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Annotation of OpenXM/src/asir-doc/parts/builtin/upoly.texi, Revision 1.1.1.1

1.1       noro        1: @node $B0lJQ?tB?9`<0$N1i;;(B,,, $BAH$_9~$_H!?t(B
                      2: @section $B0lJQ?tB?9`<0$N1i;;(B
                      3:
                      4: @menu
                      5: * umul umul_ff usquare usquare_ff utmul utmul_ff::
                      6: * kmul ksquare ktmul::
                      7: * utrunc udecomp ureverse::
                      8: * set_upkara set_uptkara set_upfft::
                      9: * uinv_as_power_series ureverse_inv_as_power_series::
                     10: * udiv urem urembymul urembymul_precomp ugcd::
                     11: @end menu
                     12:
                     13: @node umul umul_ff usquare usquare_ff utmul utmul_ff,,, $B0lJQ?tB?9`<0$N1i;;(B
                     14: @subsection @code{umul}, @code{umul_ff}, @code{usquare}, @code{usquare_ff}, @code{utmul}, @code{utmul_ff}
                     15: @findex umul
                     16: @findex umul_ff
                     17: @findex usquare
                     18: @findex usquare_ff
                     19: @findex utmul
                     20: @findex utmul_ff
                     21:
                     22: @table @t
                     23: @item umul(@var{p1},@var{p2})
                     24: @itemx umul_ff(@var{p1},@var{p2})
                     25: :: $B0lJQ?tB?9`<0$N9bB.>h;;(B
                     26: @item usquare(@var{p1})
                     27: @itemx usquare_ff(@var{p1})
                     28: :: $B0lJQ?tB?9`<0$N9bB.(B 2 $B>h;;(B
                     29: @item utmul(@var{p1},@var{p2},@var{d})
                     30: @itemx utmul_ff(@var{p1},@var{p2},@var{d})
                     31: :: $B0lJQ?tB?9`<0$N9bB.>h;;(B ($BBG$A@Z$j<!?t;XDj(B)
                     32: @end table
                     33:
                     34: @table @var
                     35: @item return
                     36: $B0lJQ?tB?9`<0(B
                     37: @item p1 p2
                     38: $B0lJQ?tB?9`<0(B
                     39: @item d
                     40: $BHsIi@0?t(B
                     41: @end table
                     42:
                     43: @itemize @bullet
                     44: @item
                     45: $B0lJQ?tB?9`<0$N>h;;$r(B, $B<!?t$K1~$8$F7h$^$k%"%k%4%j%:%`$rMQ$$$F(B
                     46: $B9bB.$K9T$&(B.
                     47: @item
                     48: @code{umul()}, @code{usquare()}, @code{utmul()} $B$O(B
                     49: $B78?t$r@0?t$H8+$J$7$F(B, $B@0?t78?t$NB?9`<0$H$7$F@Q$r5a$a$k(B.
                     50: $B78?t$,M-8BBN(B GF(p) $B$N85$N>l9g$K$O(B, $B78?t$O(B 0 $B0J>e(B p $BL$K~$N@0?t$H8+$J$5$l$k(B.
                     51: @item
                     52: @code{umul_ff()}, @code{usquare_ff()}, @code{utmul_ff()} $B$O(B,
                     53: $B78?t$rM-8BBN$N85$H8+$J$7$F(B, $BM-8BBN>e$NB?9`<0$H$7$F(B
                     54: $B@Q$r5a$a$k(B. $B$?$@$7(B, $B0z?t$,A4$/M-8BBN$N85$r4^$^$J$$>l9g$K$O(B,
                     55: $B@0?t78?t$NB?9`<0$rJV$9>l9g$b$"$k$N$G(B, $B$3$l$i$r8F$S=P$7$?7k2L(B
                     56: $B$,M-8BBN78?t$G$"$k$3$H$rJ]>Z$9$k$?$a$K$O(B
                     57: $B$"$i$+$8$a(B @code{simp_ff()} $B$G78?t$rM-8BBN$N85$KJQ49$7$F$*$/$H$h$$(B.
                     58: @item
                     59: @code{umul_ff()}, @code{usquare_ff()}, @code{utmul_ff()} $B$O(B,
                     60: GF(2^n) $B78?t$NB?9`<0$r0z?t$K<h$l$J$$(B.
                     61: @item
                     62: @code{umul()}, @code{umul_ff()} $B$N7k2L$O(B @var{p1}, @var{p2} $B$N@Q(B,
                     63: @code{usquare()}, @code{usquare_ff()} $B$N7k2L$O(B @var{p1} $B$N(B 2 $B>h(B,
                     64: @code{utmul()}, @code{utmul_ff()} $B$N7k2L$O(B @var{p1}, @var{p2} $B$N@Q(B
                     65: $B$N(B, @var{d} $B<!0J2<$NItJ,$H$J$k(B.
                     66: @item
                     67: $B$$$:$l$b(B, @code{set_upkara()} (@code{utmul}, @code{utmul_ff} $B$K$D$$$F$O(B
                     68: @code{set_uptkara()}) $B$GJV$5$l$kCM0J2<$N<!?t$KBP$7$F$ODL>o$NI.;;(B
                     69: $B7A<0$NJ}K!(B, @code{set_uufft()} $B$GJV$5$l$kCM0J2<$N<!?t$KBP$7$F$O(B Karatsuba
                     70: $BK!(B, $B$=$l0J>e$G$O(B FFT$B$*$h$SCf9q>jM>DjM}$,MQ$$$i$l$k(B. $B$9$J$o$A(B,
                     71: $B@0?t$KBP$9$k(B FFT $B$G$O$J$/(B, $B==J,B?$/$N(B 1 $B%o!<%I0JFb$NK!(B @var{mi} $B$rMQ0U$7(B,
                     72: @var{p1}, @var{p2} $B$N78?t$r(B @var{mi} $B$G3d$C$?M>$j$H$7$?$b$N$N@Q$r(B,
                     73: FFT $B$G7W;;$7(B, $B:G8e$KCf9q>jM>DjM}$G9g@.$9$k(B. $B$=$N:](B, $BM-8BBNHG$N4X?t$K(B
                     74: $B$*$$$F$O(B, $B:G8e$K4pACBN$rI=$9K!$G3F78?t$N>jM>$r7W;;$9$k$,(B, $B$3$3$G$O(B
                     75: Shoup $B$K$h$k%H%j%C%/$rMQ$$$F9bB.2=$7$F$"$k(B.
                     76: @end itemize
                     77:
                     78: @example
                     79: [176] load("fff")$
                     80: [177] cputime(1)$
                     81: 0sec(1.407e-05sec)
                     82: [178] setmod_ff(2^160-47);
                     83: 1461501637330902918203684832716283019655932542929
                     84: 0sec(0.00028sec)
                     85: [179] A=randpoly_ff(100,x)$
                     86: 0sec(0.001422sec)
                     87: [180] B=randpoly_ff(100,x)$
                     88: 0sec(0.00107sec)
                     89: [181] for(I=0;I<100;I++)A*B;
                     90: 7.77sec + gc : 8.38sec(16.15sec)
                     91: [182] for(I=0;I<100;I++)umul(A,B);
                     92: 2.24sec + gc : 1.52sec(3.767sec)
                     93: [183] for(I=0;I<100;I++)umul_ff(A,B);
                     94: 1.42sec + gc : 0.24sec(1.653sec)
                     95: [184] for(I=0;I<100;I++)usquare_ff(A);
                     96: 1.08sec + gc : 0.21sec(1.297sec)
                     97: [185] for(I=0;I<100;I++)utmul_ff(A,B,100);
                     98: 1.2sec + gc : 0.17sec(1.366sec)
                     99: [186] deg(utmul_ff(A,B,100),x);
                    100: 100
                    101: @end example
                    102:
                    103: @table @t
                    104: @item $B;2>H(B
                    105: @fref{set_upkara set_uptkara set_upfft},
                    106: @fref{kmul ksquare ktmul}.
                    107: @end table
                    108:
                    109: @node kmul ksquare ktmul,,, $B0lJQ?tB?9`<0$N1i;;(B
                    110: @subsection @code{kmul}, @code{ksquare}, @code{ktmul}
                    111: @findex kmul
                    112: @findex ksquare
                    113: @findex ktmul
                    114:
                    115: @table @t
                    116: @item kmul(@var{p1},@var{p2})
                    117: :: $B0lJQ?tB?9`<0$N9bB.>h;;(B
                    118: @item ksquare(@var{p1})
                    119: :: $B0lJQ?tB?9`<0$N9bB.(B 2 $B>h;;(B
                    120: @item ktmul(@var{p1},@var{p2},@var{d})
                    121: :: $B0lJQ?tB?9`<0$N9bB.>h;;(B ($BBG$A@Z$j<!?t;XDj(B)
                    122: @end table
                    123:
                    124: @table @var
                    125: @item return
                    126: $B0lJQ?tB?9`<0(B
                    127: @item p1 p2
                    128: $B0lJQ?tB?9`<0(B
                    129: @item d
                    130: $BHsIi@0?t(B
                    131: @end table
                    132:
                    133: @itemize @bullet
                    134: @item
                    135: $B0lJQ?tB?9`<0$N>h;;$r(B Karatsuba $BK!$G9T$&(B.
                    136: @item
                    137: $B4pK\E*$K$O(B @code{umul} $B$HF1MM$@$,(B, $B<!?t$,Bg$-$/$J$C$F$b(B
                    138: FFT $B$rMQ$$$?9bB.2=$O9T$o$J$$(B.
                    139: @item
                    140: GF(2^n) $B78?t$NB?9`<0$K$bMQ$$$k$3$H$,$G$-$k(B.
                    141: @end itemize
                    142:
                    143: @example
                    144: [0] load("code/fff");
                    145: 1
                    146: [34] setmod_ff(defpoly_mod2(160));
                    147: x^160+x^5+x^3+x^2+1
                    148: [35] A=randpoly_ff(100,x)$
                    149: [36] B=randpoly_ff(100,x)$
                    150: [37] umul(A,B)$
                    151: umul : invalid argument
                    152: return to toplevel
                    153: [37] kmul(A,B)$
                    154: @end example
                    155:
                    156: @node set_upkara set_uptkara set_upfft,,, $B0lJQ?tB?9`<0$N1i;;(B
                    157: @subsection @code{set_upkara}, @code{set_uptkara}, @code{set_upfft}
                    158: @findex set_upkara
                    159: @findex set_uptkara
                    160: @findex set_upfft
                    161:
                    162: @table @t
                    163: @item set_upkara([@var{threshold}])
                    164: @itemx set_uptkara([@var{threshold}])
                    165: @itemx set_upfft([@var{threshold}])
                    166: :: 1 $BJQ?tB?9`<0$N@Q1i;;$K$*$1$k(B N^2 , Karatsuba, FFT $B%"%k%4%j%:%`$N@ZBX$($NogCM(B
                    167: @end table
                    168:
                    169: @table @var
                    170: @item return
                    171: $B@_Dj$5$l$F$$$kCM(B
                    172: @item threshold
                    173: $BHsIi@0?t(B
                    174: @end table
                    175:
                    176: @itemize @bullet
                    177: @item
                    178: $B$$$:$l$b(B, $B0lJQ?tB?9`<0$N@Q$N7W;;$K$*$1$k(B, $B%"%k%4%j%:%`@ZBX$($NogCM$r(B
                    179: $B@_Dj$9$k(B.
                    180: @item
                    181: $B0lJQ?tB?9`<0$N@Q$O(B, $B<!?t(B N $B$,>.$5$$HO0O$G$ODL>o$N(B N^2 $B%"%k%4%j%:%`(B, $BCfDxEY(B
                    182: $B$N>l9g(B Karatsuba $B%"%k%4%j%:%`(B, $BBg$-$$>l9g$K$O(B FFT $B%"%k%4%j%:%`$G7W;;(B
                    183: $B$5$l$k(B. $B$3$N@ZBX$($N<!?t$r@_Dj$9$k(B.
                    184: @item
                    185: $B>\:Y$O(B, $B$=$l$>$l$N@Q4X?t$N9`$r;2>H$N$3$H(B.
                    186: @end itemize
                    187:
                    188: @table @t
                    189: @item $B;2>H(B
                    190: @fref{kmul ksquare ktmul},
                    191: @fref{umul umul_ff usquare usquare_ff utmul utmul_ff}.
                    192: @end table
                    193:
                    194: @node utrunc udecomp ureverse,,, $B0lJQ?tB?9`<0$N1i;;(B
                    195: @subsection @code{utrunc}, @code{udecomp}, @code{ureverse}
                    196: @findex utrunc
                    197: @findex udecomp
                    198: @findex ureverse
                    199:
                    200: @table @t
                    201: @item utrunc(@var{p},@var{d})
                    202: @itemx udecomp(@var{p},@var{d})
                    203: @itemx ureverse(@var{p})
                    204: :: $BB?9`<0$KBP$9$kA`:n(B
                    205: @end table
                    206:
                    207: @table @var
                    208: @item return
                    209: $B0lJQ?tB?9`<0$"$k$$$O0lJQ?tB?9`<0$N%j%9%H(B
                    210: @item p
                    211: $B0lJQ?tB?9`<0(B
                    212: @item d
                    213: $BHsIi@0?t(B
                    214: @end table
                    215:
                    216: @itemize @bullet
                    217: @item
                    218: @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}
                    219: (@var{p1} $B$N<!?t$O(B @var{d} $B0J2<(B) $B$HJ,2r$G$-$k(B. @code{utrunc()} $B$O(B
                    220: @var{p1} $B$rJV$7(B, @code{udecomp()} $B$O(B [@var{p1},@var{p2}] $B$rJV$9(B.
                    221: @item
                    222: @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,
                    223: @code{ureverse()} $B$O(B @var{p[e]}+@var{p[e-1]}x+... $B$rJV$9(B.
                    224: @end itemize
                    225:
                    226: @example
                    227: [132] utrunc((x+1)^10,5);
                    228: 252*x^5+210*x^4+120*x^3+45*x^2+10*x+1
                    229: [133] udecomp((x+1)^10,5);
                    230: [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]
                    231: [134] ureverse(3*x^3+x^2+2*x);
                    232: 2*x^2+x+3
                    233: @end example
                    234:
                    235: @table @t
                    236: @item $B;2>H(B
                    237: @fref{udiv urem urembymul urembymul_precomp ugcd}.
                    238: @end table
                    239:
                    240: @node uinv_as_power_series ureverse_inv_as_power_series,,, $B0lJQ?tB?9`<0$N1i;;(B
                    241: @subsection @code{uinv_as_power_series}, @code{ureverse_inv_as_power_series}
                    242: @findex uinv_as_power_series
                    243: @findex ureverse_inv_as_power_series
                    244:
                    245: @table @t
                    246: @item uinv_as_power_series(@var{p},@var{d})
                    247: @itemx ureverse_inv_as_power_series(@var{p},@var{d})
                    248: :: $BB?9`<0$rQQ5i?t$H$_$F(B, $B5U857W;;(B
                    249: @end table
                    250:
                    251: @table @var
                    252: @item return
                    253: $B0lJQ?tB?9`<0(B
                    254: @item p
                    255: $B0lJQ?tB?9`<0(B
                    256: @item d
                    257: $BHsIi@0?t(B
                    258: @end table
                    259:
                    260: @itemize @bullet
                    261: @item
                    262: @code{uinv_as_power_series(@var{p},@var{d})} $B$O(B, $BDj?t9`$,(B 0 $B$G$J$$(B
                    263: $BB?9`<0(B @var{p} $B$KBP$7(B, @var{p}@var{r}-1 $B$N:GDc<!?t$,(B @var{d}+1
                    264: $B0J>e$K$J$k$h$&$J(B $B9b!9(B @var{d} $B<!$NB?9`<0(B @var{r} $B$r5a$a$k(B.
                    265: @item
                    266: @code{ureverse_inv_as_power_series(@var{p},@var{d})} $B$O(B
                    267: @var{p} $B$N<!?t$r(B @var{e} $B$H$9$k$H$-(B, @var{p1}=@code{ureverse(@var{p},@var{e})}
                    268: $B$KBP$7$F(B @code{uinv_as_power_series(@var{p1},@var{d})} $B$r7W;;$9$k(B.
                    269: @item
                    270: @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.
                    271: @end itemize
                    272:
                    273: @example
                    274: [123] A=(x+1)^5;
                    275: x^5+5*x^4+10*x^3+10*x^2+5*x+1
                    276: [124] uinv_as_power_series(A,5);
                    277: -126*x^5+70*x^4-35*x^3+15*x^2-5*x+1
                    278: [126] A*R;
                    279: -126*x^10-560*x^9-945*x^8-720*x^7-210*x^6+1
                    280: [127] A=x^10+x^9;
                    281: x^10+x^9
                    282: [128] R=ureverse_inv_as_power_series(A,5);
                    283: -x^5+x^4-x^3+x^2-x+1
                    284: [129] ureverse(A)*R;
                    285: -x^6+1
                    286: @end example
                    287:
                    288: @table @t
                    289: @item $B;2>H(B
                    290: @fref{utrunc udecomp ureverse},
                    291: @fref{udiv urem urembymul urembymul_precomp ugcd}.
                    292: @end table
                    293:
                    294: @node udiv urem urembymul urembymul_precomp ugcd,,, $B0lJQ?tB?9`<0$N1i;;(B
                    295: @subsection @code{udiv}, @code{urem}, @code{urembymul}, @code{urembymul_precomp}, @code{ugcd}
                    296: @findex udiv
                    297: @findex urem
                    298: @findex urembymul
                    299: @findex urembymul_precomp
                    300: @findex ugcd
                    301:
                    302: @table @t
                    303: @item udiv(@var{p1},@var{p2})
                    304: @item urem(@var{p1},@var{p2})
                    305: @item urembymul(@var{p1},@var{p2})
                    306: @item urembymul_precomp(@var{p1},@var{p2},@var{inv})
                    307: @item ugcd(@var{p1},@var{p2})
                    308: :: $BB?9`<0$KBP$9$kA`:n(B
                    309: @end table
                    310:
                    311: @table @var
                    312: @item return
                    313: $B0lJQ?tB?9`<0(B
                    314: @item p1,p2,inv
                    315: $B0lJQ?tB?9`<0(B
                    316: @end table
                    317:
                    318: @itemize @bullet
                    319: @item
                    320: $B0lJQ?tB?9`<0(B @var{p1}, @var{p2} $B$KBP$7(B,
                    321: @code{udiv} $B$O>&(B, @code{urem}, @code{urembymul} $B$O>jM>(B,
                    322: @code{ugcd} $B$O(B GCD $B$rJV$9(B.
                    323: $B$3$l$i$O(B, $BL)$J0lJQ?tB?9`<0$KBP$9$k9bB.2=$r?^$C$?$b$N$G$"$k(B.
                    324: @code{urembymul} $B$O(B, @var{p2} $B$K$h$k>jM>7W;;$r(B, @var{p2} $B$N(B
                    325: $BQQ5i?t$H$7$F$N5U857W;;$*$h$S(B, $B>h;;(B 2 $B2s$KCV$-49$($?$b$N$G(B,
                    326: $B<!?t$,Bg$-$$>l9g$KM-8z$G$"$k(B.
                    327: @item @code{urembymul_precomp} $B$O(B, $B8GDj$5$l$?B?9`<0$K$h$k>jM>(B
                    328: $B7W;;$rB??t9T$&>l9g$J$I$K8z2L$rH/4x$9$k(B.
                    329: @end itemize
                    330:
                    331: @example
                    332: [177] setmod_ff(2^160-47);
                    333: 1461501637330902918203684832716283019655932542929
                    334: [178] A=randpoly_ff(200,x)$
                    335: [179] B=randpoly_ff(101,x)$
                    336: [180] cputime(1)$
                    337: 0sec(1.597e-05sec)
                    338: [181] srem(A,B)$
                    339: 0.15sec + gc : 0.15sec(0.3035sec)
                    340: [182] urem(A,B)$
                    341: 0.11sec + gc : 0.12sec(0.2347sec)
                    342: [183] urembymul(A,B)$
                    343: 0.08sec + gc : 0.09sec(0.1651sec)
                    344: [184] R=ureverse_inv_as_power_series(B,101)$
                    345: 0.04sec + gc : 0.03sec(0.063sec)
                    346: [185] urembymul_precomp(A,B,R)$
                    347: 0.03sec(0.02501sec)
                    348: @end example
                    349:
                    350: @table @t
                    351: @item $B;2>H(B
                    352: @fref{uinv_as_power_series ureverse_inv_as_power_series}.
                    353: @end table