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version 1.2, 2001/11/19 10:00:02

version 1.3, 2001/11/26 08:41:14

Line 1

% $OpenXM: OpenXM/doc/Papers/dag-noro-proc.tex,v 1.1 2001/11/19 01:02:30 noro Exp $

% $OpenXM$

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

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

% author book to be published by Springer Verlag.

Line 335 Table \ref{gbq}, $C_7$ and $McKay$ can be computed by

algorithm with the methods described in Section \ref{gbtech}. It is

obvious that $F_4$ implementation in Risa/Asir over {\bf Q} is too

immature. Nevertheless the timing of $McKay$ is greatly reduced.

Why is $F_4$ efficient in this case? The answer is in the right

Fig. \ref{f4vsbuch} explains why $F_4$ is efficient in this case.

half of Fig. \ref{f4vsbuch}. During processing S-polynomials of degree

The figure shows that

16, the Buchberger algorithm produces intermediate polynomials with

the Buchberger algorithm produces normal forms with

huge coefficients, but if we compute normal forms of these polynomials

huge coefficients for S-polynomals after the 250-th one,

by using all subsequently generated basis elements, then their

which are the computations in degree 16.

coefficients will be reduced after removing contents. As $F_4$

However, we know that the reduced basis elements have

algorithm automatically produces the reduced basis elements, the

much smaller coefficients after removing contents.

degree 16 computation is quite easy in $F_4$.

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

Line 378 FGb(estimated) & 8 &11 & 0.6 & 5 & 10 \\ \hline

\begin{figure}[hbtp]

\begin{center}

\epsfxsize=12cm

\epsffile{../compalg/ps/blenall.ps}

\epsffile{blenall.ps}

\end{center}

\caption{Maximal coefficient bit length of intermediate bases}

\label{f4vsbuch}

Line 386 FGb(estimated) & 8 &11 & 0.6 & 5 & 10 \\ \hline

\subsection{Polynomial factorization}

Table \ref{unifac} shows timing data for univariate factorization over

%Table \ref{unifac} shows timing data for univariate factorization over

{\bf Q}. $N_{i,j}$ is an irreducible polynomial which are hard to

%{\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

%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

%and $\deg(N_i) = i$ with $j$ modular factors. Risa/Asir is

disadvantageous in factoring polynomials of this type because the

%disadvantageous in factoring polynomials of this type because the

algorithm used in Risa/Asir has exponential complexity. In contrast,

%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{COCOA} and NTL-5.2\cite{NTL} show nice performances

because they implement recently developed algorithms.

%because they implement recently developed algorithms.

%\begin{table}[hbtp]

%\begin{center}

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

% & $N_{105,23}$ & $N_{120,20}$ & $N_{168,24}$ & $N_{210,54}$ \\ \hline

%Asir & 0.86 & 59 & 840 & hard \\ \hline

%Asir NormFactor & 1.6 & 2.2& 6.1& hard \\ \hline

%%Singular& hard? & hard?& hard? & hard? \\ \hline

%CoCoA 4 & 0.2 & 7.1 & 16 & 0.5 \\ \hline\hline

%NTL-5.2 & 0.16 & 0.9 & 1.4 & 0.4 \\ \hline

%\end{tabular}

%\end{center}

%\caption{Univariate factorization over {\bf Q}}

%\label{unifac}

%\end{table}

\begin{table}[hbtp]

\begin{center}

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

& $N_{105,23}$ & $N_{120,20}$ & $N_{168,24}$ & $N_{210,54}$ \\ \hline

Asir & 0.86 & 59 & 840 & hard \\ \hline

Asir NormFactor & 1.6 & 2.2& 6.1& hard \\ \hline

%Singular& hard? & hard?& hard? & hard? \\ \hline

CoCoA 4 & 0.2 & 7.1 & 16 & 0.5 \\ \hline\hline

NTL-5.2 & 0.16 & 0.9 & 1.4 & 0.4 \\ \hline

\end{tabular}

\end{center}

\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

Line 436 Maple 7& 0.5 & 18 & 967 & 48 & 1.3 \\ \hline

\label{multifac}

\end{table}

As to univariate factorization over {\bf Q},

the univariate factorizer implements only classical

algorithms and its behaviour is what one expects,

that is, it shows average performance in cases

where there are little eraneous factors, but

shows poor performance for hard to factor polynomials.

\section{OpenXM and Risa/Asir OpenXM interfaces}

\subsection{OpenXM overview}

Line 547 def gbcheck(B,V,O,Procs) {

Line 554 def gbcheck(B,V,O,Procs) {

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

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

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

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

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

also a stack and library functions to manipulate it. In order to make

full use of this interface, one has to prepare conversion functions

between CMO and the data structures proper to the application.

A function {\tt asir\_ox\_pop\_string()} is provided to convert

CMO to a human 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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