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|
// file kernel/n/c/toom.c: Toom multiplication of natural integers
/*-----------------------------------------------------------------------+
| Copyright 2005-2006, Michel Quercia (michel.quercia@prepas.org) |
| |
| This file is part of Numerix. Numerix is free software; you can |
| redistribute it and/or modify it under the terms of the GNU Lesser |
| General Public License as published by the Free Software Foundation; |
| either version 2.1 of the License, or (at your option) any later |
| version. |
| |
| The Numerix Library is distributed in the hope that it will be |
| useful, but WITHOUT ANY WARRANTY; without even the implied warranty |
| of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
| Lesser General Public License for more details. |
| |
| You should have received a copy of the GNU Lesser General Public |
| License along with the GNU MP Library; see the file COPYING. If not, |
| write to the Free Software Foundation, Inc., 59 Temple Place - |
| Suite 330, Boston, MA 02111-1307, USA. |
+-----------------------------------------------------------------------+
| |
| Multiplication de Toom |
| |
+-----------------------------------------------------------------------*/
/* +------------------+
| Multiplication |
+------------------+ */
/*
entre :
a = naturel de longueur la
b = naturel de longueur lb
c = naturel de longueur la + lb non confondu avec a ou b
contraintes : 0 < lb <= la
sortie :
c <- a*b
*/
#ifndef assembly_sn_toommul
#ifdef debug_toommul
void xn(toommul_buggy)
#else
void xn(toommul)
#endif
(chiffre *a, long la, chiffre *b, long lb, chiffre *c) {
chiffre *d,*e,*f,x;
long i,p,q,r;
/* petite multiplication -> Karatsuba */
if (lb <= toommul_lim) {xn(karamul)(a,la,b,lb,c); return;}
/* si lb > 2*ceil(la/3), dcoupage de Toom */
p = (la+2)/3; q = la - 2*p; r = lb -2*p;
if (r > 0) {
/* mmoire de travail */
d = xn(alloc_tmp)(6*p+10);
e = d + 2*p+2;
f = e + 2*p+2;
/* d <- (a0 + a1 + a2)(b0 + b1 + b2) = c0 + c1 + c2 + c3 + c4 */
c[p] = xn(add)(a,p, a+2*p,q, c);
c[2*p+1] = xn(add)(b,p, b+2*p,r, c+p+1);
xn(add)(c, p+1, a+p,p, c+2*p+2);
xn(add)(c+p+1,p+1, b+p,p, c+3*p+3);
xn(toommul)(c+2*p+2,p+1, c+3*p+3,p+1, d);
/* e <- (a0 - a1 + a2)(b0 - b1 + b2) mod BASE^2p+2 = c0 - c1 + c2 - c3 + c4 */
xn(dec)(c,p+1, a+p,p);
xn(dec)(c+p+1,p+1, b+p,p);
xn(toommul)(c,p+1, c+p+1,p+1,e);
if (c[p] == (BASE_2)+(BASE_2-1)) xn(dec)(e+p+1,p+1, c+p+1,p+1);
if (c[2*p+1] == (BASE_2)+(BASE_2-1)) xn(dec)(e+p+1,p+1, c, p+1);
/* f <- (a0 + BASE*a1 + BASE^2*a2)*(b0 + BASE*b1 + BASE^2*b2)
= c0 + BASE*c1 + BASE^2*c2 + BASE^3*c3 + BASE^4*c4 */
c[0] = a[0];
c[p+1] = xn(add)(a+p,p, a+1,p-1, c+1);
c[p+2] = xn(inc)(c+2,p, a+2*p,q);
c[p+3] = b[0];
c[2*p+4] = xn(add)(b+p,p, b+1,p-1, c+p+4);
c[2*p+5] = xn(inc)(c+p+5,p, b+2*p,r);
xn(toommul)(c,p+3, c+p+3,p+3, f);
/* c[2p..4p] <- (d+e)/2 = c0 + c2 + c4, d <- (d-e)/2 mod BASE^(2p+1) = c1 + c3 */
xn(add)(d,2*p+2, e,2*p+2, c+2*p);
for (i=2*p; i <= 4*p; i++) c[i] = (c[i+1] & 1) ? (c[i]/2) + (BASE_2) : c[i]/2;
xn(dec)(d,2*p+1, c+2*p,2*p+1);
/* c[0..2p-1] <- a0*b0 = c0, c[4*p..4p+q+r-1] <- a2*b2 = c4 */
x = c[4*p];
xn(toommul)(a,p, b,p, c);
xn(toommul)(a+2*p,q, b+2*p,r, c+4*p);
/* c[2p..4p] <- c[2p..4p] - c0 - c4 = c2 */
x -= xn(dec)(c+2*p,2*p, c,2*p);
x -= xn(dec)(c+2*p,2*p, c+4*p,q+r);
/* f <- f - c0 - BASE^2*c2 - BASE^4*c4 = BASE*c1 + BASE^3*c3 */
xn(dec)(f, 2*p+6, c, 2*p);
xn(dec)(f+2,2*p+4, c+2*p,2*p); xn(dec)(f+2*p+2,4, &x,1);
xn(dec)(f+4,2*p+2, c+4*p,q+r);
/* f <- -(f - BASE*d)/(BASE^2 - 1) = -BASE*c3 */
xn(dec)(f+1,2*p+5, d, 2*p+2);
xn(inc)(f+3,2*p+3, f+1,2*p+3);
if (f[2*p+5]) { /* on a (f[2p+5] != 0) <=> (c3 != 0) */
/* injecte c3 dans c */
if ((!xn(dec)(c+3*p,p+q+1, f+1,p+q+1)) && (r>1)) xn(inc1)(c+4*p+q+1,r-1);
/* d <- d + f/BASE = c1 */
xn(inc)(d,2*p+1, f+1,2*p+1);
}
/* injecte c1 et x dans c */
xn(inc)(c+p,3*p+q+r, d,2*p+1);
xn(inc)(c+4*p, q+r, &x,1);
xn(free_tmp)(d); /* libre la mmoire auxilliaire */
}
/* si lb <= 2*ceil(la/3), dcoupe a en tranches de lb chiffres */
else {
p = la % lb; if (p == 0) p = lb;
xn(toommul)(b,lb,a,p,c); /* 1re multiplication */
d = xn(alloc_tmp)(lb); /* tampon pour les mult. suivantes */
/* multiplie les tranches suivantes et les cumule dans c */
for (a+=p, la-=p, c+=p; la; a+=lb, la-=lb, c+=lb) {
xn(move)(c,lb,d);
xn(toommul)(a,lb, b,lb, c);
xn(inc)(c,2*lb,d,lb);
}
xn(free_tmp)(d); /* libre la mmoire auxilliaire */
}
}
#endif /* assembly_sn_toommul */
/* +---------+
| Carr |
+---------+ */
/*
entre :
a = naturel de longueur la
b = naturel de longueur 2*la, non confondu avec a
contraintes : 0 < la
sortie :
b <- a^2
*/
#ifndef assembly_sn_toomsqr
#ifdef debug_toommul
void xn(toomsqr_buggy)
#else
void xn(toomsqr)
#endif
(chiffre *a, long la, chiffre *b) {
chiffre *d,*e,*f,x;
long i,p,q;
/* petit carr -> Karatsuba */
if (la <= toomsqr_lim) {xn(karasqr)(a,la,b); return;}
/* dcoupage de Toom */
p = (la+2)/3; q = la - 2*p;
/* mmoire de travail */
d = xn(alloc_tmp)(6*p+10);
e = d + 2*p+2;
f = e + 2*p+2;
/* d <- (a0 + a1 + a2)^2 = b0 + b1 + b2 + b3 + b4 */
b[p] = xn(add)(a,p, a+2*p,q, b);
xn(add)(b, p+1, a+p,p, b+2*p+2);
xn(toomsqr)(b+2*p+2,p+1, d);
/* e <- (a0 - a1 + a2)^2 mod BASE^2p+2 = b0 - b1 + b2 - b3 + b4 */
xn(dec)(b,p+1, a+p,p);
xn(toomsqr)(b,p+1,e);
if (b[p] == (BASE_2)+(BASE_2-1)) {
xn(dec)(e+p+1,p+1, b,p+1);
xn(dec)(e+p+1,p+1, b,p+1);
}
/* f <- (a0 + BASE*a1 + BASE^2*a2)^2
= b0 + BASE*b1 + BASE^2*b2 + BASE^3*b3 + BASE^4*b4 */
b[0] = a[0];
b[p+1] = xn(add)(a+p,p, a+1,p-1, b+1);
b[p+2] = xn(inc)(b+2,p, a+2*p,q);
xn(toomsqr)(b,p+3, f);
/* b[2p..4p] <- (d+e)/2 = c0 + c2 + c4, d <- (d-e)/2 mod BASE^(2p+1) = b1 + b3 */
xn(add)(d,2*p+2, e,2*p+2, b+2*p);
for (i=2*p; i <= 4*p; i++) b[i] = (b[i+1] & 1) ? (b[i]/2) + (BASE_2) : b[i]/2;
xn(dec)(d,2*p+1, b+2*p,2*p+1);
/* b[0..2p-1] <- a0^2 = c0, b[4*p..4p+q+r-1] <- a2^2 = b4 */
x = b[4*p];
xn(toomsqr)(a,p,b);
xn(toomsqr)(a+2*p,q, b+4*p);
/* b[2p..4p] <- b[2p..4p] - b0 - b4 = b2 */
x -= xn(dec)(b+2*p,2*p, b,2*p);
x -= xn(dec)(b+2*p,2*p, b+4*p,2*q);
/* f <- f - b0 - BASE^2*b2 - BASE^4*b4 = BASE*b1 + BASE^3*b3 */
xn(dec)(f, 2*p+6, b, 2*p);
xn(dec)(f+2,2*p+4, b+2*p,2*p); xn(dec)(f+2*p+2,4, &x,1);
xn(dec)(f+4,2*p+2, b+4*p,2*q);
/* f <- -(f - BASE*d)/(BASE^2 - 1) = -BASE*b3 */
xn(dec)(f+1,2*p+5, d, 2*p+2);
xn(inc)(f+3,2*p+3, f+1,2*p+3);
if (f[2*p+5]) { /* on a (f[2p+5] != 0) <=> (b3 != 0) */
/* injecte b3 dans b */
if (!xn(dec)(b+3*p,p+q+1, f+1,p+q+1)) xn(inc1)(b+4*p+q+1,q-1);
/* d <- d + f/BASE = b1 */
xn(inc)(d,2*p+1, f+1,2*p+1);
}
/* injecte b1 et x dans b */
xn(inc)(b+p,3*p+2*q, d,2*p+1);
xn(inc)(b+4*p, 2*q, &x,1);
xn(free_tmp)(d); /* libre la mmoire auxilliaire */
}
#endif /* assembly_sn_toomsqr */
/* +------------+
| Contrle |
+------------+ */
#ifdef debug_toommul
void xn(toommul_buggy)(chiffre *a, long la, chiffre *b, long lb, chiffre *c);
void xn(toomsqr_buggy)(chiffre *a, long la, chiffre *b);
void xn(toommul)(chiffre *a, long la, chiffre *b, long lb, chiffre *c) {
chiffre *d;
/* vrifie les longueurs des arguments */
if (la < lb) xn(internal_error)("error, toommul is called with la < lb",0);
/* compare les rsultats produits par toommul_buggy et par karamul */
d = xn(alloc_tmp)(la+lb);
xn(karamul)(a,la,b,lb,d);
xn(toommul_buggy)(a,la,b,lb,c);
if (xn(cmp)(c,la+lb,d,la+lb))
xn(internal_error)("error in toommul",4,a,la,b,lb,c,la+lb,d,la+lb);
xn(free_tmp)(d);
}
void xn(toomsqr)(chiffre *a, long la, chiffre *b) {
chiffre *d;
/* compare les rsultats produits par toomsqr_buggy et par karasqr */
d = xn(alloc_tmp)(2*la);
xn(karasqr)(a,la,d);
xn(toomsqr_buggy)(a,la,b);
if (xn(cmp)(b,2*la,d,2*la))
xn(internal_error)("error in toomsqr",3,a,la,b,2*la,d,2*la);
xn(free_tmp)(d);
}
#endif /* debug_toommul */
|
