version 1.9, 2000/09/12 23:09:18 |
version 1.15, 2016/08/22 05:38:27 |
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%% $OpenXM: OpenXM/doc/OpenXM-specs/cmo-basic1.tex,v 1.8 2000/02/28 14:10:27 takayama Exp $ |
%% $OpenXM: OpenXM/doc/OpenXM-specs/cmo-basic1.tex,v 1.14 2015/08/18 02:54:05 noro Exp $ |
//&jp \section{ 数, 多項式 の CMO 表現 } |
//&jp \section{ 数, 多項式 の CMO 表現 } |
//&eg \section{ CMOexpressions for numbers and polynomials } |
//&eg \section{ CMOexpressions for numbers and polynomials } |
\label{sec:basic1} |
\label{sec:basic1} |
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#define CMO_RING_BY_NAME 26 |
#define CMO_RING_BY_NAME 26 |
#define CMO_DISTRIBUTED_POLYNOMIAL 31 |
#define CMO_DISTRIBUTED_POLYNOMIAL 31 |
#define CMO_RATIONAL 34 |
#define CMO_RATIONAL 34 |
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#define CMO_COMPLEX 35 |
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#define CMO_BIGFLOAT32 52 |
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#define CMO_INDETERMINATE 60 |
#define CMO_INDETERMINATE 60 |
#define CMO_TREE 61 |
#define CMO_TREE 61 |
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\bigbreak |
\bigbreak |
\noindent |
\noindent |
Group CMObject/Basic requires CMObject/Primitive. \\ |
Group CMObject/Basic requires CMObject/Primitive. \\ |
ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/Basic. \\ |
ZZ, QQ, Zero, Rational, Indeterminate $\in$ CMObject/Basic. \\ |
\begin{eqnarray*} |
\begin{eqnarray*} |
\mbox{Zero} &:& ({\tt CMO\_ZERO}) \\ |
\mbox{Zero} &:& ({\tt CMO\_ZERO}) \\ |
& & \mbox{ --- ユニバーサルな ゼロを表す. } \\ |
& & \mbox{ --- ユニバーサルな ゼロを表す. } \\ |
\mbox{ZZ} &:& ({\tt CMO\_ZZ},{\sl int32}\, {\rm f}, {\sl byte}\, \mbox{a[1]}, \ldots |
\mbox{ZZ} &:& ({\tt CMO\_ZZ},{\sl int32}\, {\rm f}, {\sl byte}\, \mbox{a[1]}, \ldots , |
{\sl byte}\, \mbox{a[$|$f$|$]} ) \\ |
{\sl byte}\, \mbox{a[$|$f$|$]} ) \\ |
&:& \mbox{ --- bignum をあらわす. a[i] についてはあとで説明}\\ |
&:& \mbox{ --- bignum をあらわす. a[i] についてはあとで説明}\\ |
\mbox{QQ} &:& ({\tt CMO\_QQ}, |
\mbox{QQ} &:& ({\tt CMO\_QQ}, |
Line 72 ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B |
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Line 74 ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B |
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& & \mbox{ --- 有理数 $a/b$ を表す. } \\ |
& & \mbox{ --- 有理数 $a/b$ を表す. } \\ |
\mbox{Rational} &:& ({\tt CMO\_RATIONAL}, {\sl CMObject}\, {\rm a}, {\sl CMObject}\, {\rm b}) \\ |
\mbox{Rational} &:& ({\tt CMO\_RATIONAL}, {\sl CMObject}\, {\rm a}, {\sl CMObject}\, {\rm b}) \\ |
& & \mbox{ --- $a/b$ を表す. } \\ |
& & \mbox{ --- $a/b$ を表す. } \\ |
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\mbox{Bigfloat32} &:& ({\tt CMO\_BIGFLOAT32}, |
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{\sl int32}\, {\rm prec}, {\sl int32}\, {\rm sign}, {\sl int32}\, {\rm exp}, |
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{\sl int32}\, \mbox{a[1]}, \ldots , {\sl int32}\, \mbox{a[k]} ) \\ |
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&:& \mbox{ --- bigfloat をあらわす. a[i], k についてはあとで説明}\\ |
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\mbox{Complex} &:& ({\tt CMO\_COMPLEX}, {\sl CMObject}\, {\rm re}, {\sl CMObject}\, {\rm im}) \\ |
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& & \mbox{ --- $a+b\sqrt{-1}$ を表す. } \\ |
\mbox{Indeterminate} &:& ({\tt CMO\_INDETERMINATE}, {\sl Cstring}\, {\rm v}) \\ |
\mbox{Indeterminate} &:& ({\tt CMO\_INDETERMINATE}, {\sl Cstring}\, {\rm v}) \\ |
& & \mbox{ --- 変数名 $v$ . } \\ |
& & \mbox{ --- 変数名 $v$ . } \\ |
\end{eqnarray*} |
\end{eqnarray*} |
*/ |
*/ |
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/*&eg |
/*&eg |
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\bigbreak |
\bigbreak |
\noindent |
\noindent |
Group CMObject/Basic requires CMObject/Primitive. \\ |
Group CMObject/Basic requires CMObject/Primitive. \\ |
ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/Basic. \\ |
ZZ, QQ, Zero, Rational, Indeterminate $\in$ CMObject/Basic. \\ |
\begin{eqnarray*} |
\begin{eqnarray*} |
\mbox{Zero} &:& ({\tt CMO\_ZERO}) \\ |
\mbox{Zero} &:& ({\tt CMO\_ZERO}) \\ |
& & \mbox{ --- Universal zero } \\ |
& & \mbox{ --- Universal zero } \\ |
\mbox{ZZ} &:& ({\tt CMO\_ZZ},{\sl int32}\, {\rm f}, {\sl byte}\, \mbox{a[1]}, \ldots |
\mbox{ZZ} &:& ({\tt CMO\_ZZ},{\sl int32}\, {\rm f}, {\sl byte}\, \mbox{a[1]}, \ldots , |
{\sl byte}\, \mbox{a[$|$m$|$]} ) \\ |
{\sl byte}\, \mbox{a[$|$m$|$]} ) \\ |
&:& \mbox{ --- bignum. The meaning of a[i] will be explained later.}\\ |
&:& \mbox{ --- bignum. The meaning of a[i] will be explained later.}\\ |
\mbox{QQ} &:& ({\tt CMO\_QQ}, |
\mbox{QQ} &:& ({\tt CMO\_QQ}, |
Line 94 ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B |
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Line 104 ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B |
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& & \mbox{ --- Rational number $a/b$. } \\ |
& & \mbox{ --- Rational number $a/b$. } \\ |
\mbox{Rational} &:& ({\tt CMO\_RATIONAL}, {\sl CMObject}\, {\rm a}, {\sl CMObject}\, {\rm b}) \\ |
\mbox{Rational} &:& ({\tt CMO\_RATIONAL}, {\sl CMObject}\, {\rm a}, {\sl CMObject}\, {\rm b}) \\ |
& & \mbox{ --- Rational expression $a/b$. } \\ |
& & \mbox{ --- Rational expression $a/b$. } \\ |
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\mbox{Bigfloat32} &:& ({\tt CMO\_BIGFLOAT32}, |
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{\sl int32}\, {\rm prec}, {\sl int32}\, {\rm sign}, {\sl int32}\, {\rm exp}, |
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{\sl int32}\, \mbox{a[1]}, \ldots , {\sl int32}\, \mbox{a[k]} ) \\ |
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&:& \mbox{ --- bigfloat. The meaning of a[i], k will be explained later.}\\ |
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\mbox{Complex} &:& ({\tt CMO\_COMPLEX}, {\sl CMObject}\, {\rm re}, {\sl CMObject}\, {\rm im}) \\ |
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& & \mbox{ --- Complex number $a+b\sqrt{-1}$. } \\ |
\mbox{Indeterminate} &:& ({\tt CMO\_INDETERMINATE}, {\sl Cstring}\, {\rm v}) \\ |
\mbox{Indeterminate} &:& ({\tt CMO\_INDETERMINATE}, {\sl Cstring}\, {\rm v}) \\ |
& & \mbox{ --- Variable name $v$ . } \\ |
& & \mbox{ --- Variable name $v$ . } \\ |
\end{eqnarray*} |
\end{eqnarray*} |
Line 101 ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B |
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Line 117 ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B |
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/*&C |
/*&C |
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*/ |
*/ |
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/*&C |
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*/ |
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/*&jp |
/*&jp |
Indeterminate は変数名をあらわす. |
Indeterminate は変数名をあらわす. |
v はバイト列であればなにを用いてもよいが, |
v はバイト列であればなにを用いてもよいが, |
Line 114 escape sequence を用いて実現するのは, 無理があるようで |
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Line 133 escape sequence を用いて実現するのは, 無理があるようで |
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テーブルを作成する必要があるであろう.) |
テーブルを作成する必要があるであろう.) |
*/ |
*/ |
/*&eg |
/*&eg |
Indeterminate is a name of a variable. |
The name of a variable should be expressed by using Indeterminate. |
v may be any sequence of bytes, but each system has its own |
v may be any sequence of bytes, but each system has its own |
restrictions on the names of variables. |
restrictions on the names of variables. |
Indeterminates of CMO and internal variable names must be translated |
Indeterminates of CMO and internal variable names must be translated |
in one to one correspondence. |
in one-to-one correspondence. |
*/ |
*/ |
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/*&jp |
/*&jp |
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\subsection{Indeterminate および Tree} |
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\noindent |
\noindent |
Group CMObject/Tree requires CMObject/Basic. \\ |
Group CMObject/Tree requires CMObject/Basic. \\ |
Tree, Lambda $\in$ CMObject/Basic. \\ |
Tree, Lambda $\in$ CMObject/Tree. \\ |
\begin{eqnarray*} |
\begin{eqnarray*} |
\mbox{Tree} &:& ({\tt CMO\_TREE}, {\sl Cstring}\, {\rm name}, |
\mbox{Tree} &:& ({\tt CMO\_TREE}, {\sl Cstring}\, {\rm name}, |
{\sl Cstring}\, {\rm cdname}, {\sl List}\, {\rm leaves}) \\ |
{\sl List}\, {\rm attributes}, {\sl List}\, {\rm leaves}) \\ |
& & \mbox{ --- 名前 name の定数または関数. 関数の評価はおこなわない. } \\ |
& & \mbox{ --- 名前 name の定数または関数. 関数の評価はおこなわない. } \\ |
& & \mbox{ --- cdname は空文字列でなければ name の意味が説明されている }\\ |
& & \mbox{ --- attributes は空リストでなければ name の属性を保持している. }\\ |
& & \mbox{ --- OpenMath CD (content dictionary) の名前. } \\ |
& & \mbox{ --- 属性リストは, key と 値のペアである. }\\ |
\mbox{Lambda} &:& ({\tt CMO\_LAMBDA}, {\sl List}\, {\rm args}, |
\mbox{Lambda} &:& ({\tt CMO\_LAMBDA}, {\sl List}\, {\rm args}, |
{\sl Tree} {\rm body}) \\ |
{\sl Tree} {\rm body}) \\ |
& & \mbox{ --- body を args を引数とする関数とする. } \\ |
& & \mbox{ --- body を args を引数とする関数とする. } \\ |
& & \mbox{ --- optional な引数が必要なときは, leaves の後へつづける.} \\ |
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\end{eqnarray*} |
\end{eqnarray*} |
*/ |
*/ |
/*&eg |
/*&eg |
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\subsection{Indeterminate and Tree} |
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\noindent |
\noindent |
Group CMObject/Tree requires CMObject/Basic. \\ |
Group CMObject/Tree requires CMObject/Basic. \\ |
Tree, Lambda $\in$ CMObject/Basic. \\ |
Tree, Lambda $\in$ CMObject/Tree. \\ |
\begin{eqnarray*} |
\begin{eqnarray*} |
\mbox{Tree} &:& ({\tt CMO\_TREE}, {\sl Cstring}\, {\rm name}, |
\mbox{Tree} &:& ({\tt CMO\_TREE}, {\sl Cstring}\, {\rm name}, |
{\sl Cstring}\, {\rm cdname}, {\sl List}\, {\rm leaves}) \\ |
{\sl List}\, {\rm attributes}, {\sl List}\, {\rm leaves}) \\ |
& & \mbox{ --- A function or a constant of name. Functions are not evaluated. } \\ |
& & \mbox{ --- ``name'' is the name of the node of the tree. } \\ |
& & \mbox{ --- cdname may be a null. If it is not null, it is the name of}\\ |
& & \mbox{ --- Attributes may be a null list. If it is not null, it is a list of}\\ |
& & \mbox{ --- the OpenMath CD (content dictionary). } \\ |
& & \mbox{ --- key and value pairs. } \\ |
\mbox{Lambda} &:& ({\tt CMO\_LAMBDA}, {\sl List}\, {\rm args}, |
\mbox{Lambda} &:& ({\tt CMO\_LAMBDA}, {\sl List}\, {\rm args}, |
{\sl Tree} {\rm body}) \\ |
{\sl Tree} {\rm body}) \\ |
& & \mbox{ --- a function with the arguments body. } \\ |
& & \mbox{ --- a function with the arguments body. } \\ |
& & \mbox{ --- optional arguments come after leaves.} \\ |
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\end{eqnarray*} |
\end{eqnarray*} |
*/ |
*/ |
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Line 205 Lisp の Lambda 表現と同じ. |
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Line 225 Lisp の Lambda 表現と同じ. |
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*/ |
*/ |
/*&eg |
/*&eg |
Lambda is used to define functions. |
Lambda is used to define functions. |
It is the same as the Lambda expression in Lisp. |
The notion ``lambda'' is borrowed from the language Lisp. |
*/ |
*/ |
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\noindent |
\noindent |
//&jp 例: $sin(x+e)$ の表現. |
//&jp 例: $sin(x+e)$ の表現. |
//&eg Example: the expression of $sin(x+e)$. |
//&eg Example: the expression of $sin(x+e)$. |
\begin{verbatim} |
\begin{verbatim} |
(CMO_TREE, (CMO_STRING, "sin"), (CMO_STRING, "basic"), |
(CMO_TREE, (CMO_STRING, "sin"), |
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(CMO_LIST,[size=]1,(CMO_LIST,[size=]2,(CMO_STRING, "cdname"), |
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(CMO_STRING,"basic"))) |
(CMO_LIST,[size=]1, |
(CMO_LIST,[size=]1, |
(CMO_TREE, (CMO_STRING, "plus"), (CMO_STRING, "basic"), |
(CMO_TREE, (CMO_STRING, "plus"), (CMO_STRING, "basic"), |
(CMO_LIST,[size=]2, (CMO_INDETERMINATE,"x"), |
(CMO_LIST,[size=]2, (CMO_INDETERMINATE,"x"), |
//&jp (CMO_TREE,(CMO_STRING, "e"), 自然対数の底 |
//&jp (CMO_TREE,(CMO_STRING, "e"), 自然対数の底 |
//&eg (CMO_TREE,(CMO_STRING, "e"), the base of natural logarithms |
//&eg (CMO_TREE,(CMO_STRING, "e"), the base of natural logarithms |
(CMO_STRING, "basic")) |
(CMO_LIST,[size=]1,(CMO_LIST,[size=]2,(CMO_STRING, "cdname"), |
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(CMO_STRING,"basic"))) |
)) |
)) |
) |
) |
) |
) |
\end{verbatim} |
\end{verbatim} |
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//&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. |
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//&eg Elements of the leave may be any objects including polynomials. |
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\noindent |
\noindent |
Example: |
Example: |
\begin{verbatim} |
\begin{verbatim} |
sm1> [(plus) (Basic) [(123).. (345)..]] [(class) (tree)] dc :: |
sm1> [(plus) [[(cdname) (basic)]] [(123).. (345)..]] [(class) (tree)] dc :: |
Class.tree [ $plus$ , $basic$ , [ 123 , 345 ] ] |
Class.tree [$plus$ , [[$cdname$ , $basic$ ]], [ 123 , 345 ] ] |
\end{verbatim} |
\end{verbatim} |
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\noindent |
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Example: |
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\begin{verbatim} |
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asir |
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[753] taka_cmo100_xml_form(quote(sin(x+1))); |
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<cmo_tree> <cmo_string>"sin"</cmo_string> |
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<cmo_list><cmo_int32 for="length">1</cmo_int32> |
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<cmo_list><cmo_int32 for="length">2</cmo_int32> |
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<cmo_string>"cdname"</cmo_string> |
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<cmo_string>"basic"</cmo_string> |
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</cmo_list> </cmo_list> |
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<cmo_tree> <cmo_string>"plus"</cmo_string> |
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<cmo_list><cmo_int32 for="length">1</cmo_int32> |
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<cmo_list><cmo_int32 for="length">2</cmo_int32> |
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<cmo_string>"cdname"</cmo_string> |
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<cmo_string>"basic"</cmo_string> |
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</cmo_list> </cmo_list> |
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<cmo_indeterminate> <cmo_string>"x"</cmo_string> </cmo_indeterminate> |
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<cmo_zz>1</cmo_zz> |
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</cmo_tree></cmo_tree> |
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\end{verbatim} |
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\bigbreak |
\bigbreak |
//&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. |
//&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. |
/*&eg |
/*&eg |
Let us define a group for distributed polynomials. In the following |
Let us define a group for distributed polynomials. In the following, |
DMS stands for Distributed Monomial System. |
DMS stands for Distributed Monomial System. |
*/ |
*/ |
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Line 259 $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$ |
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Line 305 $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$ |
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をあらわす.} \\ |
をあらわす.} \\ |
\mbox{Coefficient}&:& \mbox{ZZ} | \mbox{Integer32} \\ |
\mbox{Coefficient}&:& \mbox{ZZ} | \mbox{Integer32} \\ |
\mbox{Dpolynomial}&:& \mbox{Zero} \\ |
\mbox{Dpolynomial}&:& \mbox{Zero} \\ |
& & |\ ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},{\sl int32} m, \\ |
& & |\ ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},{\sl int32}\, m, \\ |
& & \ \ \mbox{DringDefinition}, |
& & \ \ \mbox{DringDefinition}, |
[\mbox{Monomial32}|\mbox{Zero}], \\ |
[\mbox{Monomial32}|\mbox{Zero}], \\ |
& &\ \ |
& &\ \ |
Line 301 $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$ |
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Line 347 $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$ |
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$x^e = x_1^{e_1} \cdots x_n^{e_n}$. } \\ |
$x^e = x_1^{e_1} \cdots x_n^{e_n}$. } \\ |
\mbox{Coefficient}&:& \mbox{ZZ} | \mbox{Integer32} \\ |
\mbox{Coefficient}&:& \mbox{ZZ} | \mbox{Integer32} \\ |
\mbox{Dpolynomial}&:& \mbox{Zero} \\ |
\mbox{Dpolynomial}&:& \mbox{Zero} \\ |
& & |\ ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},{\sl int32} m, \\ |
& & |\ ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},{\sl int32}\, m, \\ |
& & \ \ \mbox{DringDefinition}, [\mbox{Monomial32}|\mbox{Zero}], \\ |
& & \ \ \mbox{DringDefinition}, [\mbox{Monomial32}|\mbox{Zero}], \\ |
& &\ \ |
& &\ \ |
\{\mbox{Monomial32}\}) \\ |
\{\mbox{Monomial32}\}) \\ |
Line 347 are implemented on Asir and Kan. |
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Line 393 are implemented on Asir and Kan. |
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\subsection{ Zero} |
\subsection{ Zero} |
/*&jp |
/*&jp |
CMO では ゼロの表現法がなんとおりもあるが, |
CMO では ゼロの表現法がなんとおりもあることに注意. |
どのようなゼロをうけとっても, |
%% どのようなゼロをうけとっても, |
システムのゼロに変換できるべきである. |
%% システムのゼロに変換できるべきである. |
*/ |
*/ |
/*&eg |
/*&eg |
Though CMO has various representations of zero, |
Note that CMO has various representations of zero. |
each representation should be translated into zero |
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in the system. |
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*/ |
*/ |
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Line 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 |
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Line 417 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 |
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plugin/cmo-gmp.c}) CMO\_ZZ は次の形式をとる. |
plugin/cmo-gmp.c}) CMO\_ZZ は次の形式をとる. |
*/ |
*/ |
/*&eg |
/*&eg |
We describe the bignum (multi-precision integer) representation in OpenXM. |
We describe the bignum (multi-precision integer) representation |
In OpenXM {\tt CMO\_ZZ} is used to represent bignum. Its design is similar |
{\tt CMO\_ZZ} in OpenXM. |
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The format is similar |
to that in GNU MP. (cf. {\tt plugin/cmo-gmp.c} in the {\tt kan/sm1} |
to that in GNU MP. (cf. {\tt plugin/cmo-gmp.c} in the {\tt kan/sm1} |
distribution). CMO\_ZZ is defined as follows. |
distribution). CMO\_ZZ is defined as follows. |
*/ |
*/ |
Line 396 Open xxx 規約では上の CMO は以下の整数を意味する. ($R |
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Line 441 Open xxx 規約では上の CMO は以下の整数を意味する. ($R |
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/*&eg |
/*&eg |
$f$ is a 32bit integer. $b_0, \ldots, b_n$ are unsigned 32bit integers. |
$f$ is a 32bit integer. $b_0, \ldots, b_n$ are unsigned 32bit integers. |
$|f|$ is equal to $n+1$. |
$|f|$ is equal to $n+1$. |
The sign of $f$ represents that of the above CMO. As stated in Section |
The sign of $f$ represents that of the above integer to be expressed. |
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As stated in Section |
\ref{sec:basic0}, a negative 32bit integer is represented by |
\ref{sec:basic0}, a negative 32bit integer is represented by |
two's complement. |
two's complement. |
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Line 408 In OpenXM the above CMO represents the following integ |
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Line 454 In OpenXM the above CMO represents the following integ |
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\] |
\] |
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/*&jp |
/*&jp |
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\noindent 例: |
{\tt int32} を network byte order で表現 |
{\tt int32} を network byte order で表現 |
しているとすると,例えば, 整数 $14$ は CMO\_ZZ で表わすと, |
しているとすると,例えば, 整数 $14$ は CMO\_ZZ で表わすと, |
*/ |
*/ |
/*&eg |
/*&eg |
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\noindent Example: |
If we express {\tt int32} by the network byte order, |
If we express {\tt int32} by the network byte order, |
a CMO\_ZZ $14$ is expressed by |
a CMO\_ZZ $14$ is expressed by |
*/ |
*/ |
Line 468 Ring by Name を用いた場合, 現在の名前空間で変数 yyy に |
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Line 516 Ring by Name を用いた場合, 現在の名前空間で変数 yyy に |
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We treat polynomial rings and their elements as follows. |
We treat polynomial rings and their elements as follows. |
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Generic DMS ring is an $n$-variate polynomial ring $K[x_1, \ldots, x_n]$, |
Generic DMS ring is an $n$-variate polynomial ring $K[x_1, \ldots, x_n]$, |
where $K$ is some coefficient set. $K$ is unknown in advance |
where $K$ is a coefficient set. $K$ is unknown in advance |
and it is determined when coefficients of an element are received. |
and it is determined when coefficients of an element are received. |
When a server has received an element in Generic DMS ring, |
When a server has received an element in Generic DMS ring, |
the server has to translate it into the corresponding local object |
the server has to translate it into the corresponding local object |
Line 479 In {\tt kan/sm1} things are complicated. |
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Line 527 In {\tt kan/sm1} things are complicated. |
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{\tt kan/sm1} translates a DMS of N variables into an element of |
{\tt kan/sm1} translates a DMS of N variables into an element of |
the CurrentRing. |
the CurrentRing. |
If the CurrentRing is $n'$-variate and $n' < n$, then |
If the CurrentRing is $n'$-variate and $n' < n$, then |
an $n$-variate polynomial ring is newly created. Optional informations such as |
an $n$-variate polynomial ring is newly created. |
the term order are all ignored. |
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If RingByName ({\tt CMO\_RING\_BY\_NAME}, yyy) |
If RingByName ({\tt CMO\_RING\_BY\_NAME}, yyy) |
is specified as the second field of DMS, |
is specified as the second field of DMS, |
it requests a sever to use a ring object whose name is yyy |
it requests a sever to use a ring object whose name is yyy |
as the destination ring for the translation. |
as the destination ring for the translation. |
This is done in {\tt kan/sm1}. |
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*/ |
*/ |
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\medbreak \noindent |
\medbreak \noindent |
Line 536 $\in$ CMObject/RecursivePolynomial \\ |
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Line 583 $\in$ CMObject/RecursivePolynomial \\ |
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\mbox{Polynomial in 1 variable} &:& |
\mbox{Polynomial in 1 variable} &:& |
\mbox{({\tt CMO\_POLYNOMIAL\_IN\_ONE\_VARIABLE},\, {\sl int32}\, m, } \\ |
\mbox{({\tt CMO\_POLYNOMIAL\_IN\_ONE\_VARIABLE},\, {\sl int32}\, m, } \\ |
& & \quad \mbox{ Name of the main variable }, \\ |
& & \quad \mbox{ Name of the main variable }, \\ |
& & \quad \mbox{ \{ {\sl int32} e, Coefficient \}} \\ |
& & \quad \mbox{ \{ {\sl int32} e, Coefficient \}} ) \\ |
& & \mbox{ --- m はモノミアルの個数. } \\ |
& & \mbox{ --- m はモノミアルの個数. } \\ |
& & \mbox{ --- e, Coefficieint はモノミアルを表現している. } \\ |
& & \mbox{ --- e, Coefficieint はモノミアルを表現している. } \\ |
& & \mbox{ --- 順序の高い順にならべる. 普通は巾の高い順.} \\ |
& & \mbox{ --- 順序の高い順にならべる. 普通は巾の高い順.} \\ |
Line 552 $\in$ CMObject/RecursivePolynomial \\ |
|
Line 599 $\in$ CMObject/RecursivePolynomial \\ |
|
\mbox{ ( {\tt CMO\_RECURSIVE\_POLYNOMIAL}, } \\ |
\mbox{ ( {\tt CMO\_RECURSIVE\_POLYNOMIAL}, } \\ |
& & \quad \mbox{ RringDefinition, } \\ |
& & \quad \mbox{ RringDefinition, } \\ |
& & \quad |
& & \quad |
\mbox{ Polynomial in 1 variable}\, | \, \mbox{Coefficient} \\ |
\mbox{ Polynomial in 1 variable}\, | \, \mbox{Coefficient} ) \\ |
\mbox{RringDefinition} |
\mbox{RringDefinition} |
& : & \mbox{ {\sl List} v } \\ |
& : & \mbox{ {\sl List} v } \\ |
& & \quad \mbox{ --- v は, 変数名(indeterminate) のリスト. } \\ |
& & \quad \mbox{ --- v は, 変数名(indeterminate) または Tree のリスト. } \\ |
& & \quad \mbox{ --- 順序の高い順. } \\ |
& & \quad \mbox{ --- 順序の高い順. } \\ |
\end{eqnarray*} |
\end{eqnarray*} |
*/ |
*/ |
Line 564 $\in$ CMObject/RecursivePolynomial \\ |
|
Line 611 $\in$ CMObject/RecursivePolynomial \\ |
|
\mbox{Polynomial in 1 variable} &:& |
\mbox{Polynomial in 1 variable} &:& |
\mbox{({\tt CMO\_POLYNOMIAL\_IN\_ONE\_VARIABLE},\, {\sl int32}\, m, } \\ |
\mbox{({\tt CMO\_POLYNOMIAL\_IN\_ONE\_VARIABLE},\, {\sl int32}\, m, } \\ |
& & \quad \mbox{ Name of the main variable }, \\ |
& & \quad \mbox{ Name of the main variable }, \\ |
& & \quad \mbox{ \{ {\sl int32} e, Coefficient \}} \\ |
& & \quad \mbox{ \{ {\sl int32} e, Coefficient \}} ) \\ |
& & \mbox{ --- m is the number of monomials. } \\ |
& & \mbox{ --- m is the number of monomials. } \\ |
& & \mbox{ --- A pair of e and Coefficient represents a monomial. } \\ |
& & \mbox{ --- A pair of e and Coefficient represents a monomial. } \\ |
& & \mbox{ --- The pairs of e and Coefficient are sorted in the } \\ |
& & \mbox{ --- The pairs of e and Coefficient are sorted in the } \\ |
Line 582 $\in$ CMObject/RecursivePolynomial \\ |
|
Line 629 $\in$ CMObject/RecursivePolynomial \\ |
|
\mbox{ ( {\tt CMO\_RECURSIVE\_POLYNOMIAL}, } \\ |
\mbox{ ( {\tt CMO\_RECURSIVE\_POLYNOMIAL}, } \\ |
& & \quad \mbox{ RringDefinition, } \\ |
& & \quad \mbox{ RringDefinition, } \\ |
& & \quad |
& & \quad |
\mbox{ Polynomial in 1 variable}\, | \, \mbox{Coefficient} \\ |
\mbox{ Polynomial in 1 variable}\, | \, \mbox{Coefficient} ) \\ |
\mbox{RringDefinition} |
\mbox{RringDefinition} |
& : & \mbox{ {\sl List} v } \\ |
& : & \mbox{ {\sl List} v } \\ |
& & \quad \mbox{ --- v is a list of names of indeterminates. } \\ |
& & \quad \mbox{ --- v is a list of names of indeterminates or trees. } \\ |
& & \quad \mbox{ --- It is sorted in the decreasing order. } \\ |
& & \quad \mbox{ --- It is sorted in the decreasing order. } \\ |
\end{eqnarray*} |
\end{eqnarray*} |
*/ |
*/ |
|
|
$$ x^3 (1234 y^5 + 17 ) + x^1 (y^{10} + 31 y^5) $$ |
$$ x^3 (1234 y^5 + 17 ) + x^1 (y^{10} + 31 y^5) $$ |
/*&jp |
/*&jp |
をあらわす. |
をあらわす. |
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 |
%%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 |
すること. つまり, 主変数かける係数の順番. |
%%すること. つまり, 主変数かける係数の順番. |
*/ |
*/ |
/*&eg |
/*&eg |
We intend to represent non-commutative polynomials with the |
%%We intend to represent non-commutative polynomials with the |
same form. In such a case, the order of products are defined |
%%same form. In such a case, the order of products are defined |
as above, that is a power of the main variable $\times$ a coeffcient. |
%%as above, that is a power of the main variable $\times$ a coeffcient. |
|
|
*/ |
*/ |
|
|
\noindent |
\noindent |
Line 644 $\in$ CMObject/MachineDouble \\ |
|
Line 692 $\in$ CMObject/MachineDouble \\ |
|
\begin{eqnarray*} |
\begin{eqnarray*} |
\mbox{64bit machine double} &:& |
\mbox{64bit machine double} &:& |
\mbox{({\tt CMO\_64BIT\_MACHINE\_DOUBLE}, } \\ |
\mbox{({\tt CMO\_64BIT\_MACHINE\_DOUBLE}, } \\ |
& & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte}} s8)\\ |
& & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte} s8})\\ |
& & \mbox{ --- s1, $\ldots$, s8 は {\tt double} (64bit). } \\ |
& & \mbox{ --- s1, $\ldots$, s8 は {\tt double} (64bit). } \\ |
& & \mbox{ --- この表現はCPU依存である.}\\ |
& & \mbox{ --- この表現はCPU依存である.}\\ |
&& \mbox{\quad\quad mathcap に CPU 情報を付加しておく.} \\ |
&& \mbox{\quad\quad byte order negotiation を用いる.} \\ |
\mbox{Array of 64bit machine double} &:& |
\mbox{Array of 64bit machine double} &:& |
\mbox{({\tt CMO\_ARRAY\_OF\_64BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\ |
\mbox{({\tt CMO\_ARRAY\_OF\_64BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\ |
& & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte}}\, s8[1], \ldots , {\sl byte}\, s8[m])\\ |
& & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte}}\, s8[1], \ldots , {\sl byte}\, s8[m])\\ |
& & \mbox{ --- s*[1], $\ldots$ s*[m] は m 個の double (64bit) である. } \\ |
& & \mbox{ --- s*[1], $\ldots$ s*[m] は m 個の double (64bit) である. } \\ |
& & \mbox{ --- この表現はCPU依存である.}\\ |
& & \mbox{ --- この表現はCPU依存である.}\\ |
& & \mbox{ \quad\quad mathcap に CPU 情報を付加しておく.} \\ |
& & \mbox{ \quad\quad byte order negotiation を用いる.} \\ |
\mbox{128bit machine double} &:& |
\mbox{128bit machine double} &:& |
\mbox{({\tt CMO\_128BIT\_MACHINE\_DOUBLE}, } \\ |
\mbox{({\tt CMO\_128BIT\_MACHINE\_DOUBLE}, } \\ |
& & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte}} s16)\\ |
& & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte} s16})\\ |
& & \mbox{ --- s1, $\ldots$, s16 は {\tt long double} (128bit). } \\ |
& & \mbox{ --- s1, $\ldots$, s16 は {\tt long double} (128bit). } \\ |
& & \mbox{ --- この表現はCPU依存である.}\\ |
& & \mbox{ --- この表現はCPU依存である.}\\ |
&& \mbox{\quad\quad mathcap に CPU 情報を付加しておく.} \\ |
&& \mbox{\quad\quad byte order negotiation を用いる.} \\ |
\mbox{Array of 128bit machine double} &:& |
\mbox{Array of 128bit machine double} &:& |
\mbox{({\tt CMO\_ARRAY\_OF\_128BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\ |
\mbox{({\tt CMO\_ARRAY\_OF\_128BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\ |
& & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte}} s16[1], \ldots , {\sl byte} s16[m])\\ |
& & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte} s16[1], \ldots , {\sl byte} s16[m]})\\ |
& & \mbox{ --- s*[1], $\ldots$ s*[m] は m 個の long double (128bit) である. } \\ |
& & \mbox{ --- s*[1], $\ldots$ s*[m] は m 個の long double (128bit) である. } \\ |
& & \mbox{ --- この表現はCPU依存である.}\\ |
& & \mbox{ --- この表現はCPU依存である.}\\ |
& & \mbox{ \quad\quad mathcap に CPU 情報を付加しておく.} |
& & \mbox{ \quad\quad byte order negotiation を用いる.} |
\end{eqnarray*} |
\end{eqnarray*} |
*/ |
*/ |
/*&eg |
/*&eg |
\begin{eqnarray*} |
\begin{eqnarray*} |
\mbox{64bit machine double} &:& |
\mbox{64bit machine double} &:& |
\mbox{({\tt CMO\_64BIT\_MACHINE\_DOUBLE}, } \\ |
\mbox{({\tt CMO\_64BIT\_MACHINE\_DOUBLE}, } \\ |
& & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte}} s8)\\ |
& & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte} s8})\\ |
& & \mbox{ --- s1, $\ldots$, s8 は {\tt double} (64bit). } \\ |
& & \mbox{ --- s1, $\ldots$, s8 は {\tt double} (64bit). } \\ |
& & \mbox{ --- This depends on CPU.}\\ |
& & \mbox{ --- Encoding depends on CPU.}\\ |
&& \mbox{\quad\quad Add informations on CPU to the mathcap.} \\ |
&& \mbox{\quad\quad Need the byte order negotiation.} \\ |
\mbox{Array of 64bit machine double} &:& |
\mbox{Array of 64bit machine double} &:& |
\mbox{({\tt CMO\_ARRAY\_OF\_64BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\ |
\mbox{({\tt CMO\_ARRAY\_OF\_64BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\ |
& & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte}}\, s8[1], \ldots , {\sl byte}\, s8[m])\\ |
& & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte}}\, s8[1], \ldots , {\sl byte}\, s8[m])\\ |
& & \mbox{ --- s*[1], $\ldots$ s*[m] are 64bit double's. } \\ |
& & \mbox{ --- s*[1], $\ldots$ s*[m] are 64bit double's. } \\ |
& & \mbox{ --- This depends on CPU.}\\ |
& & \mbox{ --- Encoding depends on CPU.}\\ |
& & \mbox{\quad\quad Add informations on CPU to the mathcap.} \\ |
& & \mbox{\quad\quad Need the byte order negotiation.} \\ |
\mbox{128bit machine double} &:& |
\mbox{128bit machine double} &:& |
\mbox{({\tt CMO\_128BIT\_MACHINE\_DOUBLE}, } \\ |
\mbox{({\tt CMO\_128BIT\_MACHINE\_DOUBLE}, } \\ |
& & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte}} s16)\\ |
& & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte} s16})\\ |
& & \mbox{ --- s1, $\ldots$, s16 は {\tt long double} (128bit). } \\ |
& & \mbox{ --- s1, $\ldots$, s16 は {\tt long double} (128bit). } \\ |
& & \mbox{ --- This depends on CPU.}\\ |
& & \mbox{ --- Encoding depends on CPU.}\\ |
& & \mbox{\quad\quad Add informations on CPU to the mathcap.} \\ |
& & \mbox{\quad\quad Need the byte order negotiation.} \\ |
\mbox{Array of 128bit machine double} &:& |
\mbox{Array of 128bit machine double} &:& |
\mbox{({\tt CMO\_ARRAY\_OF\_128BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\ |
\mbox{({\tt CMO\_ARRAY\_OF\_128BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\ |
& & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte}} s16[1], \ldots , {\sl byte} s16[m])\\ |
& & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte} s16[1], \ldots , {\sl byte} s16[m]})\\ |
& & \mbox{ --- s*[1], $\ldots$ s*[m] are 128bit long double's. } \\ |
& & \mbox{ --- s*[1], $\ldots$ s*[m] are 128bit long double's. } \\ |
& & \mbox{ --- This depends on CPU.}\\ |
& & \mbox{ --- Encoding depends on CPU.}\\ |
& & \mbox{\quad\quad Add informations on CPU to the mathcap.} \\ |
& & \mbox{\quad\quad Need the byte order negotiation.} \\ |
\end{eqnarray*} |
\end{eqnarray*} |
*/ |
*/ |
|
|
\bigbreak |
\bigbreak |
//&jp 次に IEEE 準拠の float および Big float を定義しよう. |
|
//&eg We define float and big float conforming to the IEEE standard. |
|
\begin{verbatim} |
\begin{verbatim} |
#define CMO_BIGFLOAT 50 |
|
#define CMO_IEEE_DOUBLE_FLOAT 51 |
#define CMO_IEEE_DOUBLE_FLOAT 51 |
\end{verbatim} |
\end{verbatim} |
|
|
/*&jp |
/*&jp |
IEEE 準拠の float については, IEEE 754 double precision floating-point |
IEEE 準拠の float については, IEEE 754 double precision floating-point |
format (64 bit) の定義を見よ. |
format (64 bit) の定義を見よ. |
|
|
|
256.100006 の Intel Pentium の double64 での内部表現は |
|
{\tt cd 0c 80 43 } \\ |
|
256.100006 の PowerPC (Mac) の double64 での内部表現は |
|
{\tt 43 80 0c cd }. |
|
この例でみるように byte の順序が逆である. |
|
エンジンスタートの時の byte order negotiation で byte の順序を指定する. |
|
|
|
|
*/ |
*/ |
/*&eg |
/*&eg |
See IEEE 754 double precision floating-point (64 bit) for the details of |
See IEEE 754 double precision floating-point (64 bit) for the details of |
float conforming to the IEEE standard. |
float compliant to the IEEE standard. |
|
|
|
The internal expression of 256.100006 in the Intel Pentium is |
|
{\tt cd 0c 80 43 } \\ |
|
The internal expression of 256.100006 in the PowerPC (Mac) is |
|
{\tt 43 80 0c cd }. |
|
As you have seen in this example, |
|
the orders of the bytes are opposite each other. |
|
The byte order is specified by the byte order negotiation protocol |
|
when the engine starts. |
|
|
*/ |
*/ |
|
|
\noindent |
\subsection{Bigfloat32} |
Group CMObject/Bigfloat requires CMObject/Primitive, CMObject/Basic.\\ |
/*&jp |
Bigfloat |
int32 を基本とした bigfloat の表現方法について述べる. |
$\in$ CMObject/Bigfloat \\ |
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. |
|
*/ |
|
/*&eg |
|
This subsection describes our format for bigfloat in terms of the int32. |
|
This format is identical to the internal format of mpfr on 32 bit CPU's. |
|
*/ |
|
Ref: {\tt OpenXM/src/mpfr/bfsize/bfsize.c} |
|
|
\begin{eqnarray*} |
\begin{verbatim} |
\mbox{Bigfloat} &:& |
#define CMO_BIGFLOAT32 52 |
\mbox{({\tt CMO\_BIGFLOAT}, } \\ |
\end{verbatim} |
& & \quad \mbox{ {\sl ZZ} a , {\sl ZZ} e})\\ |
|
& & \mbox{ --- $a \times 2^e$. } \\ |
|
\end{eqnarray*} |
/*&jp |
|
Bigfloat32 は次の形式の int32 の配列である. |
|
*/ |
|
/*&eg |
|
The bigfloat32 is an array of int32 numbers of the following format. |
|
*/ |
|
|
|
\begin{center} |
|
{\sl int32}\, {\rm prec}, {\sl int32}\, {\rm sign}, {\sl int32}\, {\rm exp}, |
|
{\sl int32}\, \mbox{a[1]}, \ldots , {\sl int32}\, \mbox{a[k]} |
|
\end{center} |
|
|
|
/*&jp |
|
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) , |
|
E=exp は指数部で, 上のデータは数 |
|
$$ s (a[k]/B + a[k-1]/B^2 + ... + a[1]/B^k) 2^E $$ |
|
を表す. |
|
ここで $B=2^{32}$, |
|
$k = \lceil p/32 \rceil$ |
|
である. |
|
*/ |
|
/*&eg |
|
p=prec is the precision, s=sign is the sign(1 means positive, -1 (expressed by two's complement) is nevative), |
|
E=exp is the exponent, and the data above expresses the number |
|
$$ s (a[k]/B + a[k-1]/B^2 + ... + a[1]/B^k) 2^E. $$ |
|
Here, $B=2^{32}$, |
|
$k = \lceil p/32 \rceil$. |
|
*/ |
|
|