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Annotation of OpenXM/doc/issac2000/heterotic-network.tex, Revision 1.6

1.6     ! takayama    1: % $OpenXM: OpenXM/doc/issac2000/heterotic-network.tex,v 1.5 2000/01/15 03:23:59 takayama Exp $
1.2       takayama    2:
1.4       takayama    3: \subsection{Heterogeneous Servers}
1.2       takayama    4:
                      5: \def\pd#1{ \partial_{#1} }
1.4       takayama    6:
                      7: By using OpenXM, we can treat OpenXM servers essentially
                      8: like a subroutine.
                      9: Since OpenXM provides a universal stackmachine which does not
                     10: depend each servers,
                     11: it is relatively easy to install new servers.
                     12: We can build a new computer math system by assembling
                     13: different OpenXM servers.
                     14: It is similar to building a toy house by LEGO blocks.
                     15:
1.5       takayama   16: We will see two examples of custom made systems
1.4       takayama   17: built by OpenXM servers.
                     18:
                     19: \subsubsection{Computation of annihilating ideals by kan/sm1 and ox\_asir}
1.2       takayama   20:
                     21: Let $D = {\bf Q} \langle x_1, \ldots, x_n , \pd{1}, \ldots, \pd{n} \rangle$
                     22: be the ring of differential operators.
                     23: For a given polynomial
                     24: $ f \in {\bf Q}[x_1, \ldots, x_n] $,
                     25: the annihilating ideal of $f^{-1}$ is defined as
                     26: $$ {\rm Ann}\, f^{-1} = \{ \ell \in D \,|\,
                     27:   \ell \bullet f^{-1} = 0 \}.
                     28: $$
                     29: Here, $\bullet$ denotes the action of $D$ to functions.
                     30: The annihilating ideal can be regarded as the maximal differential
                     31: equations for the function $f^{-1}$.
                     32: An algorithm to determine generators of the annihilating ideal
1.4       takayama   33: was given by Oaku (see, e.g., \cite[5.3]{sst-book}).
1.2       takayama   34: His algorithm reduces the problem to computations of Gr\"obner bases
                     35: in $D$ and to find the maximal integral root of a polynomial.
1.4       takayama   36: This algorithm (the function {\tt annfs}) is implemented by
                     37: kan/sm1 \cite{kan}, for Gr\"obner basis computation in $D$, and
                     38: ox\_asir, to factorize polynomials to find the integral
1.2       takayama   39: roots.
1.4       takayama   40: These two OpenXM complient systems are integrated by
                     41: OpenXM protocol.
                     42:
1.3       takayama   43: For example, the following is a sm1 session to find the annihilating
1.2       takayama   44: ideal for $f = x^3 - y^2 z^2$.
                     45: \begin{verbatim}
                     46: sm1>[(x^3-y^2 z^2) (x,y,z)] annfs ::
                     47: Starting ox_asir server.
                     48: Byte order for control process is network byte order.
                     49: Byte order for engine process is network byte order.
                     50: [[-y*Dy+z*Dz, 2*x*Dx+3*y*Dy+6, -2*y*z^2*Dx-3*x^2*Dy,
1.6     ! takayama   51: -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   52:  [-1,-139968*s^7-1119744*s^6-3802464*s^5-7107264*s^4
                     53:      -7898796*s^3-5220720*s^2-1900500*s-294000]]
                     54: \end{verbatim}
                     55: The last polynomial is factored as
1.3       takayama   56: $-12(s+1)(3s+5)(3s+4)(6s+5)(6s+7)$
1.2       takayama   57: and the minimal integral root is $-1$
                     58: as shown in the output.
                     59:
1.4       takayama   60: Similarly,
                     61: an algorithm to stratify singularity
                     62: \cite{oaku-advance}
                     63: is implemented by
                     64: kan/sm1 \cite{kan}, for Gr\"obner basis computation in $D$, and
                     65: ox\_asir, for primary ideal decompositions.
1.2       takayama   66:
                     67: \subsubsection{A Course on Solving Algebraic Equations}
                     68:
                     69: Risa/asir \cite{asir} is a general computer algebra system
                     70: which is good at Gr\"obner basis computations for zero dimensional ideal
                     71: with ${\bf Q}$ coefficients.
                     72: However, it is not good at graphical presentations and
                     73: numerical methods.
1.4       takayama   74: We integrated Risa/asir, ox\_phc (based on PHC pack by Verschelde \cite{phc}
1.2       takayama   75: for the polyhedral homotopy method) and
1.4       takayama   76: ox\_gnuplot (GNUPLOT) servers
1.2       takayama   77: to teach a course on solving algebraic equations.
1.3       takayama   78: This course was presented with the text book \cite{CLO} which discusses
1.2       takayama   79: on the Gr\"obner basis method and the polyhedral homotopy method
                     80: to solve systems of algebraic equations.
1.5       takayama   81: We could teach a course
1.2       takayama   82: with a unified environment
1.5       takayama   83: controlled by asir user language, which is similar to C.
1.3       takayama   84: The following is an asir session to solve algebraic equations by calling
                     85: the PHC pack.
1.2       takayama   86: \begin{verbatim}
                     87: [257] phc([x^2+y^2-4,x*y-1]);
                     88: The detailed output is in the file tmp.output.*
                     89: The answer is in the variable Phc.
                     90: 0
                     91: [260] Phc ;
                     92: [[[-0.517638,0],[-1.93185,0]],
                     93: [[1.93185,0],[0.517638,0]],
                     94: [[-1.93185,0],[-0.517638,0]],
                     95: [[0.517638,0],[1.93185,0]]]
                     96: [261]
                     97: \end{verbatim}
                     98:
                     99:
                    100: