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1.10    ! takayama    1: % $OpenXM: OpenXM/doc/issac2000/heterotic-network.tex,v 1.9 2000/01/16 06:39:39 takayama Exp $
1.7       takayama    2: \section{Applications}
1.2       takayama    3:
1.4       takayama    4: \subsection{Heterogeneous Servers}
1.2       takayama    5:
                      6: \def\pd#1{ \partial_{#1} }
1.4       takayama    7:
                      8: By using OpenXM, we can treat OpenXM servers essentially
                      9: like a subroutine.
                     10: Since OpenXM provides a universal stackmachine which does not
                     11: depend each servers,
                     12: it is relatively easy to install new servers.
                     13: We can build a new computer math system by assembling
                     14: different OpenXM servers.
                     15: It is similar to building a toy house by LEGO blocks.
                     16:
1.10    ! takayama   17: We will see two examples of custom-made systems
1.4       takayama   18: built by OpenXM servers.
                     19:
                     20: \subsubsection{Computation of annihilating ideals by kan/sm1 and ox\_asir}
1.2       takayama   21:
                     22: Let $D = {\bf Q} \langle x_1, \ldots, x_n , \pd{1}, \ldots, \pd{n} \rangle$
                     23: be the ring of differential operators.
                     24: For a given polynomial
                     25: $ f \in {\bf Q}[x_1, \ldots, x_n] $,
                     26: the annihilating ideal of $f^{-1}$ is defined as
                     27: $$ {\rm Ann}\, f^{-1} = \{ \ell \in D \,|\,
                     28:   \ell \bullet f^{-1} = 0 \}.
                     29: $$
                     30: Here, $\bullet$ denotes the action of $D$ to functions.
                     31: The annihilating ideal can be regarded as the maximal differential
                     32: equations for the function $f^{-1}$.
                     33: An algorithm to determine generators of the annihilating ideal
1.4       takayama   34: was given by Oaku (see, e.g., \cite[5.3]{sst-book}).
1.2       takayama   35: His algorithm reduces the problem to computations of Gr\"obner bases
                     36: in $D$ and to find the maximal integral root of a polynomial.
1.4       takayama   37: This algorithm (the function {\tt annfs}) is implemented by
                     38: kan/sm1 \cite{kan}, for Gr\"obner basis computation in $D$, and
1.9       takayama   39: {\tt ox\_asir}, to factorize polynomials to find the integral
1.2       takayama   40: roots.
1.10    ! takayama   41: These two OpenXM compliant systems are integrated by
        !            42: the OpenXM protocol.
1.4       takayama   43:
1.3       takayama   44: For example, the following is a sm1 session to find the annihilating
1.2       takayama   45: ideal for $f = x^3 - y^2 z^2$.
                     46: \begin{verbatim}
                     47: sm1>[(x^3-y^2 z^2) (x,y,z)] annfs ::
                     48: Starting ox_asir server.
                     49: Byte order for control process is network byte order.
                     50: Byte order for engine process is network byte order.
                     51: [[-y*Dy+z*Dz, 2*x*Dx+3*y*Dy+6, -2*y*z^2*Dx-3*x^2*Dy,
1.6       takayama   52: -2*y^2*z*Dx-3*x^2*Dz, -2*z^3*Dx*Dz-3*x^2*Dy^2-2*z^2*Dx],
1.2       takayama   53:  [-1,-139968*s^7-1119744*s^6-3802464*s^5-7107264*s^4
                     54:      -7898796*s^3-5220720*s^2-1900500*s-294000]]
                     55: \end{verbatim}
                     56: The last polynomial is factored as
1.3       takayama   57: $-12(s+1)(3s+5)(3s+4)(6s+5)(6s+7)$
1.2       takayama   58: and the minimal integral root is $-1$
                     59: as shown in the output.
                     60:
1.4       takayama   61: Similarly,
                     62: an algorithm to stratify singularity
                     63: \cite{oaku-advance}
                     64: is implemented by
                     65: kan/sm1 \cite{kan}, for Gr\"obner basis computation in $D$, and
1.9       takayama   66: {\tt ox\_asir}, for primary ideal decompositions.
1.2       takayama   67:
                     68: \subsubsection{A Course on Solving Algebraic Equations}
                     69:
1.9       takayama   70: Risa/Asir \cite{asir} is a general computer algebra system
                     71: which can be used for Gr\"obner basis computations for zero dimensional ideal
1.2       takayama   72: with ${\bf Q}$ coefficients.
                     73: However, it is not good at graphical presentations and
                     74: numerical methods.
1.9       takayama   75: We integrated Risa/Asir, ox\_phc (based on PHC pack by Verschelde \cite{phc}
1.2       takayama   76: for the polyhedral homotopy method) and
1.4       takayama   77: ox\_gnuplot (GNUPLOT) servers
1.2       takayama   78: to teach a course on solving algebraic equations.
1.9       takayama   79: This course was presented with the text book \cite{CLO},
                     80: which discusses
1.2       takayama   81: on the Gr\"obner basis method and the polyhedral homotopy method
                     82: to solve systems of algebraic equations.
1.10    ! takayama   83: We taught the course
1.2       takayama   84: with a unified environment
1.9       takayama   85: controlled by Asir user language, which is similar to C.
                     86: The following is an Asir session to solve algebraic equations by calling
                     87: the PHC pack (Figure \ref{katsura} is the output of {\tt [292]}):
1.2       takayama   88: \begin{verbatim}
1.7       takayama   89: [287] phc(katsura(7));
1.2       takayama   90: The detailed output is in the file tmp.output.*
                     91: The answer is in the variable Phc.
                     92: 0
1.7       takayama   93: [290] B=map(first,Phc)$
                     94: [291] gnuplot_plotDots([],0)$
                     95: [292] gnuplot_plotDots(B,0)$
1.2       takayama   96: \end{verbatim}
1.7       takayama   97:
                     98: \begin{figure}[htbp]
                     99: \epsfxsize=8.5cm
                    100: \epsffile{katsura7.ps}
                    101: \caption{The first components of the solutions to the system of algebraic equations Katsura 7.}
                    102: \label{katsura}
                    103: \end{figure}
                    104:
1.2       takayama  105:
                    106:
                    107: