[BACK]Return to heterotic-network.tex CVS log [TXT][DIR] Up to [local] / OpenXM / doc / issac2000

Diff for /OpenXM/doc/issac2000/heterotic-network.tex between version 1.1 and 1.12

version 1.1, 1999/12/23 10:25:08 version 1.12, 2000/01/17 08:06:15
Line 1 
Line 1 
 % $OpenXM$  % $OpenXM: OpenXM/doc/issac2000/heterotic-network.tex,v 1.11 2000/01/17 07:06:53 noro Exp $
   \section{Applications}
   
   \subsection{Heterogeneous Servers}
   
   \def\pd#1{ \partial_{#1} }
   
   By using OpenXM, we can treat OpenXM servers essentially
   like a subroutine.
   Since OpenXM provides a universal stack machine which does not
   depend each servers,
   it is relatively easy to install new servers.
   We can build a new computer math system by assembling
   different OpenXM servers.
   It is similar to building a toy house by LEGO blocks.
   
   We will see two examples of custom-made systems
   built by OpenXM servers.
   
   \subsubsection{Computation of annihilating ideals by kan/sm1 and ox\_asir}
   
   Let $D = {\bf Q} \langle x_1, \ldots, x_n , \pd{1}, \ldots, \pd{n} \rangle$
   be the ring of differential operators.
   For a given polynomial
   $ f \in {\bf Q}[x_1, \ldots, x_n] $,
   the annihilating ideal of $f^{-1}$ is defined as
   $$ {\rm Ann}\, f^{-1} = \{ \ell \in D \,|\,
     \ell \bullet f^{-1} = 0 \}.
   $$
   Here, $\bullet$ denotes the action of $D$ to functions.
   The annihilating ideal can be regarded as the maximal differential
   equations for the function $f^{-1}$.
   An algorithm to determine generators of the annihilating ideal
   was given by Oaku (see, e.g., \cite[5.3]{sst-book}).
   His algorithm reduces the problem to computations of Gr\"obner bases
   in $D$ and to find the minimal integral root of a polynomial.
   This algorithm (the function {\tt annfs}) is implemented by
   kan/sm1 \cite{kan}, for Gr\"obner basis computation in $D$, and
   {\tt ox\_asir}, to factorize polynomials to find the integral
   roots.
   These two OpenXM compliant systems are integrated by
   the OpenXM protocol.
   
   For example, the following is a sm1 session to find the annihilating
   ideal for $f = x^3 - y^2 z^2$.
   \begin{verbatim}
   sm1>[(x^3-y^2 z^2) (x,y,z)] annfs ::
   Starting ox_asir server.
   Byte order for control process is network byte order.
   Byte order for engine process is network byte order.
   [[-y*Dy+z*Dz, 2*x*Dx+3*y*Dy+6, -2*y*z^2*Dx-3*x^2*Dy,
   -2*y^2*z*Dx-3*x^2*Dz, -2*z^3*Dx*Dz-3*x^2*Dy^2-2*z^2*Dx],
    [-1,-139968*s^7-1119744*s^6-3802464*s^5-7107264*s^4
        -7898796*s^3-5220720*s^2-1900500*s-294000]]
   \end{verbatim}
   The last polynomial is factored as
   $-12(s+1)(3s+5)(3s+4)(6s+5)(6s+7)$
   and the minimal integral root is $-1$
   as shown in the output.
   
   Similarly,
   an algorithm to stratify singularity
   \cite{oaku-advance}
   is implemented by
   kan/sm1 \cite{kan}, for Gr\"obner basis computation in $D$, and
   {\tt ox\_asir}, for primary ideal decompositions.
   
   \subsubsection{A Course on Solving Algebraic Equations}
   
   Risa/Asir \cite{asir} is a general computer algebra system
   which can be used for Gr\"obner basis computations for zero dimensional ideal
   with ${\bf Q}$ coefficients.
   However, it is not good at graphical presentations and
   numerical methods.
   We integrated Risa/Asir, ox\_phc (based on PHC pack by Verschelde \cite{phc}
   for the polyhedral homotopy method) and
   ox\_gnuplot (GNUPLOT) servers
   to teach a course on solving algebraic equations.
   This course was presented with the text book \cite{CLO},
   which discusses
   on the Gr\"obner basis method and the polyhedral homotopy method
   to solve systems of algebraic equations.
   We taught the course
   with a unified environment
   controlled by Asir user language, which is similar to C.
   The following is an Asir session to solve algebraic equations by calling
   the PHC pack (Figure \ref{katsura} is the output of {\tt [292]}):
   \begin{verbatim}
   [287] phc(katsura(7));
   The detailed output is in the file tmp.output.*
   The answer is in the variable Phc.
   0
   [290] B=map(first,Phc)$
   [291] gnuplot_plotDots([],0)$
   [292] gnuplot_plotDots(B,0)$
   \end{verbatim}
   
   \begin{figure}[htbp]
   \epsfxsize=8.5cm
   \epsffile{katsura7.ps}
   \caption{The first components of the solutions to the system of algebraic equations Katsura 7.}
   \label{katsura}
   \end{figure}
   
   
   
   
Legend:
Removed from v.1.1  
changed lines
  Added in v.1.12
FreeBSD-CVSweb <freebsd-cvsweb@FreeBSD.org>