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% $OpenXM: OpenXM/doc/Papers/dag-noro-proc.tex,v 1.8 2001/11/30 02:08:46 noro Exp $

% $OpenXM: OpenXM/doc/Papers/dag-noro-proc.tex,v 1.9 2001/12/28 06:06:15 noro Exp $

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% This is a sample input file for your contribution to a multi-

% author book to be published by Springer Verlag.

Line 117 Risa/Asir is not only a standalone computer algebra sy

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 : as a client, as a

Risa/Asir has three kinds of OpenXM interfaces :

server, and as a subroutine library. Our goals of developing Risa/Asir

OpenXM API in the Risa/Asir user language,

are as follows:

OpenXM C language API in the Risa/Asir subroutine library,

and an OpenXM server.

Our goals of developing Risa/Asir are as follows:

\begin{enumerate}

\item Providing a platform for testing new algorithms

Line 177 data on Groebner basis computation.

Line 179 data on Groebner basis computation.

\subsection{Polynomial factorization}

Here we briefly review functions on polynomial factorization. For

univariate factorization over {\bf Q}, the classical

univariate factorization over {\bf Q}, the Berlekamp-Zassenhaus

Berlekamp-Zassenhaus algorithm is implemented. Efficient algorithms

algorithm is implemented. Efficient algorithms recently proposed have

recently proposed have not yet implemented. For univariate

not yet implemented. For univariate factorization over algebraic

factorization over algebraic number fields, Trager's algorithm

number fields, Trager's algorithm \cite{TRAGER} is implemented with

\cite{TRAGER} is implemented with some modifications. Its major

some modifications. Its major applications are splitting field and

applications are splitting field and Galois group computation of

Galois group computation of polynomials over {\bf Q} \cite{ANY}. For

polynomials over {\bf Q} \cite{ANY}. For such purpose a tower of

such purpose a tower of simple extensions are suitable because factors

simple extensions are suitable because factors represented over a

represented over a simple extension often have huge coefficients. For

simple extension often have huge coefficients. For univariate

univariate factorization over finite fields, equal degree

factorization over finite fields, equal degree factorization and

factorization and Cantor-Zassenhaus algorithm are implemented. We can

Cantor-Zassenhaus algorithm are implemented. We can use various

use various representation of finite fields: $GF(p)$ with a machine

representation of finite fields: $GF(p)$ with a machine integer prime

integer prime $p$, $GF(p)$ and $GF(p^n)$ with any odd prime $p$,

$p$, $GF(p)$ and $GF(p^n)$ with any odd prime $p$, $GF(2^n)$ with a

$GF(2^n)$ with a bit-array representation of polynomials over $GF(2)$

bit-array representation of polynomials over $GF(2)$ and $GF(p^n)$

and $GF(p^n)$ with small $p^n$ represented by a primitive root. For

with small $p^n$ represented by a primitive root. For multivariate

multivariate factorization over {\bf Q}, the EZ(Extended Zassenhaus)

factorization over {\bf Q}, the classical EZ(Extended

type algorithm is implemented.

Zassenhaus) type algorithm is implemented.

\subsection{Other functions}

By applying Groebner basis computation and polynomial factorization,

Line 396 measured.

Line 397 measured.

\subsection{Groebner basis computation}

Table \ref{gbmod} and Table \ref{gbq} show timing data for Groebner

basis computation over $GF(32003)$ and over {\bf Q} respectively.

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. In the both tables, the

order is graded reverse lexicographic order.

first three rows are timings for the Buchberger algorithm, and the

As to the Buchberger algorithm over $GF(32003)$,

last two rows are timings for $F_4$ algorithm. As to the Buchberger

Singular\cite{SINGULAR}'s Buchberger algorithm implementation shows

algorithm over $GF(32003)$, Singular\cite{SINGULAR} shows the best

better performance than Risa/Asir. $F_4$ implementation in Risa/Asir

performance among the three systems. $F_4$ implementation in Risa/Asir

is outperforms both of them, but it is still several times slower than

is faster than the Buchberger algorithm implementation in Singular,

$F_4$ implementation in FGb \cite{FGB}.

but it is still several times slower than $F_4$ implementation in FGb

\cite{FGB}. In Table \ref{gbq}, Risa/Asir computed $C_7$ and $McKay$

by the Buchberger algorithm with the methods described in Section

\ref{gbhomo} and \ref{gbcont}. It is obvious that $F_4$

implementation in Risa/Asir over {\bf Q} is too immature. Nevertheless

the timing of $McKay$ is greatly reduced. Fig. \ref{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 basis elements have much

smaller coefficients after removing contents. As $F_4$ algorithm

automatically produces the reduced ones, the degree 16 computation is

quite easy in $F_4$.

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}.

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

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

basis elements have much smaller coefficients after removing contents.

As $F_4$ algorithm automatically produces the reduced ones, the degree

16 computation is quite easy in $F_4$.

\begin{table}[hbtp]

\begin{center}

\begin{tabular}{|c||c|c|c|c|c|c|c|} \hline

& $C_7$ & $C_8$ & $K_7$ & $K_8$ & $K_9$ & $K_{10}$ & $K_{11}$ \\ \hline

Asir $Buchberger$ & 31 & 1687 & 2.6 & 27 & 294 & 4309 & --- \\ \hline

Singular & 8.7 & 278 & 0.6 & 5.6 & 54 & 508 & 5510 \\ \hline

%Singular & 8.7 & 278 & 0.6 & 5.6 & 54 & 508 & 5510 \\ \hline

CoCoA 4 & 241 & $>$ 5h & 3.8 & 35 & 402 &7021 & --- \\ \hline\hline

%CoCoA 4 & 241 & $>$ 5h & 3.8 & 35 & 402 &7021 & --- \\ \hline\hline

Asir $F_4$ & 5.3 & 129 & 0.5 & 4.5 & 31 & 273 & 2641 \\ \hline

FGb(estimated) & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \\ \hline

\end{tabular}

Line 437 FGb(estimated) & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \

Line 442 FGb(estimated) & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \

\begin{table}[hbtp]

\begin{center}

\begin{tabular}{|c||c|c|c|c|c|c|} \hline

\begin{tabular}{|c||c|c|c|c|c|} \hline

& $C_7$ & $Homog. C_7$ & $C_8$ & $K_7$ & $K_8$ & $McKay$ \\ \hline

& $C_7$ & $C_8$ & $K_7$ & $K_8$ & $McKay$ \\ \hline

Asir $Buchberger$ & 389 & 594 & 54000 & 29 & 299 & 34950 \\ \hline

TR+Homo,Cont & 389 & 54000 & 29 & 299 & 34950 \\ \hline

Singular & --- & 15247 & --- & 7.6 & 79 & $>$ 20h \\ \hline

TR+Homo & --- & --- & --- & --- & --- \\ \hline

CoCoA 4 & --- & 13227 & --- & 57 & 709 & --- \\ \hline\hline

TR & --- & --- & --- & --- & --- \\ \hline

Asir $F_4$ & 989 & 456 & --- & 90 & 991 & 4939 \\ \hline

%Singular & --- & 15247 & --- & 7.6 & 79 & $>$ 20h \\ \hline

FGb(estimated) & 8 &11 & 288 & 0.6 & 5 & 10 \\ \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}

Line 461 FGb(estimated) & 8 &11 & 288 & 0.6 & 5 & 10 \\ \hline

Line 470 FGb(estimated) & 8 &11 & 288 & 0.6 & 5 & 10 \\ \hline

\end{figure}

Table \ref{minipoly} shows timing data for the minimal polynomial

computation over {\bf Q}. Singular provides a function {\tt finduni}

computations of all variables over {\bf Q} by the modular method.

for computing the minimal polynomial in each variable in ${\bf

Q}[x_1,\ldots,x_n]/I$ for zero dimensional ideal $I$. The modular

method used in Asir is efficient when the resulting minimal

polynomials have large coefficients and we can verify the fact from Table

\ref{minipoly}.

\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

%Singular & 0.9 & 846 & 307 & 60880 & --- \\ \hline

Asir & 1.5 & 182 & 12 & 164 & 3420 \\ \hline

\end{tabular}

\end{center}

Line 505 Asir & 1.5 & 182 & 12 & 164 & 3420 \\ \hline

Line 509 Asir & 1.5 & 182 & 12 & 164 & 3420 \\ \hline

%\label{unifac}

%\end{table}

Table \ref{multifac} shows timing data for multivariate

Table \ref{multifac} shows timing data for multivariate factorization

factorization over {\bf Q}.

over {\bf Q}. $W_{i,j,k}$ is a product of three multivariate

$W_{i,j,k}$ is a product of three multivariate polynomials

polynomials $Wang[i]$, $Wang[j]$, $Wang[k]$ given in a data file {\tt

$Wang[i]$, $Wang[j]$, $Wang[k]$ given in a data file

fctrdata} in Asir library directory. It is also included in Risa/Asir

{\tt fctrdata} in Asir library directory. It is also included

source tree and located in {\tt asir2000/lib}. These examples have

in Risa/Asir source tree and located in {\tt asir2000/lib}.

leading coefficients of large degree which vanish at 0 which tend to

For these examples Risa/Asir shows reasonable performance

cause so-called the leading coefficient problem the bad zero

compared with other famous systems.

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

Line 521 variables & 3 & 5 & 5 & 5 & 4 \\ \hline

Line 526 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

%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}

Line 529 Maple 7& 0.5 & 18 & 967 & 48 & 1.3 \\ \hline

Line 534 Maple 7& 0.5 & 18 & 967 & 48 & 1.3 \\ \hline

\caption{Multivariate factorization over {\bf Q}}

\label{multifac}

\end{table}

As to univariate factorization over {\bf Q},

As to univariate factorization over {\bf Q}, the univariate factorizer

the univariate factorizer implements only classical

implements old algorithms and its behavior is what one expects,

algorithms and its behavior is what one expects,

that is, it shows average performance in cases where there are little

that is, it shows average performance in cases

extraneous factors, but shows poor performance for hard to factor

where there are little extraneous factors, but

polynomials with many extraneous factors.

shows poor performance for hard to factor polynomials with

many extraneous factors.

\section{OpenXM and Risa/Asir OpenXM interfaces}

Line 565 the method to reset a server is carefully designed and

Line 568 the method to reset a server is carefully designed and

a robust way of using servers both for interactive and non-interactive

purposes.

\subsection{OpenXM client interface of {\tt asir}}

\subsection{OpenXM API in Risa/Asir user language}

Risa/Asir is a main client in OpenXM package. The application {\tt

asir} can access to OpenXM servers via several built-in interface

Line 644 def gbcheck(B,V,O,Procs) {

Line 647 def gbcheck(B,V,O,Procs) {

}

\end{verbatim}

\subsection{Asir OpenXM library {\tt libasir.a}}

\subsection{OpenXM C language API in {\tt libasir.a}}

Asir OpenXM library {\tt libasir.a} contains functions simulating the

Risa/Asir subroutine library {\tt libasir.a} contains functions

stack machine commands supported in {\tt ox\_asir}. By linking {\tt

simulating the stack machine commands supported in {\tt ox\_asir}. By

libasir.a} an application can use the same functions as in {\tt

linking {\tt libasir.a} an application can use the same functions as

ox\_asir} without accessing to {\tt ox\_asir} via TCP/IP. There is

in {\tt ox\_asir} without accessing to {\tt ox\_asir} via

also a stack, which can be manipulated by the library functions. In

TCP/IP. There is also a stack, which can be manipulated by the library

order to make full use of this interface, one has to prepare

functions. In order to make full use of this interface, one has to

conversion functions between CMO and the data structures proper to the

prepare conversion functions between CMO and the data structures

application itself. A function {\tt asir\_ox\_pop\_string()} is

proper to the application itself. A function {\tt

provided to convert CMO to a human readable form, which may be

asir\_ox\_pop\_string()} is provided to convert CMO to a human

sufficient for a simple use of this interface.

readable form, which may be sufficient for a simple use of this

interface.

\section{Concluding remarks}

We have shown the current status of Risa/Asir and its OpenXM

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