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Up to [local] / OpenXM / src / asir-doc / parts / builtin

Imported OpenXM/src/asir-doc. 

@node $B0lJQ?tB?9`<0$N1i;;(B,,, $BAH$_9~$_H!?t(B
@section $B0lJQ?tB?9`<0$N1i;;(B

@menu
* umul umul_ff usquare usquare_ff utmul utmul_ff::
* kmul ksquare ktmul::
* utrunc udecomp ureverse::
* set_upkara set_uptkara set_upfft::
* uinv_as_power_series ureverse_inv_as_power_series::
* udiv urem urembymul urembymul_precomp ugcd::
@end menu

@node umul umul_ff usquare usquare_ff utmul utmul_ff,,, $B0lJQ?tB?9`<0$N1i;;(B
@subsection @code{umul}, @code{umul_ff}, @code{usquare}, @code{usquare_ff}, @code{utmul}, @code{utmul_ff}
@findex umul
@findex umul_ff
@findex usquare
@findex usquare_ff
@findex utmul
@findex utmul_ff

@table @t
@item umul(@var{p1},@var{p2}) 
@itemx umul_ff(@var{p1},@var{p2}) 
:: $B0lJQ?tB?9`<0$N9bB.>h;;(B
@item usquare(@var{p1})
@itemx usquare_ff(@var{p1})
:: $B0lJQ?tB?9`<0$N9bB.(B 2 $B>h;;(B
@item utmul(@var{p1},@var{p2},@var{d}) 
@itemx utmul_ff(@var{p1},@var{p2},@var{d}) 
:: $B0lJQ?tB?9`<0$N9bB.>h;;(B ($BBG$A@Z$j<!?t;XDj(B)
@end table

@table @var
@item return
$B0lJQ?tB?9`<0(B
@item p1 p2
$B0lJQ?tB?9`<0(B
@item d
$BHsIi@0?t(B
@end table

@itemize @bullet
@item
$B0lJQ?tB?9`<0$N>h;;$r(B, $B<!?t$K1~$8$F7h$^$k%"%k%4%j%:%`$rMQ$$$F(B
$B9bB.$K9T$&(B. 
@item
@code{umul()}, @code{usquare()}, @code{utmul()} $B$O(B
$B78?t$r@0?t$H8+$J$7$F(B, $B@0?t78?t$NB?9`<0$H$7$F@Q$r5a$a$k(B. 
$B78?t$,M-8BBN(B GF(p) $B$N85$N>l9g$K$O(B, $B78?t$O(B 0 $B0J>e(B p $BL$K~$N@0?t$H8+$J$5$l$k(B. 
@item
@code{umul_ff()}, @code{usquare_ff()}, @code{utmul_ff()} $B$O(B,
$B78?t$rM-8BBN$N85$H8+$J$7$F(B, $BM-8BBN>e$NB?9`<0$H$7$F(B
$B@Q$r5a$a$k(B. $B$?$@$7(B, $B0z?t$,A4$/M-8BBN$N85$r4^$^$J$$>l9g$K$O(B, 
$B@0?t78?t$NB?9`<0$rJV$9>l9g$b$"$k$N$G(B, $B$3$l$i$r8F$S=P$7$?7k2L(B
$B$,M-8BBN78?t$G$"$k$3$H$rJ]>Z$9$k$?$a$K$O(B
$B$"$i$+$8$a(B @code{simp_ff()} $B$G78?t$rM-8BBN$N85$KJQ49$7$F$*$/$H$h$$(B. 
@item
@code{umul_ff()}, @code{usquare_ff()}, @code{utmul_ff()} $B$O(B, 
GF(2^n) $B78?t$NB?9`<0$r0z?t$K<h$l$J$$(B. 
@item
@code{umul()}, @code{umul_ff()} $B$N7k2L$O(B @var{p1}, @var{p2} $B$N@Q(B, 
@code{usquare()}, @code{usquare_ff()} $B$N7k2L$O(B @var{p1} $B$N(B 2 $B>h(B, 
@code{utmul()}, @code{utmul_ff()} $B$N7k2L$O(B @var{p1}, @var{p2} $B$N@Q(B
$B$N(B, @var{d} $B<!0J2<$NItJ,$H$J$k(B. 
@item
$B$$$:$l$b(B, @code{set_upkara()} (@code{utmul}, @code{utmul_ff} $B$K$D$$$F$O(B
@code{set_uptkara()}) $B$GJV$5$l$kCM0J2<$N<!?t$KBP$7$F$ODL>o$NI.;;(B
$B7A<0$NJ}K!(B, @code{set_uufft()} $B$GJV$5$l$kCM0J2<$N<!?t$KBP$7$F$O(B Karatsuba
$BK!(B, $B$=$l0J>e$G$O(B FFT$B$*$h$SCf9q>jM>DjM}$,MQ$$$i$l$k(B. $B$9$J$o$A(B, 
$B@0?t$KBP$9$k(B FFT $B$G$O$J$/(B, $B==J,B?$/$N(B 1 $B%o!<%I0JFb$NK!(B @var{mi} $B$rMQ0U$7(B, 
@var{p1}, @var{p2} $B$N78?t$r(B @var{mi} $B$G3d$C$?M>$j$H$7$?$b$N$N@Q$r(B, 
FFT $B$G7W;;$7(B, $B:G8e$KCf9q>jM>DjM}$G9g@.$9$k(B. $B$=$N:](B, $BM-8BBNHG$N4X?t$K(B
$B$*$$$F$O(B, $B:G8e$K4pACBN$rI=$9K!$G3F78?t$N>jM>$r7W;;$9$k$,(B, $B$3$3$G$O(B
Shoup $B$K$h$k%H%j%C%/$rMQ$$$F9bB.2=$7$F$"$k(B. 
@end itemize

@example
[176] load("fff")$
[177] cputime(1)$
0sec(1.407e-05sec)
[178] setmod_ff(2^160-47);
1461501637330902918203684832716283019655932542929
0sec(0.00028sec)
[179] A=randpoly_ff(100,x)$
0sec(0.001422sec)
[180] B=randpoly_ff(100,x)$
0sec(0.00107sec)
[181] for(I=0;I<100;I++)A*B;
7.77sec + gc : 8.38sec(16.15sec)
[182] for(I=0;I<100;I++)umul(A,B);
2.24sec + gc : 1.52sec(3.767sec)
[183] for(I=0;I<100;I++)umul_ff(A,B);
1.42sec + gc : 0.24sec(1.653sec)
[184] for(I=0;I<100;I++)usquare_ff(A);  
1.08sec + gc : 0.21sec(1.297sec)
[185] for(I=0;I<100;I++)utmul_ff(A,B,100);
1.2sec + gc : 0.17sec(1.366sec)
[186] deg(utmul_ff(A,B,100),x);
100
@end example

@table @t
@item $B;2>H(B
@fref{set_upkara set_uptkara set_upfft},
@fref{kmul ksquare ktmul}.
@end table

@node kmul ksquare ktmul,,, $B0lJQ?tB?9`<0$N1i;;(B
@subsection @code{kmul}, @code{ksquare}, @code{ktmul}
@findex kmul
@findex ksquare
@findex ktmul

@table @t
@item kmul(@var{p1},@var{p2}) 
:: $B0lJQ?tB?9`<0$N9bB.>h;;(B
@item ksquare(@var{p1})
:: $B0lJQ?tB?9`<0$N9bB.(B 2 $B>h;;(B
@item ktmul(@var{p1},@var{p2},@var{d}) 
:: $B0lJQ?tB?9`<0$N9bB.>h;;(B ($BBG$A@Z$j<!?t;XDj(B)
@end table

@table @var
@item return
$B0lJQ?tB?9`<0(B
@item p1 p2
$B0lJQ?tB?9`<0(B
@item d
$BHsIi@0?t(B
@end table

@itemize @bullet
@item
$B0lJQ?tB?9`<0$N>h;;$r(B Karatsuba $BK!$G9T$&(B. 
@item
$B4pK\E*$K$O(B @code{umul} $B$HF1MM$@$,(B, $B<!?t$,Bg$-$/$J$C$F$b(B
FFT $B$rMQ$$$?9bB.2=$O9T$o$J$$(B. 
@item
GF(2^n) $B78?t$NB?9`<0$K$bMQ$$$k$3$H$,$G$-$k(B. 
@end itemize

@example
[0] load("code/fff");
1
[34] setmod_ff(defpoly_mod2(160));
x^160+x^5+x^3+x^2+1
[35] A=randpoly_ff(100,x)$
[36] B=randpoly_ff(100,x)$
[37] umul(A,B)$
umul : invalid argument
return to toplevel
[37] kmul(A,B)$
@end example

@node set_upkara set_uptkara set_upfft,,, $B0lJQ?tB?9`<0$N1i;;(B
@subsection @code{set_upkara}, @code{set_uptkara}, @code{set_upfft}
@findex set_upkara
@findex set_uptkara
@findex set_upfft

@table @t
@item set_upkara([@var{threshold}])
@itemx set_uptkara([@var{threshold}])
@itemx set_upfft([@var{threshold}])
:: 1 $BJQ?tB?9`<0$N@Q1i;;$K$*$1$k(B N^2 , Karatsuba, FFT $B%"%k%4%j%:%`$N@ZBX$($NogCM(B
@end table

@table @var
@item return
$B@_Dj$5$l$F$$$kCM(B
@item threshold
$BHsIi@0?t(B
@end table

@itemize @bullet
@item
$B$$$:$l$b(B, $B0lJQ?tB?9`<0$N@Q$N7W;;$K$*$1$k(B, $B%"%k%4%j%:%`@ZBX$($NogCM$r(B
$B@_Dj$9$k(B. 
@item
$B0lJQ?tB?9`<0$N@Q$O(B, $B<!?t(B N $B$,>.$5$$HO0O$G$ODL>o$N(B N^2 $B%"%k%4%j%:%`(B, $BCfDxEY(B
$B$N>l9g(B Karatsuba $B%"%k%4%j%:%`(B, $BBg$-$$>l9g$K$O(B FFT $B%"%k%4%j%:%`$G7W;;(B
$B$5$l$k(B. $B$3$N@ZBX$($N<!?t$r@_Dj$9$k(B. 
@item
$B>\:Y$O(B, $B$=$l$>$l$N@Q4X?t$N9`$r;2>H$N$3$H(B. 
@end itemize

@table @t
@item $B;2>H(B
@fref{kmul ksquare ktmul},
@fref{umul umul_ff usquare usquare_ff utmul utmul_ff}.
@end table

@node utrunc udecomp ureverse,,, $B0lJQ?tB?9`<0$N1i;;(B
@subsection @code{utrunc}, @code{udecomp}, @code{ureverse}
@findex utrunc
@findex udecomp
@findex ureverse

@table @t
@item utrunc(@var{p},@var{d}) 
@itemx udecomp(@var{p},@var{d}) 
@itemx ureverse(@var{p})
:: $BB?9`<0$KBP$9$kA`:n(B
@end table

@table @var
@item return
$B0lJQ?tB?9`<0$"$k$$$O0lJQ?tB?9`<0$N%j%9%H(B
@item p
$B0lJQ?tB?9`<0(B
@item d
$BHsIi@0?t(B
@end table

@itemize @bullet
@item
@var{p} $B$NJQ?t$r(B x $B$H$9$k(B. $B$3$N$H$-(B @var{p} = @var{p1}+x^(d+1)@var{p2}
(@var{p1} $B$N<!?t$O(B @var{d} $B0J2<(B) $B$HJ,2r$G$-$k(B. @code{utrunc()} $B$O(B
@var{p1} $B$rJV$7(B, @code{udecomp()} $B$O(B [@var{p1},@var{p2}] $B$rJV$9(B. 
@item
@var{p} $B$N<!?t$r(B @var{e} $B$H$7(B, @var{i} $B<!$N78?t$r(B @var{p[i]} $B$H$9$l$P(B, 
@code{ureverse()} $B$O(B @var{p[e]}+@var{p[e-1]}x+... $B$rJV$9(B. 
@end itemize

@example
[132] utrunc((x+1)^10,5);
252*x^5+210*x^4+120*x^3+45*x^2+10*x+1
[133] udecomp((x+1)^10,5);
[252*x^5+210*x^4+120*x^3+45*x^2+10*x+1,x^4+10*x^3+45*x^2+120*x+210]
[134] ureverse(3*x^3+x^2+2*x);
2*x^2+x+3
@end example

@table @t
@item $B;2>H(B
@fref{udiv urem urembymul urembymul_precomp ugcd}.
@end table

@node uinv_as_power_series ureverse_inv_as_power_series,,, $B0lJQ?tB?9`<0$N1i;;(B
@subsection @code{uinv_as_power_series}, @code{ureverse_inv_as_power_series}
@findex uinv_as_power_series
@findex ureverse_inv_as_power_series

@table @t
@item uinv_as_power_series(@var{p},@var{d}) 
@itemx ureverse_inv_as_power_series(@var{p},@var{d})
:: $BB?9`<0$rQQ5i?t$H$_$F(B, $B5U857W;;(B
@end table

@table @var
@item return
$B0lJQ?tB?9`<0(B
@item p
$B0lJQ?tB?9`<0(B
@item d
$BHsIi@0?t(B
@end table

@itemize @bullet
@item
@code{uinv_as_power_series(@var{p},@var{d})} $B$O(B, $BDj?t9`$,(B 0 $B$G$J$$(B
$BB?9`<0(B @var{p} $B$KBP$7(B, @var{p}@var{r}-1 $B$N:GDc<!?t$,(B @var{d}+1
$B0J>e$K$J$k$h$&$J(B $B9b!9(B @var{d} $B<!$NB?9`<0(B @var{r} $B$r5a$a$k(B. 
@item
@code{ureverse_inv_as_power_series(@var{p},@var{d})} $B$O(B
@var{p} $B$N<!?t$r(B @var{e} $B$H$9$k$H$-(B, @var{p1}=@code{ureverse(@var{p},@var{e})}
$B$KBP$7$F(B @code{uinv_as_power_series(@var{p1},@var{d})} $B$r7W;;$9$k(B. 
@item
@code{rembymul_precomp()} $B$N0z?t$H$7$FMQ$$$k>l9g(B, @code{ureverse_inv_as_power_series()} $B$N7k2L$r$=$N$^$^MQ$$$k$3$H$,$G$-$k(B. 
@end itemize

@example
[123] A=(x+1)^5;                 
x^5+5*x^4+10*x^3+10*x^2+5*x+1
[124] uinv_as_power_series(A,5); 
-126*x^5+70*x^4-35*x^3+15*x^2-5*x+1
[126] A*R;
-126*x^10-560*x^9-945*x^8-720*x^7-210*x^6+1
[127] A=x^10+x^9;
x^10+x^9
[128] R=ureverse_inv_as_power_series(A,5);
-x^5+x^4-x^3+x^2-x+1
[129] ureverse(A)*R;
-x^6+1
@end example

@table @t
@item $B;2>H(B
@fref{utrunc udecomp ureverse},
@fref{udiv urem urembymul urembymul_precomp ugcd}.
@end table

@node udiv urem urembymul urembymul_precomp ugcd,,, $B0lJQ?tB?9`<0$N1i;;(B
@subsection @code{udiv}, @code{urem}, @code{urembymul}, @code{urembymul_precomp}, @code{ugcd}
@findex udiv
@findex urem
@findex urembymul
@findex urembymul_precomp
@findex ugcd

@table @t
@item udiv(@var{p1},@var{p2}) 
@item urem(@var{p1},@var{p2}) 
@item urembymul(@var{p1},@var{p2}) 
@item urembymul_precomp(@var{p1},@var{p2},@var{inv}) 
@item ugcd(@var{p1},@var{p2}) 
:: $BB?9`<0$KBP$9$kA`:n(B
@end table

@table @var
@item return
$B0lJQ?tB?9`<0(B
@item p1,p2,inv
$B0lJQ?tB?9`<0(B
@end table

@itemize @bullet
@item
$B0lJQ?tB?9`<0(B @var{p1}, @var{p2} $B$KBP$7(B, 
@code{udiv} $B$O>&(B, @code{urem}, @code{urembymul} $B$O>jM>(B, 
@code{ugcd} $B$O(B GCD $B$rJV$9(B. 
$B$3$l$i$O(B, $BL)$J0lJQ?tB?9`<0$KBP$9$k9bB.2=$r?^$C$?$b$N$G$"$k(B. 
@code{urembymul} $B$O(B, @var{p2} $B$K$h$k>jM>7W;;$r(B, @var{p2} $B$N(B
$BQQ5i?t$H$7$F$N5U857W;;$*$h$S(B, $B>h;;(B 2 $B2s$KCV$-49$($?$b$N$G(B, 
$B<!?t$,Bg$-$$>l9g$KM-8z$G$"$k(B. 
@item @code{urembymul_precomp} $B$O(B, $B8GDj$5$l$?B?9`<0$K$h$k>jM>(B
$B7W;;$rB??t9T$&>l9g$J$I$K8z2L$rH/4x$9$k(B. 
@end itemize

@example
[177] setmod_ff(2^160-47);
1461501637330902918203684832716283019655932542929
[178] A=randpoly_ff(200,x)$
[179] B=randpoly_ff(101,x)$
[180] cputime(1)$
0sec(1.597e-05sec)
[181] srem(A,B)$
0.15sec + gc : 0.15sec(0.3035sec)
[182] urem(A,B)$
0.11sec + gc : 0.12sec(0.2347sec)
[183] urembymul(A,B)$          
0.08sec + gc : 0.09sec(0.1651sec)
[184] R=ureverse_inv_as_power_series(B,101)$
0.04sec + gc : 0.03sec(0.063sec)
[185] urembymul_precomp(A,B,R)$             
0.03sec(0.02501sec)
@end example

@table @t
@item $B;2>H(B
@fref{uinv_as_power_series ureverse_inv_as_power_series}.
@end table