Annotation of OpenXM/doc/issac2000/heterotic-network.tex, Revision 1.8
1.8 ! takayama 1: % $OpenXM: OpenXM/doc/issac2000/heterotic-network.tex,v 1.7 2000/01/15 06:11:17 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:
1.5 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
39: ox\_asir, to factorize polynomials to find the integral
1.2 takayama 40: roots.
1.4 takayama 41: These two OpenXM complient systems are integrated by
42: OpenXM protocol.
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
66: ox\_asir, for primary ideal decompositions.
1.2 takayama 67:
68: \subsubsection{A Course on Solving Algebraic Equations}
69:
70: Risa/asir \cite{asir} is a general computer algebra system
71: which is good at Gr\"obner basis computations for zero dimensional ideal
72: with ${\bf Q}$ coefficients.
73: However, it is not good at graphical presentations and
74: numerical methods.
1.4 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.
1.3 takayama 79: This course was presented with the text book \cite{CLO} which discusses
1.2 takayama 80: on the Gr\"obner basis method and the polyhedral homotopy method
81: to solve systems of algebraic equations.
1.5 takayama 82: We could teach a course
1.2 takayama 83: with a unified environment
1.5 takayama 84: controlled by asir user language, which is similar to C.
1.3 takayama 85: The following is an asir session to solve algebraic equations by calling
1.8 ! takayama 86: the PHC pack (see Figure \ref{katsura} too):
1.2 takayama 87: \begin{verbatim}
1.7 takayama 88: [287] phc(katsura(7));
1.2 takayama 89: The detailed output is in the file tmp.output.*
90: The answer is in the variable Phc.
91: 0
1.7 takayama 92: [290] B=map(first,Phc)$
93: [291] gnuplot_plotDots([],0)$
94: [292] gnuplot_plotDots(B,0)$
1.2 takayama 95: \end{verbatim}
1.7 takayama 96:
97: \begin{figure}[htbp]
98: \epsfxsize=8.5cm
99: \epsffile{katsura7.ps}
100: \caption{The first components of the solutions to the system of algebraic equations Katsura 7.}
101: \label{katsura}
102: \end{figure}
103:
1.2 takayama 104:
105:
106:
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