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1.15    ! takayama    1: %% $OpenXM: OpenXM/doc/OpenXM-specs/cmo-basic1.tex,v 1.14 2015/08/18 02:54:05 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
1.14      noro       17: #define     CMO_COMPLEX       35
1.1       noro       18:
1.15    ! takayama   19: #define     CMO_BIGFLOAT32    52
1.1       noro       20:
                     21: #define     CMO_INDETERMINATE  60
                     22: #define     CMO_TREE           61
                     23: #define     CMO_LAMBDA         62    /* for function definition */
                     24: \end{verbatim}
                     25:
                     26: */
                     27:
                     28: /*&jp
1.4       noro       29: 以下, グループ CMObject/Basic, CMObject/Tree
1.1       noro       30: および CMObject/DistributedPolynomial
                     31: に属する CMObject の形式を説明する.
                     32:
1.5       noro       33: \noindent
                     34: {\tt OpenXM/src/ox\_toolkit} にある {\tt bconv} をもちいると
                     35: CMO expression を binary format に変換できるので,
                     36: これを参考にするといい.
1.1       noro       37: */
                     38: /*&eg
                     39: In the sequel, we will explain on the groups
1.4       noro       40: CMObject/Basic, CMObject/Tree
1.1       noro       41: and CMObject/DistributedPolynomial.
1.5       noro       42:
                     43: \noindent
                     44: The program {\tt bconv} at {\tt OpenXM/src/ox\_toolkit}
                     45: translates
                     46: CMO expressions into binary formats.
                     47: It is convinient to understand the binary formats explained in
                     48: this section.
1.1       noro       49: */
                     50:
1.5       noro       51: /*&C
                     52: \noindent Example:
                     53: \begin{verbatim}
                     54: bash$ ./bconv
                     55: > (CMO_ZZ,123123);
                     56: 00 00 00 14 00 00 00 01 00 01 e0 f3
                     57: \end{verbatim}
                     58: */
1.1       noro       59: /*&jp
                     60:
                     61: \bigbreak
                     62: \noindent
1.4       noro       63: Group CMObject/Basic requires CMObject/Primitive. \\
1.12      takayama   64: ZZ, QQ, Zero, Rational, Indeterminate $\in$ CMObject/Basic. \\
1.1       noro       65: \begin{eqnarray*}
                     66: \mbox{Zero} &:& ({\tt CMO\_ZERO}) \\
                     67: & & \mbox{ --- ユニバーサルな ゼロを表す. } \\
1.12      takayama   68: \mbox{ZZ}         &:& ({\tt CMO\_ZZ},{\sl int32}\, {\rm f}, {\sl byte}\, \mbox{a[1]}, \ldots ,
1.9       noro       69: {\sl byte}\, \mbox{a[$|$f$|$]} ) \\
1.1       noro       70: &:& \mbox{ --- bignum をあらわす. a[i] についてはあとで説明}\\
1.9       noro       71: \mbox{QQ}        &:& ({\tt CMO\_QQ},
                     72:                       {\sl int32}\, {\rm m}, {\sl byte}\, \mbox{a[1]}, \ldots, {\sl byte}\, \mbox{a[$|$m$|$]},
                     73:                       {\sl int32}\, {\rm n}, {\sl byte}\, \mbox{b[1]}, \ldots, {\sl byte}\, \mbox{b[$|$n$|$]})\\
1.1       noro       74: & & \mbox{ --- 有理数 $a/b$ を表す. } \\
                     75: \mbox{Rational}        &:& ({\tt CMO\_RATIONAL}, {\sl CMObject}\, {\rm a}, {\sl CMObject}\, {\rm b}) \\
                     76: & & \mbox{ ---  $a/b$ を表す. } \\
1.15    ! takayama   77: \mbox{Bigfloat32}         &:& ({\tt CMO\_BIGFLOAT32},
        !            78: {\sl int32}\, {\rm prec}, {\sl int32}\, {\rm sign}, {\sl int32}\, {\rm exp},
        !            79: {\sl int32}\, \mbox{a[1]}, \ldots , {\sl int32}\, \mbox{a[k]} ) \\
        !            80: &:& \mbox{ --- bigfloat をあらわす. a[i], k についてはあとで説明}\\
1.14      noro       81: \mbox{Complex}        &:& ({\tt CMO\_COMPLEX}, {\sl CMObject}\, {\rm re}, {\sl CMObject}\, {\rm im}) \\
                     82: & & \mbox{ ---  $a+b\sqrt{-1}$ を表す. } \\
1.1       noro       83: \mbox{Indeterminate}        &:& ({\tt CMO\_INDETERMINATE}, {\sl Cstring}\, {\rm v}) \\
                     84: & & \mbox{ --- 変数名 $v$ . } \\
                     85: \end{eqnarray*}
                     86: */
1.12      takayama   87:
                     88:
1.1       noro       89: /*&eg
                     90:
                     91: \bigbreak
                     92: \noindent
1.4       noro       93: Group CMObject/Basic requires CMObject/Primitive. \\
1.12      takayama   94: ZZ, QQ, Zero, Rational, Indeterminate $\in$ CMObject/Basic. \\
1.1       noro       95: \begin{eqnarray*}
                     96: \mbox{Zero} &:& ({\tt CMO\_ZERO}) \\
                     97: & & \mbox{ --- Universal zero } \\
1.12      takayama   98: \mbox{ZZ}         &:& ({\tt CMO\_ZZ},{\sl int32}\, {\rm f}, {\sl byte}\, \mbox{a[1]}, \ldots ,
1.9       noro       99: {\sl byte}\, \mbox{a[$|$m$|$]} ) \\
1.1       noro      100: &:& \mbox{ --- bignum. The meaning of a[i] will be explained later.}\\
1.9       noro      101: \mbox{QQ}        &:& ({\tt CMO\_QQ},
                    102:                       {\sl int32}\, {\rm m}, {\sl byte}\, \mbox{a[1]}, \ldots, {\sl byte}\, \mbox{a[$|$m$|$]},
                    103:                       {\sl int32}\, {\rm n}, {\sl byte}\, \mbox{b[1]}, \ldots, {\sl byte}\, \mbox{b[$|$n$|$]})\\
1.1       noro      104: & & \mbox{ --- Rational number $a/b$. } \\
                    105: \mbox{Rational}        &:& ({\tt CMO\_RATIONAL}, {\sl CMObject}\, {\rm a}, {\sl CMObject}\, {\rm b}) \\
                    106: & & \mbox{ ---  Rational expression $a/b$. } \\
1.15    ! takayama  107: \mbox{Bigfloat32}         &:& ({\tt CMO\_BIGFLOAT32},
        !           108: {\sl int32}\, {\rm prec}, {\sl int32}\, {\rm sign}, {\sl int32}\, {\rm exp},
        !           109: {\sl int32}\, \mbox{a[1]}, \ldots , {\sl int32}\, \mbox{a[k]} ) \\
        !           110: &:& \mbox{ --- bigfloat. The meaning of a[i], k will be explained later.}\\
1.14      noro      111: \mbox{Complex}        &:& ({\tt CMO\_COMPLEX}, {\sl CMObject}\, {\rm re}, {\sl CMObject}\, {\rm im}) \\
                    112: & & \mbox{ ---  Complex number $a+b\sqrt{-1}$. } \\
1.1       noro      113: \mbox{Indeterminate}        &:& ({\tt CMO\_INDETERMINATE}, {\sl Cstring}\, {\rm v}) \\
                    114: & & \mbox{ --- Variable name $v$ . } \\
                    115: \end{eqnarray*}
                    116: */
                    117: /*&C
                    118:
                    119: */
1.12      takayama  120: /*&C
                    121:
                    122: */
1.1       noro      123:
                    124: /*&jp
                    125: Indeterminate は変数名をあらわす.
                    126: v はバイト列であればなにを用いてもよいが,
                    127: 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.
                    128: 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
                    129: 実装しないといけない.
                    130: (これを
                    131: {\tt Dx} は {\tt \#dx} と変換するなどの
                    132: escape sequence を用いて実現するのは, 無理があるようである.
                    133: テーブルを作成する必要があるであろう.)
                    134: */
                    135: /*&eg
1.12      takayama  136: The name of a variable should be expressed by using Indeterminate.
1.1       noro      137: v may be any sequence of bytes, but each system has its own
                    138: restrictions on the names of variables.
                    139: Indeterminates of CMO and internal variable names must be translated
1.12      takayama  140: in one-to-one correspondence.
1.1       noro      141: */
                    142:
1.12      takayama  143:
1.1       noro      144: /*&jp
1.12      takayama  145: \subsection{Indeterminate および Tree}
1.1       noro      146:
                    147: \noindent
1.4       noro      148: Group CMObject/Tree requires CMObject/Basic. \\
1.12      takayama  149: Tree, Lambda $\in$ CMObject/Tree. \\
1.1       noro      150: \begin{eqnarray*}
                    151: \mbox{Tree}        &:& ({\tt CMO\_TREE}, {\sl Cstring}\, {\rm name},
1.11      takayama  152:  {\sl List}\, {\rm attributes}, {\sl List}\, {\rm leaves}) \\
1.1       noro      153: & & \mbox{ --- 名前 name の定数または関数. 関数の評価はおこなわない. } \\
1.11      takayama  154: & & \mbox{ --- attributes は空リストでなければ name の属性を保持している. }\\
                    155: & & \mbox{ --- 属性リストは, key と 値のペアである. }\\
1.1       noro      156: \mbox{Lambda}        &:& ({\tt CMO\_LAMBDA}, {\sl List}\, {\rm args},
                    157:                           {\sl Tree} {\rm body}) \\
                    158: & & \mbox{ --- body を args を引数とする関数とする. } \\
                    159: \end{eqnarray*}
                    160: */
                    161: /*&eg
1.12      takayama  162: \subsection{Indeterminate and Tree}
1.1       noro      163:
                    164: \noindent
1.4       noro      165: Group CMObject/Tree requires CMObject/Basic. \\
1.12      takayama  166: Tree, Lambda $\in$ CMObject/Tree. \\
1.1       noro      167: \begin{eqnarray*}
                    168: \mbox{Tree}        &:& ({\tt CMO\_TREE}, {\sl Cstring}\, {\rm name},
1.11      takayama  169:  {\sl List}\, {\rm attributes}, {\sl List}\, {\rm leaves}) \\
1.12      takayama  170: & & \mbox{ --- ``name'' is the name of the node of the tree. } \\
                    171: & & \mbox{ --- Attributes may be a null list. If it is not null, it is a list of}\\
1.11      takayama  172: & & \mbox{ --- key and value pairs. } \\
1.1       noro      173: \mbox{Lambda}        &:& ({\tt CMO\_LAMBDA}, {\sl List}\, {\rm args},
                    174:                           {\sl Tree} {\rm body}) \\
                    175: & & \mbox{ --- a function with the arguments body. } \\
                    176: \end{eqnarray*}
                    177: */
                    178:
                    179: /*&jp
                    180: 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.
                    181: たとえば, $\sin(x+e)$ は,
                    182: {\tt (sin, (plus, x, e))}
                    183: なる Tree であらわすのが一般的である.
                    184: 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,
                    185: {\tt ox\_BEGIN\_BLOCK}, {\tt ox\_END\_BLOCK} で評価を抑制するのが
                    186: 一つの方法である (cf. Postscript の {\tt \{ }, {\tt \} }).
                    187: たとえば上の方法では
                    188: {\tt x, e, plus, sin } を begin block, end block でかこめばよろしい.
                    189: 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,
                    190: 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
                    191: ない.
                    192: したがって,
                    193: 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.
                    194: 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
                    195: はるかに容易である.
                    196: なお, Tree は, Open Math では, Symbol, Application のメカニズムに相当する.
                    197: */
                    198: /*&eg
                    199: In many computer algebra systems, mathematical expressions are usually
                    200: expressed in terms of a tree structure.
                    201: For example,
                    202: $\sin(x+e)$ is expressed as
                    203: {\tt (sin, (plus, x, e))}
                    204: as a tree.
1.4       noro      205: Tree may be expressed by putting the expression between
1.3       noro      206: {\tt SM\_beginBlock} and {\tt SM\_endBlock}, which are
                    207: stack machine commands for delayed evaluation.
                    208: (cf. {\tt \{ }, {\tt \} } in PostScript).
                    209: However it makes the implementation of stack machines complicated.
                    210: It is desirable that CMObject is independent of OX stack machine.
                    211: Therefore we introduce an OpenMath like tree representation for CMO
1.7       noro      212: Tree object.
1.3       noro      213: This method allows us to implement tree structure easily
                    214: on individual OpenXM systems.
                    215: Note that CMO Tree corresponds to Symbol and Application in OpenMath.
1.1       noro      216: */
                    217:
                    218:
                    219: /*&C
                    220:
                    221: */
                    222: /*&jp
                    223: Lambda は関数を定義するための関数である.
                    224: Lisp の Lambda 表現と同じ.
1.3       noro      225: */
                    226: /*&eg
                    227: Lambda is used to define functions.
1.12      takayama  228: The notion ``lambda'' is borrowed from the language Lisp.
1.3       noro      229: */
1.1       noro      230:
                    231: \noindent
1.3       noro      232: //&jp 例: $sin(x+e)$ の表現.
                    233: //&eg Example: the expression of $sin(x+e)$.
1.1       noro      234: \begin{verbatim}
1.11      takayama  235: (CMO_TREE, (CMO_STRING, "sin"),
                    236:     (CMO_LIST,[size=]1,(CMO_LIST,[size=]2,(CMO_STRING, "cdname"),
                    237:                                           (CMO_STRING,"basic")))
1.1       noro      238:     (CMO_LIST,[size=]1,
                    239:         (CMO_TREE, (CMO_STRING, "plus"), (CMO_STRING, "basic"),
                    240:             (CMO_LIST,[size=]2, (CMO_INDETERMINATE,"x"),
1.6       noro      241: //&jp                  (CMO_TREE,(CMO_STRING, "e"),  自然対数の底
                    242: //&eg                  (CMO_TREE,(CMO_STRING, "e"),  the base of natural logarithms
1.12      takayama  243:     (CMO_LIST,[size=]1,(CMO_LIST,[size=]2,(CMO_STRING, "cdname"),
                    244:                                           (CMO_STRING,"basic")))
1.1       noro      245:         ))
                    246:     )
                    247: )
                    248: \end{verbatim}
1.10      takayama  249: //&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.
                    250: //&eg  Elements of the leave may be any objects including polynomials.
1.1       noro      251:
                    252: \noindent
                    253: Example:
                    254: \begin{verbatim}
1.12      takayama  255: sm1> [(plus) [[(cdname) (basic)]] [(123).. (345)..]] [(class) (tree)] dc ::
                    256: Class.tree [$plus$ , [[$cdname$ , $basic$ ]], [ 123 , 345 ]  ]
1.1       noro      257: \end{verbatim}
                    258:
1.12      takayama  259: \noindent
                    260: Example:
                    261: \begin{verbatim}
                    262: asir
                    263: [753] taka_cmo100_xml_form(quote(sin(x+1)));
                    264: <cmo_tree>  <cmo_string>"sin"</cmo_string>
                    265:  <cmo_list><cmo_int32 for="length">1</cmo_int32>
                    266:    <cmo_list><cmo_int32 for="length">2</cmo_int32>
                    267:      <cmo_string>"cdname"</cmo_string>
                    268:      <cmo_string>"basic"</cmo_string>
                    269:    </cmo_list> </cmo_list>
                    270: <cmo_tree>    <cmo_string>"plus"</cmo_string>
                    271:   <cmo_list><cmo_int32 for="length">1</cmo_int32>
                    272:     <cmo_list><cmo_int32 for="length">2</cmo_int32>
                    273:       <cmo_string>"cdname"</cmo_string>
                    274:       <cmo_string>"basic"</cmo_string>
                    275:     </cmo_list> </cmo_list>
                    276:  <cmo_indeterminate> <cmo_string>"x"</cmo_string>  </cmo_indeterminate>
                    277:  <cmo_zz>1</cmo_zz>
                    278: </cmo_tree></cmo_tree>
                    279: \end{verbatim}
1.1       noro      280:
                    281:
                    282: \bigbreak
1.3       noro      283: //&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      284: /*&eg
1.12      takayama  285: Let us define a group for distributed polynomials. In the following,
1.4       noro      286: DMS stands for Distributed Monomial System.
                    287: */
1.1       noro      288:
                    289: \medbreak
                    290: \noindent
1.4       noro      291: Group CMObject/DistributedPolynomials requires CMObject/Primitive,
                    292: CMObject/Basic. \\
1.1       noro      293: Monomial, Monomial32, Coefficient, Dpolynomial, DringDefinition,
                    294: Generic DMS ring, RingByName, DMS of N variables $\in$
                    295: CMObject/DistributedPolynomials. \\
1.3       noro      296: /*&jp
1.1       noro      297: \begin{eqnarray*}
                    298: \mbox{Monomial} &:& \mbox{Monomial32}\, |\, \mbox{Zero} \\
                    299: \mbox{Monomial32}&:& ({\tt CMO\_MONOMIAL32}, {\sl int32}\, n,
                    300: {\sl int32}\, \mbox{e[1]}, \ldots,
                    301: {\sl int32}\, \mbox{e[n]}, \\
                    302: & & \ \mbox{Coefficient}) \\
                    303: & & \mbox{ --- e[i] で, $n$ 変数 monomial
                    304: $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$
                    305: をあらわす.} \\
                    306: \mbox{Coefficient}&:& \mbox{ZZ} | \mbox{Integer32} \\
                    307: \mbox{Dpolynomial}&:& \mbox{Zero} \\
1.12      takayama  308: & & |\ ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},{\sl int32}\, m, \\
1.1       noro      309: & & \ \ \mbox{DringDefinition},
                    310: [\mbox{Monomial32}|\mbox{Zero}], \\
                    311: & &\ \
                    312: \{\mbox{Monomial32}\}) \\
                    313: & &\mbox{--- m はモノミアルの個数である.}\\
                    314: \mbox{DringDefinition}
                    315: &:& \mbox{DMS of N variables} \\
                    316: & & |\ \mbox{RingByName} \\
                    317: & & |\ \mbox{Generic DMS ring} \\
                    318: & & \mbox{ --- 分散表現多項式環の定義. } \\
                    319: \mbox{Generic DMS ring}
1.2       noro      320: &:& \mbox{({\tt CMO\_DMS\_GENERIC}) --- 新版はこちら}\\
1.1       noro      321: \mbox{RingByName}&:& ({\tt CMO\_RING\_BY\_NAME}, {\sl Cstring}\  {\rm s}) \\
                    322: & & \mbox{ --- 名前 s で, 格納された ring 定義.} \\
                    323: \mbox{DMS of N variables}
                    324: &:& ({\tt CMO\_DMS\_OF\_N\_VARIABLES}, \\
                    325: & & \ ({\tt CMO\_LIST}, {\sl int32}\, \mbox{m},
                    326: {\sl Integer32}\,  \mbox{n}, {\sl Integer32}\,\mbox{p} \\
                    327: & & \ \ [,{\sl object}\,\mbox{s}, {\sl Cstring}\,\mbox{c},
                    328:           {\sl List}\, \mbox{vlist},
                    329: {\sl List}\, \mbox{wvec}, {\sl List}\, \mbox{outord}]) \\
                    330: & & \mbox{ --- m はあとに続く要素の数} \\
                    331: & & \mbox{ --- n は変数の数, p は 標数} \\
                    332: & & \mbox{ --- s は ring の名前} \\
                    333: & & \mbox{ --- c は係数環, QQ, ZZ の場合は文字列で QQ, ZZ と書く.} \\
                    334: & & \mbox{ --- vlist は Indeterminate のリスト(新版). 多項式環の変数リスト} \\
                    335: & & \mbox{ --- wvec は order をきめる weight vector,} \\
                    336: & & \mbox{ --- outord は出力するときの変数順序.} \\
                    337: \end{eqnarray*}
1.3       noro      338: */
                    339: /*&eg
                    340: \begin{eqnarray*}
                    341: \mbox{Monomial} &:& \mbox{Monomial32}\, |\, \mbox{Zero} \\
                    342: \mbox{Monomial32}&:& ({\tt CMO\_MONOMIAL32}, {\sl int32}\, n,
                    343:                       {\sl int32}\, \mbox{e[1]}, \ldots,
                    344:                       {\sl int32}\, \mbox{e[n]}, \\
                    345:                  & & \ \mbox{Coefficient}) \\
                    346:                  & & \mbox{ --- e[i] is the exponent $e_i$ of the monomial
                    347:                       $x^e = x_1^{e_1} \cdots x_n^{e_n}$. } \\
                    348: \mbox{Coefficient}&:& \mbox{ZZ} | \mbox{Integer32} \\
                    349: \mbox{Dpolynomial}&:& \mbox{Zero} \\
1.12      takayama  350:                  & & |\ ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},{\sl int32}\, m, \\
1.3       noro      351:                  & & \ \ \mbox{DringDefinition}, [\mbox{Monomial32}|\mbox{Zero}], \\
                    352:                  & &\ \
                    353:                     \{\mbox{Monomial32}\})  \\
                    354:                  & &\mbox{--- m is equal to the number of monomials.}\\
                    355: \mbox{DringDefinition}
                    356:                  &:& \mbox{DMS of N variables} \\
                    357:                  & & |\ \mbox{RingByName} \\
                    358:                  & & |\ \mbox{Generic DMS ring} \\
                    359:                  & & \mbox{ --- definition of the ring of distributed polynomials. } \\
                    360: \mbox{Generic DMS ring}
                    361:                  &:& ({\tt CMO\_DMS\_GENERIC}) \\
                    362: \mbox{RingByName}&:& ({\tt CMO\_RING\_BY\_NAME}, {\sl Cstring} s) \\
1.4       noro      363:                  & & \mbox{ --- The ring definition referred by the name ``s''.} \\
1.3       noro      364: \mbox{DMS of N variables}
                    365:                  &:& ({\tt CMO\_DMS\_OF\_N\_VARIABLES}, \\
                    366:                  & & \ ({\tt CMO\_LIST}, {\sl int32}\, \mbox{m},
                    367:                   {\sl Integer32}\,  \mbox{n}, {\sl Integer32}\, \mbox{p} \\
                    368:                  & & \ \ [,{\sl Cstring}\,\mbox{s}, {\sl List}\, \mbox{vlist},
                    369:                           {\sl List}\, \mbox{wvec}, {\sl List}\, \mbox{outord}]) \\
                    370:                  & & \mbox{ --- m is the number of elements.} \\
                    371:                  & & \mbox{ --- n is the number of variables, p is the characteristic} \\
                    372:                  & & \mbox{ --- s is the name of the ring, vlist is the list of variables.} \\
                    373:                  & & \mbox{ --- wvec is the weight vector.} \\
                    374:                  & & \mbox{ --- outord is the order of variables to output.} \\
                    375: \end{eqnarray*}
                    376: */
1.1       noro      377:
1.3       noro      378: /*&jp
1.1       noro      379: RingByName や DMS of N variables はなくても, DMS を定義できる.
                    380: 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
                    381: あってもかまわない.
                    382:
                    383: 以下, 以上の CMObject  にたいする,
                    384: xxx = asir, kan の振舞いを記述する.
1.3       noro      385: */
                    386: /*&eg
                    387: Note that it is possible to define DMS without RingByName and
                    388: DMS of N variables.
                    389:
                    390: In the following we describe how the above CMObjects
                    391: are implemented on Asir and Kan.
                    392: */
1.1       noro      393:
                    394: \subsection{ Zero}
1.3       noro      395: /*&jp
1.12      takayama  396: CMO では ゼロの表現法がなんとおりもあることに注意.
                    397: %% どのようなゼロをうけとっても,
                    398: %% システムのゼロに変換できるべきである.
1.3       noro      399: */
                    400: /*&eg
1.12      takayama  401: Note that CMO has various representations of zero.
1.3       noro      402: */
1.1       noro      403:
                    404:
1.3       noro      405: //&jp \subsection{ 整数 ZZ }
                    406: //&eg \subsection{ Integer ZZ }
1.1       noro      407:
                    408: \begin{verbatim}
                    409: #define     CMO_ZZ          20
                    410: \end{verbatim}
                    411:
1.3       noro      412: /*&jp
                    413: 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
                    414: 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
                    415: 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
                    416: いている.  (cf. {\tt kan/sm1} の配布ディレクトリのなかの {\tt
                    417: plugin/cmo-gmp.c}) CMO\_ZZ は次の形式をとる.
                    418: */
                    419: /*&eg
1.12      takayama  420: We describe the bignum (multi-precision integer) representation
                    421: {\tt CMO\_ZZ} in OpenXM.
                    422: The format is similar
1.3       noro      423: to that in GNU MP. (cf. {\tt plugin/cmo-gmp.c} in the {\tt kan/sm1}
                    424: distribution). CMO\_ZZ is defined as follows.
                    425: */
1.1       noro      426:
                    427: \begin{tabular}{|c|c|c|c|c|}
                    428: \hline
                    429: {\tt int32 CMO\_ZZ} & {\tt int32 $f$} & {\tt int32 $b_0$} & $\cdots$ &
                    430: {\tt int32 $b_{n}$} \\
                    431: \hline
1.3       noro      432: \end{tabular}
                    433:
                    434: /*&jp
                    435: $f$ は32bit整数である.  $b_0, \ldots, b_n$ は unsigned int32 である.
                    436: $|f|$ は $n+1$ である.  この CMO の符号は $f$ の符号で定める.  前述し
                    437: たように, 32bit整数の負数は 2 の補数表現で表される.
                    438:
                    439: Open xxx 規約では上の CMO は以下の整数を意味する. ($R = 2^{32}$)
                    440: */
                    441: /*&eg
                    442: $f$ is a 32bit integer. $b_0, \ldots, b_n$ are unsigned 32bit integers.
                    443: $|f|$ is equal to $n+1$.
1.12      takayama  444: The sign of $f$ represents that of the above integer to be expressed.
                    445: As stated in Section
1.3       noro      446: \ref{sec:basic0}, a negative 32bit integer is represented by
                    447: two's complement.
                    448:
                    449: In OpenXM the above CMO represents the following integer. ($R = 2^{32}$.)
                    450: */
1.1       noro      451:
                    452: \[
                    453: \mbox{sgn}(f)\times (b_0 R^{0}+ b_1 R^{1} + \cdots + b_{n-1}R^{n-1} + b_n R^n).
                    454: \]
1.3       noro      455:
                    456: /*&jp
1.12      takayama  457: \noindent  例:
1.3       noro      458: {\tt int32} を network byte order で表現
                    459: しているとすると,例えば, 整数 $14$ は CMO\_ZZ で表わすと,
                    460: */
                    461: /*&eg
1.12      takayama  462: \noindent Example:
1.3       noro      463: If we express {\tt int32} by the network byte order,
                    464: a CMO\_ZZ $14$ is expressed by
                    465: */
1.1       noro      466: \[
                    467: \mbox{(CMO\_ZZ, 1, 0, 0, 0, e)},
                    468: \]
1.3       noro      469: //&jp と表わす. これはバイト列では
1.4       noro      470: //&eg The corresponding byte sequence is
1.1       noro      471: \[
                    472: \mbox{\tt 00 00 00 14 00 00 00 01 00 00 00 0e}
                    473: \]
1.3       noro      474: //&jp となる.
1.1       noro      475:
                    476:
1.3       noro      477: //&jp なお ZZ の 0 ( (ZZ) 0 と書く ) は, {\tt (CMO\_ZZ, 00,00,00,00)} と表現する.
                    478: //&eg Note that CMO\_ZZ 0 is expressed by {\tt (CMO\_ZZ, 00,00,00,00)}.
1.1       noro      479:
                    480:
1.3       noro      481: //&jp \subsection{ 分散表現多項式 Dpolynomial }
                    482: //&eg \subsection{ Distributed polynomial Dpolynomial }
1.1       noro      483:
1.3       noro      484: /*&jp
1.1       noro      485: 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.
                    486:
                    487: Generic DMS ring に属する元は,
                    488: 変数を $n$ 個持つ 適当な係数集合 $K$ を持つ多項式環 $K[x_1, \ldots, x_n]$
                    489: の元である.
                    490: 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
                    491: わかる.
                    492: 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
                    493: サーバの対応する Object  に変換しないといけない.
                    494: この変換の仕方は, 各サーバ毎にきめる.
                    495:
                    496: 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.
                    497: \noroa{ でも, order はどうなるの? }
                    498:
                    499: {\tt kan/sm1} の場合は事情は複雑である.
                    500: {\tt kan/sm1} は, Generic DMS ring にあたる クラスをもたない.
                    501: 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.
                    502: したがって, {\tt kan/sm1} では, DMS of N variables が来た場合,
                    503: これを CurrentRing の元として読み込む.  CurrentRing の変数の数が $n'$
                    504: 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.
                    505: Order その他の optional 情報はすべて無視する.
                    506:
                    507: DMS の 2 番目のフィールドで,
                    508: 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
                    509: 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.
                    510: {\tt kan/sm1} の場合, 環の定義は ring object として格納されており,
                    511: 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
                    512: 多項式をこの ring の元として格納できる.
1.3       noro      513: */
1.1       noro      514:
1.3       noro      515: /*&eg
                    516: We treat polynomial rings and their elements as follows.
1.1       noro      517:
1.4       noro      518: Generic DMS ring is an $n$-variate polynomial ring $K[x_1, \ldots, x_n]$,
1.12      takayama  519: where $K$ is a coefficient set. $K$ is unknown in advance
1.4       noro      520: and it is determined when coefficients of an element are received.
                    521: When a server has received an element in Generic DMS ring,
1.3       noro      522: the server has to translate it into the corresponding local object
                    523: on the server. Each server has its own translation scheme.
                    524: In Asir such an element are translated into a distributed polynomial.
                    525: In {\tt kan/sm1} things are complicated.
1.4       noro      526: {\tt kan/sm1} does not have any class corresponding to Generic DMS ring.
1.3       noro      527: {\tt kan/sm1} translates a DMS of N variables into an element of
                    528: the CurrentRing.
                    529: If the CurrentRing is $n'$-variate and $n' < n$, then
1.12      takayama  530: an $n$-variate polynomial ring is newly created.
                    531:
1.3       noro      532:
1.4       noro      533: If RingByName ({\tt CMO\_RING\_BY\_NAME}, yyy)
1.3       noro      534: is specified as the second field of DMS,
                    535: it requests a sever to use a ring object whose name is yyy
                    536: as the destination ring for the translation.
                    537: */
1.1       noro      538:
                    539: \medbreak \noindent
1.3       noro      540: //&jp {\bf Example}: (すべての数の表記は 16 進表記)
                    541: //&eg {\bf Example}: (all numbers are represented in hexadecimal notation)
1.1       noro      542: {\footnotesize \begin{verbatim}
                    543: Z/11Z [6 variables]
                    544: (kxx/cmotest.sm1) run
                    545: [(x,y) ring_of_polynomials ( ) elimination_order 11 ] define_ring ;
                    546: (3x^2 y). cmo /ff set ;
                    547: [(cmoLispLike) 1] extension ;
                    548: ff ::
                    549: Class.CMO CMO StandardEncoding: size = 52, size/sizeof(int) = 13,
                    550: tag=CMO_DISTRIBUTED_POLYNOMIAL
                    551:
                    552:   0  0  0 1f  0  0  0  1  0  0  0 18  0  0  0 13  0  0  0  6
                    553:   0  0  0  0  0  0  0  2  0  0  0  0  0  0  0  0  0  0  0  1
                    554:   0  0  0  0  0  0  0  2  0  0  0  3
                    555:
                    556: ff omc ::
                    557:  (CMO_DISTRIBUTED_POLYNOMIAL[1f],[size=]1,(CMO_DMS_GENERIC[18],),
                    558:   (CMO_MONOMIAL32[13],3*x^2*y),),
                    559: \end{verbatim} }
1.3       noro      560: /*&jp
                    561: $ 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.
                    562: */
                    563: /*&eg
                    564: $3 x^2 y$ is regarded as an element of a six-variate polynomial ring.
                    565: */
1.1       noro      566:
                    567:
1.3       noro      568: //&jp \subsection{再帰表現多項式の定義}
                    569: //&eg \subsection{Recursive polynomials}
1.1       noro      570:
                    571: \begin{verbatim}
                    572: #define CMO_RECURSIVE_POLYNOMIAL        27
                    573: #define CMO_POLYNOMIAL_IN_ONE_VARIABLE  33
                    574: \end{verbatim}
                    575:
1.4       noro      576: Group CMObject/RecursivePolynomial requires CMObject/Primitive, CMObject/Basic.\\
1.1       noro      577: Polynomial in 1 variable, Coefficient, Name of the main variable,
                    578: Recursive Polynomial, Ring definition for recursive polynomials
                    579: $\in$ CMObject/RecursivePolynomial \\
                    580:
1.3       noro      581: /*&jp
1.1       noro      582: \begin{eqnarray*}
                    583: \mbox{Polynomial in 1 variable} &:&
                    584: \mbox{({\tt CMO\_POLYNOMIAL\_IN\_ONE\_VARIABLE},\, {\sl int32}\, m, } \\
                    585: & & \quad \mbox{ Name of the main variable }, \\
1.12      takayama  586: & & \quad \mbox{ \{ {\sl int32} e, Coefficient \}} ) \\
1.1       noro      587: & & \mbox{ --- m はモノミアルの個数. } \\
                    588: & & \mbox{ --- e, Coefficieint はモノミアルを表現している. } \\
                    589: & & \mbox{ --- 順序の高い順にならべる. 普通は巾の高い順.} \\
                    590: & & \mbox{ ---  e は 1変数多項式の巾をあらわす. } \\
                    591: \mbox{Coefficient} &:& \mbox{ ZZ} \,|\, \mbox{ QQ } \,|\,
                    592: \mbox{ integer32  } \,|\,
                    593: \mbox{ Polynomial in 1 variable } \\
                    594: & & \quad \,|\, \mbox{Tree} \,|\, \mbox{Zero} \,|\,\mbox{Dpolynomial}\\
                    595: \mbox{Name of the main variable } &:&
                    596: \mbox{ {\sl int32} v }   \\
                    597: & & \mbox{ --- v は 変数番号 (0 からはじまる) を表す. } \\
                    598: \mbox{Recursive Polynomial} &:&
                    599: \mbox{ ( {\tt CMO\_RECURSIVE\_POLYNOMIAL}, } \\
1.3       noro      600: & & \quad \mbox{ RringDefinition, } \\
1.1       noro      601: & & \quad
1.12      takayama  602: \mbox{ Polynomial in 1 variable}\, | \, \mbox{Coefficient} )  \\
1.3       noro      603: \mbox{RringDefinition}
1.1       noro      604: & : &  \mbox{ {\sl List} v } \\
1.10      takayama  605: & & \quad \mbox{ --- v は, 変数名(indeterminate) または Tree のリスト. } \\
1.1       noro      606: & & \quad \mbox{ --- 順序の高い順. } \\
                    607: \end{eqnarray*}
1.3       noro      608: */
                    609: /*&eg
                    610: \begin{eqnarray*}
                    611: \mbox{Polynomial in 1 variable} &:&
                    612: \mbox{({\tt CMO\_POLYNOMIAL\_IN\_ONE\_VARIABLE},\, {\sl int32}\, m, } \\
                    613: & & \quad \mbox{ Name of the main variable }, \\
1.12      takayama  614: & & \quad \mbox{ \{ {\sl int32} e, Coefficient \}} ) \\
1.4       noro      615: & & \mbox{ --- m is the number of monomials. } \\
                    616: & & \mbox{ --- A pair of e and Coefficient represents a monomial. } \\
1.3       noro      617: & & \mbox{ --- The pairs of e and Coefficient are sorted in the } \\
                    618: & & \mbox{ \quad decreasing order, usually with respect to e.} \\
                    619: & & \mbox{ ---  e denotes an exponent of a monomial with respect to } \\
                    620: & & \mbox{ \quad the main variable. } \\
                    621: \mbox{Coefficient} &:& \mbox{ ZZ} \,|\, \mbox{ QQ } \,|\,
                    622: \mbox{ integer32  } \,|\,
                    623: \mbox{ Polynomial in 1 variable } \\
                    624: & & \quad \,|\, \mbox{Tree} \,|\, \mbox{Zero} \,|\,\mbox{Dpolynomial}\\
                    625: \mbox{Name of the main variable } &:&
                    626: \mbox{ {\sl int32} v }   \\
                    627: & & \mbox{ --- v denotes a variable number. } \\
                    628: \mbox{Recursive Polynomial} &:&
                    629: \mbox{ ( {\tt CMO\_RECURSIVE\_POLYNOMIAL}, } \\
                    630: & & \quad \mbox{ RringDefinition, } \\
                    631: & & \quad
1.12      takayama  632: \mbox{ Polynomial in 1 variable}\, | \, \mbox{Coefficient} )  \\
1.3       noro      633: \mbox{RringDefinition}
                    634: & : &  \mbox{ {\sl List} v } \\
1.10      takayama  635: & & \quad \mbox{ --- v is a list of names of indeterminates or trees. } \\
1.3       noro      636: & & \quad \mbox{ --- It is sorted in the decreasing order. } \\
                    637: \end{eqnarray*}
                    638: */
1.1       noro      639: \bigbreak
                    640: \noindent
1.3       noro      641: Example:
1.1       noro      642: \begin{verbatim}
                    643: (CMO_RECURSIEVE_POLYNOMIAL, ("x","y"),
                    644: (CMO_POLYNOMIAL_IN_ONE_VARIABLE, 2,      0,  <--- "x"
                    645:   3, (CMO_POLYNOMIAL_IN_ONE_VARIABLE, 2, 1,  <--- "y"
                    646:        5, 1234,
                    647:        0, 17),
                    648:   1, (CMO_POLYNOMIAL_IN_ONE_VARIABLE, 2, 1,  <--- "y"
                    649:        10, 1,
                    650:        5, 31)))
                    651: \end{verbatim}
1.3       noro      652: //&jp これは,
                    653: //&eg This represents
                    654: $$   x^3 (1234 y^5 + 17 ) +  x^1 (y^{10} + 31 y^5)  $$
                    655: /*&jp
1.1       noro      656: をあらわす.
1.12      takayama  657: %%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
                    658: %%すること. つまり, 主変数かける係数の順番.
1.3       noro      659: */
                    660: /*&eg
1.12      takayama  661: %%We intend to represent non-commutative polynomials with the
                    662: %%same form. In such a case, the order of products are defined
                    663: %%as above, that is a power of the main variable $\times$ a coeffcient.
                    664:
1.3       noro      665: */
1.1       noro      666:
                    667: \noindent
                    668: \begin{verbatim}
                    669: sm1
                    670: sm1>(x^2-h). [(class) (recursivePolynomial)] dc /ff set ;
                    671: sm1>ff ::
                    672: Class.recursivePolynomial h * ((-1)) + (x^2  * (1))
                    673: \end{verbatim}
                    674:
1.3       noro      675: //&jp \subsection{CPU依存の double }
                    676: //&eg \subsection{CPU dependent double}
1.1       noro      677:
                    678: \begin{verbatim}
                    679: #define CMO_64BIT_MACHINE_DOUBLE   40
                    680: #define CMO_ARRAY_OF_64BIT_MACHINE_DOUBLE  41
                    681: #define CMO_128BIT_MACHINE_DOUBLE   42
                    682: #define CMO_ARRAY_OF_128BIT_MACHINE_DOUBLE  43
                    683: \end{verbatim}
                    684:
                    685: \noindent
1.4       noro      686: Group CMObject/MachineDouble requires CMObject/Primitive.\\
1.1       noro      687: 64bit machine double, Array of 64bit machine double
                    688: 128bit machine double, Array of 128bit machine double
                    689: $\in$ CMObject/MachineDouble \\
                    690:
1.3       noro      691: /*&jp
1.1       noro      692: \begin{eqnarray*}
                    693: \mbox{64bit machine double} &:&
                    694: \mbox{({\tt CMO\_64BIT\_MACHINE\_DOUBLE}, } \\
1.12      takayama  695: & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte} s8})\\
1.1       noro      696: & & \mbox{ --- s1, $\ldots$, s8 は {\tt double} (64bit). } \\
                    697: & & \mbox{ --- この表現はCPU依存である.}\\
1.13      takayama  698: &&  \mbox{\quad\quad byte order negotiation を用いる.} \\
1.1       noro      699: \mbox{Array of 64bit machine double} &:&
                    700: \mbox{({\tt CMO\_ARRAY\_OF\_64BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\
                    701: & & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte}}\, s8[1], \ldots , {\sl byte}\, s8[m])\\
                    702: & & \mbox{ --- s*[1], $\ldots$ s*[m] は m 個の double (64bit) である. } \\
                    703: & & \mbox{ --- この表現はCPU依存である.}\\
1.13      takayama  704: & & \mbox{ \quad\quad byte order negotiation を用いる.} \\
1.1       noro      705: \mbox{128bit machine double} &:&
                    706: \mbox{({\tt CMO\_128BIT\_MACHINE\_DOUBLE}, } \\
1.12      takayama  707: & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte} s16})\\
1.1       noro      708: & & \mbox{ --- s1, $\ldots$, s16 は {\tt long double} (128bit). } \\
                    709: & & \mbox{ --- この表現はCPU依存である.}\\
1.13      takayama  710: &&  \mbox{\quad\quad byte order negotiation を用いる.} \\
1.1       noro      711: \mbox{Array of 128bit machine double} &:&
                    712: \mbox{({\tt CMO\_ARRAY\_OF\_128BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\
1.12      takayama  713: & & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte} s16[1], \ldots , {\sl byte} s16[m]})\\
1.1       noro      714: & & \mbox{ --- s*[1], $\ldots$ s*[m] は m 個の long double (128bit) である. } \\
                    715: & & \mbox{ --- この表現はCPU依存である.}\\
1.13      takayama  716: & & \mbox{ \quad\quad byte order negotiation を用いる.}
1.1       noro      717: \end{eqnarray*}
1.3       noro      718: */
                    719: /*&eg
                    720: \begin{eqnarray*}
                    721: \mbox{64bit machine double} &:&
                    722: \mbox{({\tt CMO\_64BIT\_MACHINE\_DOUBLE}, } \\
1.12      takayama  723: & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte} s8})\\
1.3       noro      724: & & \mbox{ --- s1, $\ldots$, s8 は {\tt double} (64bit). } \\
1.12      takayama  725: & & \mbox{ --- Encoding depends on CPU.}\\
1.13      takayama  726: &&  \mbox{\quad\quad Need the byte order negotiation.} \\
1.3       noro      727: \mbox{Array of 64bit machine double} &:&
                    728: \mbox{({\tt CMO\_ARRAY\_OF\_64BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\
                    729: & & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte}}\, s8[1], \ldots , {\sl byte}\, s8[m])\\
                    730: & & \mbox{ --- s*[1], $\ldots$ s*[m] are 64bit double's. } \\
1.12      takayama  731: & & \mbox{ --- Encoding depends on CPU.}\\
1.13      takayama  732: & & \mbox{\quad\quad Need the byte order negotiation.} \\
1.3       noro      733: \mbox{128bit machine double} &:&
                    734: \mbox{({\tt CMO\_128BIT\_MACHINE\_DOUBLE}, } \\
1.12      takayama  735: & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte} s16})\\
1.3       noro      736: & & \mbox{ --- s1, $\ldots$, s16 は {\tt long double} (128bit). } \\
1.12      takayama  737: & & \mbox{ --- Encoding depends on CPU.}\\
1.13      takayama  738: & & \mbox{\quad\quad Need the byte order negotiation.} \\
1.3       noro      739: \mbox{Array of 128bit machine double} &:&
                    740: \mbox{({\tt CMO\_ARRAY\_OF\_128BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\
1.12      takayama  741: & & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte} s16[1], \ldots , {\sl byte} s16[m]})\\
1.3       noro      742: & & \mbox{ --- s*[1], $\ldots$ s*[m] are 128bit long double's. } \\
1.12      takayama  743: & & \mbox{ --- Encoding depends on CPU.}\\
1.13      takayama  744: & & \mbox{\quad\quad Need the byte order negotiation.} \\
1.3       noro      745: \end{eqnarray*}
                    746: */
1.1       noro      747:
                    748: \bigbreak
1.12      takayama  749:
1.1       noro      750: \begin{verbatim}
                    751: #define CMO_IEEE_DOUBLE_FLOAT 51
                    752: \end{verbatim}
                    753:
1.3       noro      754: /*&jp
                    755: IEEE 準拠の float については, IEEE 754 double precision floating-point
                    756: format (64 bit) の定義を見よ.
1.13      takayama  757:
                    758: 256.100006 の Intel Pentium の double64 での内部表現は
                    759: {\tt cd 0c 80 43 } \\
                    760: 256.100006 の PowerPC (Mac) の double64 での内部表現は
                    761: {\tt 43 80 0c cd }.
                    762: この例でみるように byte の順序が逆である.
                    763: エンジンスタートの時の byte order negotiation で byte の順序を指定する.
                    764:
                    765:
1.3       noro      766: */
                    767: /*&eg
1.5       noro      768: See IEEE 754 double precision floating-point (64 bit) for the details of
1.12      takayama  769: float compliant to the IEEE standard.
1.13      takayama  770:
                    771: The internal expression of 256.100006 in the Intel Pentium  is
                    772: {\tt cd 0c 80 43 } \\
                    773: The internal expression of 256.100006 in the PowerPC (Mac) is
                    774: {\tt 43 80 0c cd }.
                    775: As you have seen in this example,
                    776: the orders of the bytes are opposite each other.
                    777: The byte order is specified by the byte order negotiation protocol
                    778: when the engine starts.
                    779:
1.3       noro      780: */
1.1       noro      781:
1.15    ! takayama  782: \subsection{Bigfloat32}
        !           783: /*&jp
        !           784: int32 を基本とした bigfloat の表現方法について述べる.
        !           785: この形式は mpfr を 32bit CPU で使用した時の内部表現と共通である.
        !           786: */
        !           787: /*&eg
        !           788: This subsection describes our format for bigfloat in terms of the int32.
        !           789: This format is identical to the internal format of mpfr on 32 bit CPU's.
        !           790: */
        !           791: Ref: {\tt OpenXM/src/mpfr/bfsize/bfsize.c}
        !           792:
        !           793: \begin{verbatim}
        !           794: #define     CMO_BIGFLOAT32          52
        !           795: \end{verbatim}
        !           796:
        !           797:
        !           798: /*&jp
        !           799: Bigfloat32 は次の形式の int32 の配列である.
        !           800: */
        !           801: /*&eg
        !           802: The bigfloat32 is an array of int32 numbers of the following format.
        !           803: */
        !           804:
        !           805: \begin{center}
        !           806: {\sl int32}\, {\rm prec}, {\sl int32}\, {\rm sign}, {\sl int32}\, {\rm exp},
        !           807: {\sl int32}\, \mbox{a[1]}, \ldots , {\sl int32}\, \mbox{a[k]}
        !           808: \end{center}
        !           809:
        !           810: /*&jp
        !           811: p=prec は精度, s=sign は符号(1 が正の数,  -1 (2の補数表現)が負の数) ,
        !           812: E=exp は指数部で, 上のデータは数
        !           813: $$ s (a[k]/B + a[k-1]/B^2 + ... + a[1]/B^k) 2^E  $$
        !           814: を表す.
        !           815: ここで $B=2^{32}$,
        !           816: $k = \lceil p/32 \rceil$
        !           817: である.
        !           818: */
        !           819: /*&eg
        !           820: p=prec is the precision, s=sign is the sign(1 means positive,  -1 (expressed by two's complement) is nevative),
        !           821: E=exp is the exponent, and the data above expresses the number
        !           822: $$ s (a[k]/B + a[k-1]/B^2 + ... + a[1]/B^k) 2^E.  $$
        !           823: Here, $B=2^{32}$,
        !           824: $k = \lceil p/32 \rceil$.
        !           825: */
1.1       noro      826: