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Diff for /OpenXM/doc/compalg/poly.tex between version 1.3 and 1.4

version 1.3, 2000/10/03 01:44:03 version 1.4, 2000/11/21 08:01:11
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 %$OpenXM: OpenXM/doc/compalg/poly.tex,v 1.2 2000/03/28 02:02:30 noro Exp $  %$OpenXM: OpenXM/doc/compalg/poly.tex,v 1.3 2000/10/03 01:44:03 noro Exp $
 \chapter{$BB?9`<0(B}  \chapter{$BB?9`<0(B}
   
 \section{$BB?9`<0$NI=8=(B}  \section{$BB?9`<0$NI=8=(B}
Line 89  Input : $f, g \in K[x]$, $g\neq 0$\\
Line 89  Input : $f, g \in K[x]$, $g\neq 0$\\
 Output : $f = qg+r$, $q,r \in K[x]$, $\deg(r) < \deg(g)$ $B$J$k(B $q,r$\\  Output : $f = qg+r$, $q,r \in K[x]$, $\deg(r) < \deg(g)$ $B$J$k(B $q,r$\\
 $q\leftarrow 0$, \quad $r\leftarrow f$\\  $q\leftarrow 0$, \quad $r\leftarrow f$\\
 while \= ( $\deg(r)\ge \deg(g)$ ) \{\\  while \= ( $\deg(r)\ge \deg(g)$ ) \{\\
 \> $t \leftarrow \lc(r)/\lc(q)\cdot x^{\deg(r)-\deg(g)}$\\  \> $t \leftarrow \lc(r)/\lc(g)\cdot x^{\deg(r)-\deg(g)}$\\
 \> $r \leftarrow r-tg$,\quad $q \leftarrow q+t$\\  \> $r \leftarrow r-tg$,\quad $q \leftarrow q+t$\\
 \}\\  \}\\
 return $(q,r)$  return $(q,r)$
Line 123  $f=f_1x^m+f_2,\quad g=g_1x^m+g_2 \quad (\deg(f_1),\deg
Line 123  $f=f_1x^m+f_2,\quad g=g_1x^m+g_2 \quad (\deg(f_1),\deg
 \end{center}  \end{center}
 $B$H=q$/$H(B,  $B$H=q$/$H(B,
 $$fg=(f_1g_1x^{2m}+((f_1-f_2)(g_2-g_1)+f_1g_1+f_2g_2)x^m+f_2g_2$$  $$fg=(f_1g_1x^{2m}+((f_1-f_2)(g_2-g_1)+f_1g_1+f_2g_2)x^m+f_2g_2$$
 $B$3$3$G(B $fg$ $B$N7W;;%3%9%H$O(B, $B9b!9(B $3T(m-1)+3\cdot 2^mA$, $B$9$J$o$A(B  $B$3$3$G(B $fg$ $B$N7W;;%3%9%H$O(B $3T(m-1)+4\cdot 2^mA$, $B$9$J$o$A(B
 $T(m) \le 3T(m-1)+3\cdot 2^mA \quad (m\le 1).$  $T(m) = 3T(m-1)+4\cdot 2^mA \quad (m\le 1).$
 $B$5$i$K(B, $T(0)=M$ $B$+$i(B  $B$5$i$K(B, $T(0)=M$ $B$+$i(B
 $T(m) \le (M+6A)3^m-6A\cdot 2^m.$ \qed  $T(m) = (M+8A)3^m-8A\cdot 2^m.$ \qed
   
 $B$3$NL?Bj$h$j(B, $n$ $B<!<0$I$&$7$N@Q$O(B, $O(n^{\log_23})$ $B$N7W;;NL$G7W;;$G$-$k(B  $B$3$NL?Bj$h$j(B, $n$ $B<!<0$I$&$7$N@Q$O(B, $O(n^{\log_23})$ $B$N7W;;NL$G7W;;$G$-$k(B
 $B$3$H$,J,$+$k(B. $B99$K>\$7$/(B, $BDL>o$N(B $O(n^2)$ $B%"%k%4%j%:%`$HHf3S$7$F$_$h$&(B.  $B$3$H$,J,$+$k(B. $B99$K>\$7$/(B, $BDL>o$N(B $O(n^2)$ $B%"%k%4%j%:%`$HHf3S$7$F$_$h$&(B.
 $2^m-1$ $B<!<0$I$&$7$NDL>o$N%"%k%4%j%:%`$K$h$k7W;;%3%9%H$r(B $T_0(m)$ $B$H$9$l$P(B,  $2^m-1$ $B<!<0$I$&$7$NDL>o$N%"%k%4%j%:%`$K$h$k7W;;%3%9%H$r(B $T_0(m)$ $B$H$9$l$P(B,
 $T_0(m)=M2^{2m}+A(2^m-1)^2$  $T_0(m)=M2^{2m}+A(2^m-1)^2$
 $B$G$"$k(B.  $B$G$"$k(B.
 $$T_0(m)-T(m) \ge M2^{2m}+A(2^m-1)^2-((M+6A)3^m-6A\cdot 2^m)$$  $$T_0(m)-T(m) \ge M2^{2m}+A(2^m-1)^2-((M+8A)3^m-8A\cdot 2^m)$$
 $B$G(B, $B1&JU$N(B $m=0,\cdots,7$ $B$KBP$9$kCM$O<!$N$h$&$K$J$k(B.  $B$G(B, $B1&JU$N(B $m=0,\cdots,6$ $B$KBP$9$kCM$O<!$N$h$&$K$J$k(B.
   
 \begin{center}  \begin{center}
 \begin{tabular}{|c||c|c|c|c|c|c|c|} \hline  \begin{tabular}{|c||c|c|c|c|c|c|c|} \hline
 $m$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline  $m$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline
 $B1&JU(B & 0 & M-5A & 7M-21A & 37M-65A & 175M-165A & 781M-305A & 3367M-21A \\ \hline  $B1&JU(B & 0 & M-7A & 7M-31A & 37M-103A & 175M-195A & 781M-727A & 3367M-1351A \\ \hline
 \end{tabular}  \end{tabular}
 \end{center}  \end{center}
   
 $B0lHL$K(B, $M>A$ $B$@$+$i(B $m \ge 4$ $B$9$J$o$A(B 15 $B<!<00J>e$N@Q$KBP$7$F$O(B, Karatsuba  $B0lHL$K(B, $M>A$ $B$@$+$i(B $m \ge 5$ $B$9$J$o$A(B 31 $B<!<00J>e$N@Q$KBP$7$F$O(B, Karatsuba
 $BK!$O>o$K9bB.$G$"$j(B, $M$ $B$,(B $A$ $B$KHf$Y$FBg$-$$>l9g$[$I(B, $BDc$$<!?t$+$i(B Karatsuba  $BK!$O>o$K9bB.$G$"$j(B, $M$ $B$,(B $A$ $B$KHf$Y$FBg$-$$>l9g$[$I(B, $BDc$$<!?t$+$i(B Karatsuba
 $BK!$,8z2LE*$G$"$k$3$H$,J,$+$k(B. $B99$K(B, $B7W;;NL$N%*!<%@$N0c$$(B  $BK!$,8z2LE*$G$"$k$3$H$,J,$+$k(B. $B99$K(B, $B7W;;NL$N%*!<%@$N0c$$(B
 ($O(n^2)$ $B$H(B  $O(n^{\log_23})$) $B$K$h$j(B, $B9b<!Dx(B Karatsuba $BK!$,9bB.$K$J$j(B,  ($O(n^2)$ $B$H(B  $O(n^{\log_23})$) $B$K$h$j(B, $B9b<!Dx(B Karatsuba $BK!$,9bB.$K$J$j(B,
Line 153  $m$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline
Line 153  $m$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline
 \begin{center}  \begin{center}
 \begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|} \hline  \begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|} \hline
 $m$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline  $m$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline
 $B7W;;%3%9%HHf(B & 1 & 1 & 0.84 & 0.67 & 0.53 & 0.41 & 0.31 & 0.24 & 0.18 & 0.13 & 0.10 \\ \hline  $B7W;;%3%9%HHf(B & 1 & 1.1 & 0.96 & 0.78 & 0.61 & 0.48 & 0.37 & 0.28 & 0.21 & 0.16 & 0.12 \\ \hline
 \end{tabular}  \end{tabular}
 \end{center}  \end{center}
   
 $B$9$J$o$A(B, 100 $B<!$G(B 4 $BG\(B, 1000 $B<!<0$G(B 10 $BG\DxEY$N:9$,$D$/$3$H$K$J$k(B.  $B$9$J$o$A(B, 100 $B<!$G(B 1.5 $BG\(B, 1000 $B<!<0$G(B 8 $BG\DxEY$N:9$,$D$/$3$H$K$J$k(B.
 Karatsuba $BK!$K$*$1$k(B, $BB?9`<0$NJ,3d(B, $B4X?t8F$S=P$7$N<j4V$J$I$,(B  Karatsuba $BK!$K$*$1$k(B, $BB?9`<0$NJ,3d(B, $B4X?t8F$S=P$7$N<j4V$J$I$,(B
 $B$+$+$k$?$a(B, $B<B:]$K$O$3$N?t;z$rC#@.$9$k$3$H$OFq$7$$$,(B, $BB?9`<0$N@Q$K(B  $B$+$+$k$?$a(B, $B<B:]$K$O$3$N?t;z$rC#@.$9$k$3$H$OFq$7$$$,(B, $BB?9`<0$N@Q$K(B
 $B4X$7$F$O(B, Karatsuba $BK!$O==J,<BMQE*$G$"$k$H8@$($k(B.  $B4X$7$F$O(B, Karatsuba $BK!$O==J,<BMQE*$G$"$k$H8@$($k(B.
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