=================================================================== RCS file: /home/cvs/OpenXM/doc/OpenXM-specs/cmo-basic1.tex,v retrieving revision 1.10 retrieving revision 1.15 diff -u -p -r1.10 -r1.15 --- OpenXM/doc/OpenXM-specs/cmo-basic1.tex 2001/04/10 11:56:29 1.10 +++ OpenXM/doc/OpenXM-specs/cmo-basic1.tex 2016/08/22 05:38:27 1.15 @@ -1,4 +1,4 @@ -%% $OpenXM: OpenXM/doc/OpenXM-specs/cmo-basic1.tex,v 1.9 2000/09/12 23:09:18 noro 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 } //&eg \section{ CMOexpressions for numbers and polynomials } \label{sec:basic1} @@ -14,7 +14,9 @@ #define CMO_RING_BY_NAME 26 #define CMO_DISTRIBUTED_POLYNOMIAL 31 #define CMO_RATIONAL 34 +#define CMO_COMPLEX 35 +#define CMO_BIGFLOAT32 52 #define CMO_INDETERMINATE 60 #define CMO_TREE 61 @@ -59,11 +61,11 @@ bash$ ./bconv \bigbreak \noindent Group CMObject/Basic requires CMObject/Primitive. \\ -ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/Basic. \\ +ZZ, QQ, Zero, Rational, Indeterminate $\in$ CMObject/Basic. \\ \begin{eqnarray*} \mbox{Zero} &:& ({\tt CMO\_ZERO}) \\ & & \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$|$]} ) \\ &:& \mbox{ --- bignum をあらわす. a[i] についてはあとで説明}\\ \mbox{QQ} &:& ({\tt CMO\_QQ}, @@ -72,20 +74,28 @@ ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B & & \mbox{ --- 有理数 $a/b$ を表す. } \\ \mbox{Rational} &:& ({\tt CMO\_RATIONAL}, {\sl CMObject}\, {\rm a}, {\sl CMObject}\, {\rm b}) \\ & & \mbox{ --- $a/b$ を表す. } \\ +\mbox{Bigfloat32} &:& ({\tt CMO\_BIGFLOAT32}, +{\sl int32}\, {\rm prec}, {\sl int32}\, {\rm sign}, {\sl int32}\, {\rm exp}, +{\sl int32}\, \mbox{a[1]}, \ldots , {\sl int32}\, \mbox{a[k]} ) \\ +&:& \mbox{ --- bigfloat をあらわす. a[i], k についてはあとで説明}\\ +\mbox{Complex} &:& ({\tt CMO\_COMPLEX}, {\sl CMObject}\, {\rm re}, {\sl CMObject}\, {\rm im}) \\ +& & \mbox{ --- $a+b\sqrt{-1}$ を表す. } \\ \mbox{Indeterminate} &:& ({\tt CMO\_INDETERMINATE}, {\sl Cstring}\, {\rm v}) \\ & & \mbox{ --- 変数名 $v$ . } \\ \end{eqnarray*} */ + + /*&eg \bigbreak \noindent Group CMObject/Basic requires CMObject/Primitive. \\ -ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/Basic. \\ +ZZ, QQ, Zero, Rational, Indeterminate $\in$ CMObject/Basic. \\ \begin{eqnarray*} \mbox{Zero} &:& ({\tt CMO\_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$|$]} ) \\ &:& \mbox{ --- bignum. The meaning of a[i] will be explained later.}\\ \mbox{QQ} &:& ({\tt CMO\_QQ}, @@ -94,6 +104,12 @@ ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B & & \mbox{ --- Rational number $a/b$. } \\ \mbox{Rational} &:& ({\tt CMO\_RATIONAL}, {\sl CMObject}\, {\rm a}, {\sl CMObject}\, {\rm b}) \\ & & \mbox{ --- Rational expression $a/b$. } \\ +\mbox{Bigfloat32} &:& ({\tt CMO\_BIGFLOAT32}, +{\sl int32}\, {\rm prec}, {\sl int32}\, {\rm sign}, {\sl int32}\, {\rm exp}, +{\sl int32}\, \mbox{a[1]}, \ldots , {\sl int32}\, \mbox{a[k]} ) \\ +&:& \mbox{ --- bigfloat. The meaning of a[i], k will be explained later.}\\ +\mbox{Complex} &:& ({\tt CMO\_COMPLEX}, {\sl CMObject}\, {\rm re}, {\sl CMObject}\, {\rm im}) \\ +& & \mbox{ --- Complex number $a+b\sqrt{-1}$. } \\ \mbox{Indeterminate} &:& ({\tt CMO\_INDETERMINATE}, {\sl Cstring}\, {\rm v}) \\ & & \mbox{ --- Variable name $v$ . } \\ \end{eqnarray*} @@ -101,7 +117,10 @@ ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B /*&C */ +/*&C +*/ + /*&jp 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, @@ -114,45 +133,46 @@ escape sequence を用いて実現するのは, 無理があるようで テーブルを作成する必要があるであろう.) */ /*&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 restrictions on the names of variables. Indeterminates of CMO and internal variable names must be translated -in one to one correspondence. +in one-to-one correspondence. */ + /*&jp +\subsection{Indeterminate および Tree} \noindent Group CMObject/Tree requires CMObject/Basic. \\ -Tree, Lambda $\in$ CMObject/Basic. \\ +Tree, Lambda $\in$ CMObject/Tree. \\ \begin{eqnarray*} \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{ --- cdname は空文字列でなければ name の意味が説明されている }\\ -& & \mbox{ --- OpenMath CD (content dictionary) の名前. } \\ +& & \mbox{ --- attributes は空リストでなければ name の属性を保持している. }\\ +& & \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}, {\sl Tree} {\rm body}) \\ & & \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.} \\ \end{eqnarray*} */ /*&eg +\subsection{Indeterminate and Tree} \noindent Group CMObject/Tree requires CMObject/Basic. \\ -Tree, Lambda $\in$ CMObject/Basic. \\ +Tree, Lambda $\in$ CMObject/Tree. \\ \begin{eqnarray*} \mbox{Tree} &:& ({\tt CMO\_TREE}, {\sl Cstring}\, {\rm name}, - {\sl Cstring}\, {\rm cdname}, {\sl List}\, {\rm leaves}) \\ -& & \mbox{ --- A function or a constant of name. Functions are not evaluated. } \\ -& & \mbox{ --- cdname may be a null. If it is not null, it is the name of}\\ -& & \mbox{ --- the OpenMath CD (content dictionary). } \\ + {\sl List}\, {\rm attributes}, {\sl List}\, {\rm leaves}) \\ +& & \mbox{ --- ``name'' is the name of the node of the tree. } \\ +& & \mbox{ --- Attributes may be a null list. If it is not null, it is a list of}\\ +& & \mbox{ --- key and value pairs. } \\ \mbox{Lambda} &:& ({\tt CMO\_LAMBDA}, {\sl List}\, {\rm args}, {\sl Tree} {\rm body}) \\ & & \mbox{ --- a function with the arguments body. } \\ -& & \mbox{ --- optional arguments come after leaves.} \\ \end{eqnarray*} */ @@ -205,20 +225,23 @@ Lisp の Lambda 表現と同じ. */ /*&eg 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. */ \noindent //&jp 例: $sin(x+e)$ の表現. //&eg Example: the expression of $sin(x+e)$. \begin{verbatim} -(CMO_TREE, (CMO_STRING, "sin"), (CMO_STRING, "basic"), +(CMO_TREE, (CMO_STRING, "sin"), + (CMO_LIST,[size=]1,(CMO_LIST,[size=]2,(CMO_STRING, "cdname"), + (CMO_STRING,"basic"))) (CMO_LIST,[size=]1, (CMO_TREE, (CMO_STRING, "plus"), (CMO_STRING, "basic"), (CMO_LIST,[size=]2, (CMO_INDETERMINATE,"x"), //&jp (CMO_TREE,(CMO_STRING, "e"), 自然対数の底 //&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"), + (CMO_STRING,"basic"))) )) ) ) @@ -229,16 +252,37 @@ It is the same as the Lambda expression in Lisp. \noindent Example: \begin{verbatim} -sm1> [(plus) (Basic) [(123).. (345)..]] [(class) (tree)] dc :: -Class.tree [ $plus$ , $basic$ , [ 123 , 345 ] ] +sm1> [(plus) [[(cdname) (basic)]] [(123).. (345)..]] [(class) (tree)] dc :: +Class.tree [$plus$ , [[$cdname$ , $basic$ ]], [ 123 , 345 ] ] \end{verbatim} +\noindent +Example: +\begin{verbatim} +asir +[753] taka_cmo100_xml_form(quote(sin(x+1))); + "sin" + 1 + 2 + "cdname" + "basic" + + "plus" + 1 + 2 + "cdname" + "basic" + + "x" + 1 + +\end{verbatim} \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. /*&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. */ @@ -261,7 +305,7 @@ $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$ をあらわす.} \\ \mbox{Coefficient}&:& \mbox{ZZ} | \mbox{Integer32} \\ \mbox{Dpolynomial}&:& \mbox{Zero} \\ -& & |\ ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},{\sl int32} m, \\ +& & |\ ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},{\sl int32}\, m, \\ & & \ \ \mbox{DringDefinition}, [\mbox{Monomial32}|\mbox{Zero}], \\ & &\ \ @@ -303,7 +347,7 @@ $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$ $x^e = x_1^{e_1} \cdots x_n^{e_n}$. } \\ \mbox{Coefficient}&:& \mbox{ZZ} | \mbox{Integer32} \\ \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{Monomial32}\}) \\ @@ -349,14 +393,12 @@ are implemented on Asir and Kan. \subsection{ Zero} /*&jp -CMO では ゼロの表現法がなんとおりもあるが, -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. +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, +%% システムのゼロに変換できるべきである. */ /*&eg -Though CMO has various representations of zero, -each representation should be translated into zero -in the system. +Note that CMO has various representations of zero. */ @@ -375,8 +417,9 @@ 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 plugin/cmo-gmp.c}) CMO\_ZZ は次の形式をとる. */ /*&eg -We describe the bignum (multi-precision integer) representation in OpenXM. -In OpenXM {\tt CMO\_ZZ} is used to represent bignum. Its design is similar +We describe the bignum (multi-precision integer) representation +{\tt CMO\_ZZ} in OpenXM. +The format is similar to that in GNU MP. (cf. {\tt plugin/cmo-gmp.c} in the {\tt kan/sm1} distribution). CMO\_ZZ is defined as follows. */ @@ -398,7 +441,8 @@ Open xxx 規約では上の CMO は以下の整数を意味する. ($R /*&eg $f$ is a 32bit integer. $b_0, \ldots, b_n$ are unsigned 32bit integers. $|f|$ 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. +As stated in Section \ref{sec:basic0}, a negative 32bit integer is represented by two's complement. @@ -410,10 +454,12 @@ In OpenXM the above CMO represents the following integ \] /*&jp +\noindent 例: {\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, */ /*&eg +\noindent Example: If we express {\tt int32} by the network byte order, a CMO\_ZZ $14$ is expressed by */ @@ -470,7 +516,7 @@ Ring by Name を用いた場合, 現在の名前空間で変数 yyy に We treat polynomial rings and their elements as follows. 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. When a server has received an element in Generic DMS ring, the server has to translate it into the corresponding local object @@ -481,14 +527,13 @@ In {\tt kan/sm1} things are complicated. {\tt kan/sm1} translates a DMS of N variables into an element of the CurrentRing. If the CurrentRing is $n'$-variate and $n' < n$, then -an $n$-variate polynomial ring is newly created. Optional informations such as -the term order are all ignored. +an $n$-variate polynomial ring is newly created. + If RingByName ({\tt CMO\_RING\_BY\_NAME}, yyy) is specified as the second field of DMS, it requests a sever to use a ring object whose name is yyy as the destination ring for the translation. -This is done in {\tt kan/sm1}. */ \medbreak \noindent @@ -538,7 +583,7 @@ $\in$ CMObject/RecursivePolynomial \\ \mbox{Polynomial in 1 variable} &:& \mbox{({\tt CMO\_POLYNOMIAL\_IN\_ONE\_VARIABLE},\, {\sl int32}\, m, } \\ & & \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{ --- 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.} \\ @@ -554,7 +599,7 @@ $\in$ CMObject/RecursivePolynomial \\ \mbox{ ( {\tt CMO\_RECURSIVE\_POLYNOMIAL}, } \\ & & \quad \mbox{ RringDefinition, } \\ & & \quad -\mbox{ Polynomial in 1 variable}\, | \, \mbox{Coefficient} \\ +\mbox{ Polynomial in 1 variable}\, | \, \mbox{Coefficient} ) \\ \mbox{RringDefinition} & : & \mbox{ {\sl List} v } \\ & & \quad \mbox{ --- v は, 変数名(indeterminate) または Tree のリスト. } \\ @@ -566,7 +611,7 @@ $\in$ CMObject/RecursivePolynomial \\ \mbox{Polynomial in 1 variable} &:& \mbox{({\tt CMO\_POLYNOMIAL\_IN\_ONE\_VARIABLE},\, {\sl int32}\, m, } \\ & & \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{ --- A pair of e and Coefficient represents a monomial. } \\ & & \mbox{ --- The pairs of e and Coefficient are sorted in the } \\ @@ -584,7 +629,7 @@ $\in$ CMObject/RecursivePolynomial \\ \mbox{ ( {\tt CMO\_RECURSIVE\_POLYNOMIAL}, } \\ & & \quad \mbox{ RringDefinition, } \\ & & \quad -\mbox{ Polynomial in 1 variable}\, | \, \mbox{Coefficient} \\ +\mbox{ Polynomial in 1 variable}\, | \, \mbox{Coefficient} ) \\ \mbox{RringDefinition} & : & \mbox{ {\sl List} v } \\ & & \quad \mbox{ --- v is a list of names of indeterminates or trees. } \\ @@ -609,13 +654,14 @@ Example: $$ x^3 (1234 y^5 + 17 ) + x^1 (y^{10} + 31 y^5) $$ /*&jp 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 -すること. つまり, 主変数かける係数の順番. +%%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. */ /*&eg -We intend to represent non-commutative polynomials with the -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. +%%We intend to represent non-commutative polynomials with the +%%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. + */ \noindent @@ -646,84 +692,135 @@ $\in$ CMObject/MachineDouble \\ \begin{eqnarray*} \mbox{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{ --- 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{({\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])\\ & & \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{ \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{({\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{ --- 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{({\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{ --- 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*} */ /*&eg \begin{eqnarray*} \mbox{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{ --- This depends on CPU.}\\ -&& \mbox{\quad\quad Add informations on CPU to the mathcap.} \\ +& & \mbox{ --- Encoding depends on CPU.}\\ +&& \mbox{\quad\quad Need the byte order negotiation.} \\ \mbox{Array of 64bit machine double} &:& \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])\\ & & \mbox{ --- s*[1], $\ldots$ s*[m] are 64bit double's. } \\ -& & \mbox{ --- This depends on CPU.}\\ -& & \mbox{\quad\quad Add informations on CPU to the mathcap.} \\ +& & \mbox{ --- Encoding depends on CPU.}\\ +& & \mbox{\quad\quad Need the byte order negotiation.} \\ \mbox{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{ --- This depends on CPU.}\\ -& & \mbox{\quad\quad Add informations on CPU to the mathcap.} \\ +& & \mbox{ --- Encoding depends on CPU.}\\ +& & \mbox{\quad\quad Need the byte order negotiation.} \\ \mbox{Array of 128bit machine double} &:& \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{ --- This depends on CPU.}\\ -& & \mbox{\quad\quad Add informations on CPU to the mathcap.} \\ +& & \mbox{ --- Encoding depends on CPU.}\\ +& & \mbox{\quad\quad Need the byte order negotiation.} \\ \end{eqnarray*} */ \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} -#define CMO_BIGFLOAT 50 #define CMO_IEEE_DOUBLE_FLOAT 51 \end{verbatim} /*&jp 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. + +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 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 -Group CMObject/Bigfloat requires CMObject/Primitive, CMObject/Basic.\\ -Bigfloat -$\in$ CMObject/Bigfloat \\ +\subsection{Bigfloat32} +/*&jp +int32 を基本とした bigfloat の表現方法について述べる. +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*} -\mbox{Bigfloat} &:& -\mbox{({\tt CMO\_BIGFLOAT}, } \\ -& & \quad \mbox{ {\sl ZZ} a , {\sl ZZ} e})\\ -& & \mbox{ --- $a \times 2^e$. } \\ -\end{eqnarray*} +\begin{verbatim} +#define CMO_BIGFLOAT32 52 +\end{verbatim} + + +/*&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, $B=2^{32}$, +$k = \lceil p/32 \rceil$. +*/ +