629: //&jp \subsection{CPU依存の double }
630: //&eg \subsection{CPU dependent double}
1.1 noro 631:
632: \begin{verbatim}
633: #define CMO_64BIT_MACHINE_DOUBLE 40
634: #define CMO_ARRAY_OF_64BIT_MACHINE_DOUBLE 41
635: #define CMO_128BIT_MACHINE_DOUBLE 42
636: #define CMO_ARRAY_OF_128BIT_MACHINE_DOUBLE 43
637: \end{verbatim}
638:
639: \noindent
1.4 noro 640: Group CMObject/MachineDouble requires CMObject/Primitive.\\
1.1 noro 641: 64bit machine double, Array of 64bit machine double
642: 128bit machine double, Array of 128bit machine double
643: $\in$ CMObject/MachineDouble \\
644:
1.3 noro 645: /*&jp
1.1 noro 646: \begin{eqnarray*}
647: \mbox{64bit machine double} &:&
648: \mbox{({\tt CMO\_64BIT\_MACHINE\_DOUBLE}, } \\
649: & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte}} s8)\\
650: & & \mbox{ --- s1, $\ldots$, s8 は {\tt double} (64bit). } \\
651: & & \mbox{ --- この表現はCPU依存である.}\\
652: && \mbox{\quad\quad mathcap に CPU 情報を付加しておく.} \\
653: \mbox{Array of 64bit machine double} &:&
654: \mbox{({\tt CMO\_ARRAY\_OF\_64BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\
655: & & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte}}\, s8[1], \ldots , {\sl byte}\, s8[m])\\
656: & & \mbox{ --- s*[1], $\ldots$ s*[m] は m 個の double (64bit) である. } \\
657: & & \mbox{ --- この表現はCPU依存である.}\\
658: & & \mbox{ \quad\quad mathcap に CPU 情報を付加しておく.} \\
659: \mbox{128bit machine double} &:&
660: \mbox{({\tt CMO\_128BIT\_MACHINE\_DOUBLE}, } \\
661: & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte}} s16)\\
662: & & \mbox{ --- s1, $\ldots$, s16 は {\tt long double} (128bit). } \\
663: & & \mbox{ --- この表現はCPU依存である.}\\
664: && \mbox{\quad\quad mathcap に CPU 情報を付加しておく.} \\
665: \mbox{Array of 128bit machine double} &:&
666: \mbox{({\tt CMO\_ARRAY\_OF\_128BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\
667: & & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte}} s16[1], \ldots , {\sl byte} s16[m])\\
668: & & \mbox{ --- s*[1], $\ldots$ s*[m] は m 個の long double (128bit) である. } \\
669: & & \mbox{ --- この表現はCPU依存である.}\\
670: & & \mbox{ \quad\quad mathcap に CPU 情報を付加しておく.}
671: \end{eqnarray*}
1.3 noro 672: */
673: /*&eg
674: \begin{eqnarray*}
675: \mbox{64bit machine double} &:&
676: \mbox{({\tt CMO\_64BIT\_MACHINE\_DOUBLE}, } \\
677: & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte}} s8)\\
678: & & \mbox{ --- s1, $\ldots$, s8 は {\tt double} (64bit). } \\
679: & & \mbox{ --- This depends on CPU.}\\
680: && \mbox{\quad\quad Add informations on CPU to the mathcap.} \\
681: \mbox{Array of 64bit machine double} &:&
682: \mbox{({\tt CMO\_ARRAY\_OF\_64BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\
683: & & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte}}\, s8[1], \ldots , {\sl byte}\, s8[m])\\
684: & & \mbox{ --- s*[1], $\ldots$ s*[m] are 64bit double's. } \\
685: & & \mbox{ --- This depends on CPU.}\\
686: & & \mbox{\quad\quad Add informations on CPU to the mathcap.} \\
687: \mbox{128bit machine double} &:&
688: \mbox{({\tt CMO\_128BIT\_MACHINE\_DOUBLE}, } \\
689: & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte}} s16)\\
690: & & \mbox{ --- s1, $\ldots$, s16 は {\tt long double} (128bit). } \\
691: & & \mbox{ --- This depends on CPU.}\\
692: & & \mbox{\quad\quad Add informations on CPU to the mathcap.} \\
693: \mbox{Array of 128bit machine double} &:&
694: \mbox{({\tt CMO\_ARRAY\_OF\_128BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\
695: & & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte}} s16[1], \ldots , {\sl byte} s16[m])\\
696: & & \mbox{ --- s*[1], $\ldots$ s*[m] are 128bit long double's. } \\
697: & & \mbox{ --- This depends on CPU.}\\
698: & & \mbox{\quad\quad Add informations on CPU to the mathcap.} \\
699: \end{eqnarray*}
700: */
1.1 noro 701:
702: \bigbreak
1.3 noro 703: //&jp 次に IEEE 準拠の float および Big float を定義しよう.
1.5 noro 704: //&eg We define float and big float conforming to the IEEE standard.
1.1 noro 705: \begin{verbatim}
706: #define CMO_BIGFLOAT 50
707: #define CMO_IEEE_DOUBLE_FLOAT 51
708: \end{verbatim}
709:
1.3 noro 710: /*&jp
711: IEEE 準拠の float については, IEEE 754 double precision floating-point
712: format (64 bit) の定義を見よ.
713: */
714: /*&eg
1.5 noro 715: See IEEE 754 double precision floating-point (64 bit) for the details of
716: float conforming to the IEEE standard.
1.3 noro 717: */
1.1 noro 718:
719: \noindent
1.4 noro 720: Group CMObject/Bigfloat requires CMObject/Primitive, CMObject/Basic.\\
1.1 noro 721: Bigfloat
722: $\in$ CMObject/Bigfloat \\
723:
724: \begin{eqnarray*}
725: \mbox{Bigfloat} &:&
726: \mbox{({\tt CMO\_BIGFLOAT}, } \\
727: & & \quad \mbox{ {\sl ZZ} a , {\sl ZZ} e})\\
1.3 noro 728: & & \mbox{ --- $a \times 2^e$. } \\
1.1 noro 729: \end{eqnarray*}
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