=================================================================== RCS file: /home/cvs/OpenXM/doc/Papers/Attic/dag-noro-proc.tex,v retrieving revision 1.10 retrieving revision 1.11 diff -u -p -r1.10 -r1.11 --- OpenXM/doc/Papers/Attic/dag-noro-proc.tex 2002/01/04 06:06:09 1.10 +++ OpenXM/doc/Papers/Attic/dag-noro-proc.tex 2002/02/25 01:02:14 1.11 @@ -1,4 +1,4 @@ -% $OpenXM: OpenXM/doc/Papers/dag-noro-proc.tex,v 1.9 2001/12/28 06:06:15 noro Exp $ +% $OpenXM: OpenXM/doc/Papers/dag-noro-proc.tex,v 1.10 2002/01/04 06:06:09 noro Exp $ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % This is a sample input file for your contribution to a multi- % author book to be published by Springer Verlag. @@ -60,6 +60,7 @@ \usepackage{epsfig} \def\cont{{\rm cont}} \def\GCD{{\rm GCD}} +\def\Q{{\bf Q}} % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -87,128 +88,77 @@ \maketitle % typesets the title of the contribution -\begin{abstract} -Risa/Asir is software for polynomial computation. It has been -developed for testing experimental polynomial algorithms, and now it -acts also as a main component in the OpenXM package \cite{OPENXM}. -OpenXM is an infrastructure for exchanging mathematical -data. It defines a client-server architecture for parallel and -distributed computation. In this article we present an overview of -Risa/Asir and review several techniques for improving performances of -Groebner basis computation over {\bf Q}. We also show Risa/Asir's -OpenXM interfaces and their usages. -\end{abstract} +%\begin{abstract} +%Risa/Asir is software for polynomial computation. It has been +%developed for testing experimental polynomial algorithms, and now it +%acts also as a main component in the OpenXM package \cite{noro:OPENXM}. +%OpenXM is an infrastructure for exchanging mathematical +%data. It defines a client-server architecture for parallel and +%distributed computation. In this article we present an overview of +%Risa/Asir and review several techniques for improving performances of +%Groebner basis computation over {\bf Q}. We also show Risa/Asir's +%OpenXM interfaces and their usages. +%\end{abstract} -\section{A computer algebra system Risa/Asir} +\section{Introduction} -\subsection{What is Risa/Asir?} +%Risa/Asir は, 数, 多項式などに対する演算を実装する engine, +%ユーザ言語を実装する parser and interpreter および, +%他の application との interaction のための OpenXM interface からなる +%computer algebra system である. +Risa/Asir is a computer algebra system which consists of an engine for +operations on numbers and polynomials, a parser and an interpreter for +the user language, and a communication interface called OpenXM API for +interaction with other applications. +%engine では, 数, 多項式などの arithmetics および, 多項式 +%GCD, 因数分解, グレブナ基底計算が実装されている. これらは組み込み関数 +%としてユーザ言語から呼び出される. +The engine implements fundamental arithmetics on numbers and polynomials, +polynomial GCD, polynomial factorizations and Groebner basis computations, +etc. These can be called from the user language as builtin functions. +%Risa/Asir のユーザ言語は C 言語 like な文法をもち, 変数の型宣言が +%ない, リスト処理および自動 garbage collection つきのインタプリタ +%言語である. ユーザ言語プログラムは parser により中間言語に +%変換され, interpreter により解釈実行される. interpreter には +%gdb 風の debugger が組み込まれている. +The user language has C-like syntax, without type declarations +of variables, with list processing and with automatic garbage collection. +The interpreter is equipped with a {\tt gdb}-like debugger. +%これらの機能は OpenXM interface を通して他の application からも使用可 +%能である. OpenXM \cite{noro:RFC100} は数学ソフトウェアの client-server +%型の相互呼び出しのための プロトコルである. +These functions can be called from other applications via OpenXM API. +OpenXM \cite{noro:RFC100} is a protocol for client-server +communications between mathematical software. We are distributing +OpenXM package \cite{noro:OPENXM}, which is a collection of various +clients and servers comlient to the OpenXM protocol specification. -Risa/Asir \cite{RISA} is software mainly for polynomial -computation. Its major functions are polynomial factorization and -Groebner basis computation, whose core parts are implemented as -built-in functions. Some higher algorithms such as primary ideal -decomposition or Galois group computation are built on them by the -user language called Asir language. Asir language can be regarded as C -language without type declaration of variables, with list processing, -and with automatic garbage collection. A built-in {\tt gdb}-like user -language debugger is available. Risa/Asir is open source and the -source code and binaries are available via {\tt ftp} or {\tt CVS}. -Risa/Asir is not only a standalone computer algebra system but also a -main component in OpenXM package \cite{OPENXM}, which is a collection -of various software compliant to OpenXM protocol specification. -OpenXM is an infrastructure for exchanging mathematical data and -Risa/Asir has three kinds of OpenXM interfaces : -OpenXM API in the Risa/Asir user language, -OpenXM C language API in the Risa/Asir subroutine library, -and an OpenXM server. -Our goals of developing Risa/Asir are as follows: +%Risa/Asir は多項式因数分解, ガロア群計算 \cite{noro:ANY}, グレブナ基底 +%計算 \cite{noro:NM,noro:NY}, 準素イデアル分解 \cite{noro:SY}, 暗号 +%\cite{noro:IKNY} における実験的アルゴリズム をテストするためのプラット +%フォームとして開発されてきた. また, OpenXM API を用いて parallel +%distributed computation の実験にも用いられている. 根幹をなすのは多項 +%式因数分解およびグレブナ基底計算である. 本稿では, 特に, グレブナ基底 +%計算に関して, その基本および {\bf Q} 上での計算の困難を克服するための +%さまざまな工夫およびその効果について述べる. また, Risa/Asir は OpenXM +%package における主要な component の一つである. Risa/Asir を client あ +%るいは server として用いる分散並列計算について, 実例をもとに解説する. +Risa/Asir has been used for implementing and testing experimental +algorithms such as polynomial factorizations, splitting field and +Galois group computations \cite{noro:ANY}, Groebner basis computations +\cite{noro:REPL,noro:NOYO} primary ideal decomposition \cite{noro:SY} +and cryptgraphy \cite{noro:IKNY}. In these applications the important +funtions are polynomial factorization and Groebner basis +computation. We focus on Groebner basis computation and we review its +fundamentals and vaious efforts for improving efficiency especially +over $\Q$. Risa/Asir is also a main component of OpenXM package and +it has been used for parallel distributed computation with OpenXM API. +We will explain how one can execute parallel distributed computation +by using Risa/Asir as a client or a server. -\begin{enumerate} -\item Providing a platform for testing new algorithms +\section{Efficient Groebner basis computation over {\bf Q}} +\label{tab:gbtech} -Risa/Asir has been a platform for testing experimental algorithms in -polynomial factorization, Groebner basis computation, -cryptography and quantifier elimination. As to Groebner basis, we have -been mainly interested in problems over {\bf Q} and we tried applying -various modular techniques to overcome difficulties caused by huge -intermediate coefficients. We have had several results and they have -been implemented in Risa/Asir with other known methods. - -\item General purpose open system - -We need a lot of functions to make Risa/Asir a general purpose -computer algebra system. In recent years we can make use of various high -performance applications or libraries as free software. We wrapped -such software as OpenXM servers and we started to release a collection -of such servers and clients as the OpenXM package in 1997. Risa/Asir -is now a main client in the package. - -\item Environment for parallel and distributed computation - -The ancestor of OpenXM is a protocol designed for doing parallel -distributed computations by connecting multiple Risa/Asir's over -TCP/IP. OpenXM is also designed to provide an environment for -efficient parallel distributed computation. Currently only -client-server communication is available, but we are preparing a -specification OpenXM-RFC 102 allowing client-client communication, -which will enable us to execute wider range of parallel algorithms -requiring collective operations efficiently. -\end{enumerate} - -\subsection{Groebner basis and the related computation} - -Currently Risa/Asir can only deal with polynomial ring. Operations on -modules over polynomial rings have not yet supported. However, both -commutative polynomial rings and Weyl algebra are supported and one -can compute Groebner basis in both rings over {\bf Q}, fields of -rational functions and finite fields. In the early stage of our -development, our effort was mainly devoted to improve the efficiency -of computation over {\bf Q}. Our main tool is modular -computation. For Buchberger algorithm we adopted the trace lifting -algorithm by Traverso \cite{TRAV} and elaborated it by applying our -theory on a correspondence between Groebner basis and its modular -image \cite{NOYO}. We also combine the trace lifting with -homogenization to stabilize selection strategies, which enables us to -compute several examples efficiently which are hard to compute without -such a combination. Our modular method can be applied to the change -of ordering algorithm\cite{FGLM} and rational univariate -representation \cite{RUR}. We also made a test implementation of -$F_4$ algorithm \cite{F4}. In the later section we will show timing -data on Groebner basis computation. - -\subsection{Polynomial factorization} - -Here we briefly review functions on polynomial factorization. For -univariate factorization over {\bf Q}, the Berlekamp-Zassenhaus -algorithm is implemented. Efficient algorithms recently proposed have -not yet implemented. For univariate factorization over algebraic -number fields, Trager's algorithm \cite{TRAGER} is implemented with -some modifications. Its major applications are splitting field and -Galois group computation of polynomials over {\bf Q} \cite{ANY}. For -such purpose a tower of simple extensions are suitable because factors -represented over a simple extension often have huge coefficients. For -univariate factorization over finite fields, equal degree -factorization and Cantor-Zassenhaus algorithm are implemented. We can -use various representation of finite fields: $GF(p)$ with a machine -integer prime $p$, $GF(p)$ and $GF(p^n)$ with any odd prime $p$, -$GF(2^n)$ with a bit-array representation of polynomials over $GF(2)$ -and $GF(p^n)$ with small $p^n$ represented by a primitive root. For -multivariate factorization over {\bf Q}, the EZ(Extended Zassenhaus) -type algorithm is implemented. - -\subsection{Other functions} -By applying Groebner basis computation and polynomial factorization, -we have implemented several higher level algorithms. A typical -application is primary ideal decomposition of polynomial ideals over -{\bf Q}, which needs both functions. Shimoyama-Yokoyama algorithm -\cite{SY} for primary decomposition is written in the user language. -Splitting field and Galois group computation \cite{ANY} are closely -related and are also important applications of polynomial -factorization. - -\section{Techniques for efficient Groebner basis computation over {\bf Q}} -\label{gbtech} - In this section we review several practical techniques to improve Groebner basis computation over {\bf Q}, which are easily implemented but may not be well known. @@ -261,12 +211,13 @@ it practical. The following are major improvements: \item Useless pair detection We don't have to process all the pairs in $D$ and several useful -criteria for detecting useless pairs were proposed. +criteria for detecting useless pairs were proposed (cf. \cite{noro:BW}). \item Selection strategy The selection of $\{f,g\}$ greatly affects the subsequent computation. -The typical strategies are the normal startegy and the sugar strategy. +The typical strategies are the normal startegy \cite{noro:BUCH} +and the sugar strategy \cite{noro:SUGAR}. The latter was proposed for efficient computation under a non degree-compatible order. @@ -274,18 +225,18 @@ degree-compatible order. Even if we apply several criteria, it is difficult to detect all pairs whose S-polynomials are reduced to zero, and the cost to process them -often occupies a major part in the whole computation. The trace algorithms -were proposed to reduce such cost. This will be explained in more detail -in Section \ref{gbhomo}. +often occupies a major part in the whole computation. The trace +algorithms \cite{noro:TRAV} were proposed to reduce such cost. This +will be explained in more detail in Section \ref{sec:gbhomo}. \item Change of ordering For elimination, we need a Groebner basis with respect to a non -degree-compatible order, but it is often hard to compute it by -the Buchberger algorithm. If the ideal is zero dimensional, we -can apply a change of ordering algorithm for a Groebner basis -with respect to any order and we can obtain a Groebner basis -with respect to a desired order. +degree-compatible order, but it is often hard to compute it by the +Buchberger algorithm. If the ideal is zero dimensional, we can apply a +change of ordering algorithm \cite{noro:FGLM} for a Groebner basis +with respect to any order and we can obtain a Groebner basis with +respect to a desired order. \end{itemize} By implementing these techniques, one can obtain Groebner bases for @@ -294,10 +245,10 @@ problems with these classical tools. In the subsequent we show several methods for further improvements. \subsection{Combination of homogenization and trace lifting} -\label{gbhomo} +\label{sec:gbhomo} -Traverso's trace lifting algorithm can be -formulated in an abstract form as follows (c.f. \cite{FPARA}). +The trace lifting algorithm can be +formulated in an abstract form as follows (c.f. \cite{noro:FPARA}). \begin{tabbing} Input : a finite subset $F \subset {\bf Z}[X]$\\ Output : a Groebner basis $G$ of $Id(F)$ with respect to a term order $<$\\ @@ -336,7 +287,7 @@ lots of redundant elements can be removed. Let $I$ be a zero-dimensional ideal in $R={\bf Q}[x_1,\ldots,x_n]$. Then the minimal polynomial $m(x_i)$ of a variable $x_i$ in $R/I$ can -be computed by a partial FGLM \cite{FGLM}, but it often takes long +be computed by a partial FGLM \cite{noro:FGLM}, but it often takes long time if one searches $m(x_i)$ incrementally over {\bf Q}. In this case we can apply a simple modular method to compute the minimal polynomial. @@ -355,16 +306,16 @@ $GF(p)$ because $\phi_p(G)$ is a Groebner basis. Once candidate of $\deg(m(x_i))$, $m(x_i)$ can be determined by solving a system of linear equations via the method of indeterminate coefficient, and it can be solved efficiently by $p$-adic method. -Arguments on \cite{NOYO} ensures that $m(x_i)$ is what we want if it +Arguments on \cite{noro:NOYO} ensures that $m(x_i)$ is what we want if it exists. Note that the full FGLM can also be computed by the same method. \subsection{Integer contents reduction} -\label{gbcont} +\label{sec:gbcont} In some cases the cost to remove integer contents during normal form computations is dominant. We can remove the content of an integral -polynomial $f$ efficiently by the following method \cite{REPL}. +polynomial $f$ efficiently by the following method \cite{noro:REPL}. \begin{tabbing} Input : an integral polynomial $f$\\ Output : a pair $(\cont(f),f/\cont(f))$\\ @@ -386,38 +337,35 @@ $g_0$ with high accuracy. Then other components are ea %cost for reading basis elements from disk is often negligible because %of the cost for coefficient computations. -\section{Risa/Asir performance} +\subsection{Performances of Groebner basis computation} -We show timing data on Risa/Asir for Groebner basis computation -and polynomial factorization. The measurements were made on -a PC with PentiumIII 1GHz and 1Gbyte of main memory. Timings -are given in seconds. In the tables `---' means it was not -measured. +We show timing data on Risa/Asir for Groebner basis computation. The +measurements were made on a PC with PentiumIII 1GHz and 1Gbyte of main +memory. Timings are given in seconds. In the tables `---' means it was +not measured. $C_n$ is the cyclic $n$ system and $K_n$ is the Katsura +$n$ system, both are famous bench mark problems \cite{noro:BENCH}. $McKay$ +\cite{noro:REPL} is a system whose Groebner basis is hard to compute over +{\bf Q}. In Risa/Asir we have a test implemention of $F_4$ algorithm +\cite{noro:F4} and we also show its current performance. The term order is +graded reverse lexicographic order. -\subsection{Groebner basis computation} +Table \ref{tab:gbmod} shows timing data for Groebner basis computation +over $GF(32003)$. $F_4$ implementation in Risa/Asir outperforms +Buchberger algorithm implementation, but it is still several times +slower than $F_4$ implementation in FGb \cite{noro:FGB}. -Table \ref{gbmod} and Table \ref{gbq} show timing data for Groebner -basis computation over $GF(32003)$ and over {\bf Q} respectively. In -Table \ref{gbq} we compare the timing data under various configuration -of algorithms: with/without trace lifting, homogenization and contents -reduction. -$C_n$ is the cyclic $n$ system and $K_n$ is the Katsura $n$ system, -both are famous bench mark problems \cite{BENCH}. We also measured -the timing for $McKay$ system over {\bf Q} \cite{REPL}. the term -order is graded reverse lexicographic order. -As to the Buchberger algorithm over $GF(32003)$, -Singular\cite{SINGULAR}'s Buchberger algorithm implementation shows -better performance than Risa/Asir. $F_4$ implementation in Risa/Asir -is outperforms both of them, but it is still several times slower than -$F_4$ implementation in FGb \cite{FGB}. - -Table \ref{gbq} shows that it is difficult or practically impossible -to compute Groebner bases of $C_7$, $C_8$ and $McKay$ without -the methods described in Section \ref{gbhomo} and \ref{gbcont}. - +Table \ref{tab:gbq} shows timing data for Groebner basis computation over +$\Q$, where we compare the timing data under various configuration of +algorithms. {\bf TR}, {\bf Homo}, {\bf Cont} means trace lifting, +homogenization and contents reduction respectively. +\ref{tab:gbq} also shows timings of minimal polynomial +computation for zero-dimensional ideals. Table \ref{tab:gbq} shows that +it is difficult or practically impossible to compute Groebner bases of +$C_7$, $C_8$ and $McKay$ without the methods described in Section +\ref{sec:gbhomo} and \ref{sec:gbcont}. Though $F_4$ implementation in Risa/Asir over {\bf Q} is still experimental, the timing of $McKay$ is greatly reduced. -Fig. \ref{f4vsbuch} explains why $F_4$ is efficient in this case. The +Fig. \ref{tab:f4vsbuch} explains why $F_4$ is efficient in this case. The figure shows that the Buchberger algorithm produces normal forms with huge coefficients for S-polynomials after the 250-th one, which are the computations in degree 16. However, we know that the reduced @@ -437,26 +385,24 @@ FGb(estimated) & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \ \end{tabular} \end{center} \caption{Groebner basis computation over $GF(32003)$} -\label{gbmod} +\label{tab:gbmod} \end{table} - \begin{table}[hbtp] \begin{center} \begin{tabular}{|c||c|c|c|c|c|} \hline & $C_7$ & $C_8$ & $K_7$ & $K_8$ & $McKay$ \\ \hline -TR+Homo,Cont & 389 & 54000 & 29 & 299 & 34950 \\ \hline +TR+Homo+Cont & 389 & 54000 & 29 & 299 & 34950 \\ \hline TR+Homo & --- & --- & --- & --- & --- \\ \hline -TR & --- & --- & --- & --- & --- \\ \hline +TR & --- & --- & --- & --- & --- \\ \hline \hline +Minipoly & --- & --- & --- & --- & N/A \\ \hline %Singular & --- & 15247 & --- & 7.6 & 79 & $>$ 20h \\ \hline %CoCoA 4 & --- & 13227 & --- & 57 & 709 & --- \\ \hline\hline %Asir $F_4$ & 989 & 456 & --- & 90 & 991 & 4939 \\ \hline %FGb(estimated) & 8 &11 & 288 & 0.6 & 5 & 10 \\ \hline \end{tabular} - -(TR : trace lifting; Homo : homogenization; Cont : contents reduction) \end{center} -\caption{Groebner basis computation over {\bf Q}} -\label{gbq} +\caption{Groebner basis and minimal polynomial computation over {\bf Q}} +\label{tab:gbq} \end{table} \begin{figure}[hbtp] @@ -466,32 +412,32 @@ TR & --- & --- & --- & --- & --- \\ \hline \epsffile{blen.ps} \end{center} \caption{Maximal coefficient bit length of intermediate bases} -\label{f4vsbuch} +\label{tab:f4vsbuch} \end{figure} -Table \ref{minipoly} shows timing data for the minimal polynomial -computations of all variables over {\bf Q} by the modular method. -\begin{table}[hbtp] -\begin{center} -\begin{tabular}{|c||c|c|c|c|c|} \hline - & $C_6$ & $C_7$ & $K_6$ & $K_7$ & $K_8$ \\ \hline +%Table \ref{minipoly} shows timing data for the minimal polynomial +%computations of all variables over {\bf Q} by the modular method. +%\begin{table}[hbtp] +%\begin{center} +%\begin{tabular}{|c||c|c|c|c|c|} \hline +% & $C_6$ & $C_7$ & $K_6$ & $K_7$ & $K_8$ \\ \hline %Singular & 0.9 & 846 & 307 & 60880 & --- \\ \hline -Asir & 1.5 & 182 & 12 & 164 & 3420 \\ \hline -\end{tabular} -\end{center} -\caption{Minimal polynomial computation} -\label{minipoly} -\end{table} +%Asir & 1.5 & 182 & 12 & 164 & 3420 \\ \hline +%\end{tabular} +%\end{center} +%\caption{Minimal polynomial computation} +%\label{minipoly} +%\end{table} -\subsection{Polynomial factorization} - +%\subsection{Polynomial factorization} +% %Table \ref{unifac} shows timing data for univariate factorization over %{\bf Q}. $N_{i,j}$ is an irreducible polynomial which are hard to %factor by the classical algorithm. $N_{i,j}$ is a norm of a polynomial %and $\deg(N_i) = i$ with $j$ modular factors. Risa/Asir is %disadvantageous in factoring polynomials of this type because the %algorithm used in Risa/Asir has exponential complexity. In contrast, -%CoCoA 4\cite{COCOA} and NTL-5.2\cite{NTL} show nice performances +%CoCoA 4\cite{noro:COCOA} and NTL-5.2\cite{noro:NTL} show nice performances %because they implement recently developed algorithms. % %\begin{table}[hbtp] @@ -508,37 +454,37 @@ Asir & 1.5 & 182 & 12 & 164 & 3420 \\ \hline %\caption{Univariate factorization over {\bf Q}} %\label{unifac} %\end{table} - -Table \ref{multifac} shows timing data for multivariate factorization -over {\bf Q}. $W_{i,j,k}$ is a product of three multivariate -polynomials $Wang[i]$, $Wang[j]$, $Wang[k]$ given in a data file {\tt -fctrdata} in Asir library directory. It is also included in Risa/Asir -source tree and located in {\tt asir2000/lib}. These examples have -leading coefficients of large degree which vanish at 0 which tend to -cause so-called the leading coefficient problem the bad zero -problem. Risa/Asir's implementation carefully treats such cases and it -shows reasonable performance compared with other famous systems. -\begin{table}[hbtp] -\begin{center} -\begin{tabular}{|c||c|c|c|c|c|} \hline - & $W_{1,2,3}$ & $W_{4,5,6}$ & $W_{7,8,9}$ & $W_{10,11,12}$ & $W_{13,14,15}$ \\ \hline -variables & 3 & 5 & 5 & 5 & 4 \\ \hline -monomials & 905 & 41369 & 51940 & 30988 & 3344 \\ \hline\hline -Asir & 0.2 & 4.7 & 14 & 17 & 0.4 \\ \hline +% +%Table \ref{multifac} shows timing data for multivariate factorization +%over {\bf Q}. $W_{i,j,k}$ is a product of three multivariate +%polynomials $Wang[i]$, $Wang[j]$, $Wang[k]$ given in a data file {\tt +%fctrdata} in Asir library directory. It is also included in Risa/Asir +%source tree and located in {\tt asir2000/lib}. These examples have +%leading coefficients of large degree which vanish at 0 which tend to +%cause so-called the leading coefficient problem the bad zero +%problem. Risa/Asir's implementation carefully treats such cases and it +%shows reasonable performance compared with other famous systems. +%\begin{table}[hbtp] +%\begin{center} +%\begin{tabular}{|c||c|c|c|c|c|} \hline +% & $W_{1,2,3}$ & $W_{4,5,6}$ & $W_{7,8,9}$ & $W_{10,11,12}$ & $W_{13,14,15}$ \\ \hline +%variables & 3 & 5 & 5 & 5 & 4 \\ \hline +%monomials & 905 & 41369 & 51940 & 30988 & 3344 \\ \hline\hline +%Asir & 0.2 & 4.7 & 14 & 17 & 0.4 \\ \hline %Singular& $>$15min & --- & ---& ---& ---\\ \hline %CoCoA 4 & 5.2 & $>$15min & $>$15min & $>$15min & 117 \\ \hline\hline -Mathematica 4& 0.2 & 16 & 23 & 36 & 1.1 \\ \hline -Maple 7& 0.5 & 18 & 967 & 48 & 1.3 \\ \hline -\end{tabular} -\end{center} -\caption{Multivariate factorization over {\bf Q}} -\label{multifac} -\end{table} -As to univariate factorization over {\bf Q}, the univariate factorizer -implements old algorithms and its behavior is what one expects, -that is, it shows average performance in cases where there are little -extraneous factors, but shows poor performance for hard to factor -polynomials with many extraneous factors. +%Mathematica 4& 0.2 & 16 & 23 & 36 & 1.1 \\ \hline +%Maple 7& 0.5 & 18 & 967 & 48 & 1.3 \\ \hline +%\end{tabular} +%\end{center} +%\caption{Multivariate factorization over {\bf Q}} +%\label{multifac} +%\end{table} +%As to univariate factorization over {\bf Q}, the univariate factorizer +%implements old algorithms and its behavior is what one expects, +%that is, it shows average performance in cases where there are little +%extraneous factors, but shows poor performance for hard to factor +%polynomials with many extraneous factors. \section{OpenXM and Risa/Asir OpenXM interfaces} @@ -546,10 +492,10 @@ polynomials with many extraneous factors. OpenXM stands for Open message eXchange protocol for Mathematics. From the viewpoint of protocol design, it can be regarded as a child -of OpenMath \cite{OPENMATH}. However our approach is somewhat +of OpenMath \cite{noro:OPENMATH}. However our approach is somewhat different. Our main purpose is to provide an environment for integrating {\it existing} mathematical software systems. OpenXM -RFC-100 \cite{RFC100} defines a client-server architecture. Under +RFC-100 \cite{noro:RFC100} defines a client-server architecture. Under this specification, a client invokes an OpenXM ({\it OX}) server. The client can send OpenXM ({\it OX}) messages to the server. OX messages consist of {\it data} and {\it command}. Data is encoded according to @@ -566,7 +512,7 @@ hybrid server. OpenXM RFC-100 also defines methods for session management. In particular the method to reset a server is carefully designed and it provides a robust way of using servers both for interactive and non-interactive -purposes. +purposes. \subsection{OpenXM API in Risa/Asir user language} @@ -662,106 +608,120 @@ readable form, which may be sufficient for a simple us interface. \section{Concluding remarks} -We have shown the current status of Risa/Asir and its OpenXM -interfaces. As a result of our policy of development, it is true that -Risa/Asir does not have abundant functions. However it is a completely -open system and its total performance is not bad. Especially on -Groebner basis computation over {\bf Q}, many techniques for improving -practical performances have been implemented. As the OpenXM interface -specification is completely documented, we can easily add another -function to Risa/Asir by wrapping an existing software system as an OX -server, and other clients can call functions in Risa/Asir by -implementing the OpenXM client interface. With the remote debugging -and the function to reset servers, one will be able to enjoy parallel -and distributed computation with OpenXM facilities. +%We have shown the current status of Risa/Asir and its OpenXM +%interfaces. As a result of our policy of development, it is true that +%Risa/Asir does not have abundant functions. However it is a completely +%open system and its total performance is not bad. Especially on +%Groebner basis computation over {\bf Q}, many techniques for improving +%practical performances have been implemented. As the OpenXM interface +%specification is completely documented, we can easily add another +%function to Risa/Asir by wrapping an existing software system as an OX +%server, and other clients can call functions in Risa/Asir by +%implementing the OpenXM client interface. With the remote debugging +%and the function to reset servers, one will be able to enjoy parallel +%and distributed computation with OpenXM facilities. % +We have shown that many techniques for +improving practical performances are implemented in Risa/Asir's +Groebner basis engine. Though another important function, the +polynomial factorizer only implements classical algorithms, its +performance is comparable with or superior to that of Maple or +Mathematica and is still practically useful. By preparing OpenXM +interface or simply linking the Asir OpenXM library, one can call +these efficient functions from any application. Risa/Asir is a +completely open system. It is open source software +and the OpenXM interface specification is completely documented, one +can easily write interfaces to call functions in Risa/Asir and one +will be able to enjoy parallel and distributed computation. + + \begin{thebibliography}{7} % \addcontentsline{toc}{section}{References} -\bibitem{ANY} +\bibitem{noro:ANY} Anay, H., Noro, M., Yokoyama, K. (1996) Computation of the Splitting fields and the Galois Groups of Polynomials. Algorithms in Algebraic geometry and Applications, Birkh\"auser (Proceedings of MEGA'94), 29--50. -\bibitem{FPARA} +\bibitem{noro:FPARA} Jean-Charles Faug\`ere (1994) Parallelization of Groebner basis. Proceedings of PASCO'94, 124--132. -\bibitem{F4} +\bibitem{noro:F4} Jean-Charles Faug\`ere (1999) A new efficient algorithm for computing Groebner bases ($F_4$). Journal of Pure and Applied Algebra (139) 1-3 , 61--88. -\bibitem{FGLM} +\bibitem{noro:FGLM} Faug\`ere, J.-C. et al. 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