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|
// file kernel/n/c/sqrt_n2.c: O(n^2) square root 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. |
+-----------------------------------------------------------------------+
| |
| Racine carre quadratique |
| |
+-----------------------------------------------------------------------*/
/* +-----------------------------+
| Racine carre quadratique |
+-----------------------------+ */
/*
entre :
a = naturel de longueur la
b = naturel de longueur la/2
contraintes :
la > 0, la pair, BASE/16 <= a[la-1] < BASE/4
a,b non confondus
sortie :
b <- 2*floor(sqrt(a))
a <- a - b^2/4
*/
#ifndef assembly_sn_sqrt_n2
#ifdef debug_sqrt_n2
void xn(sqrt_n2_buggy)
#else
void xn(sqrt_n2)
#endif
(chiffre *a, long la, chiffre *b) {
#ifdef zdouble
chiffre u,v;
ndouble x;
zdouble z;
long i,lb;
/* avance a et b aux chiffres de tte */
a += la-2; b += la/2 - 1;
/* b[0] <- 2*floor(sqrt(a[0]+BASE*a[1])), a <- a - b^2/4 */
x = (ndouble)a[0] + ((ndouble)a[1] << HW);
for (u=BASE_2, v=(u+x/u)/2; v < u; u=v, v=(u+x/u)/2);
a[0] = x - u*u;
a[1] = 0;
b[0] = u << 1;
/* calcule les chiffres suivants par divisions */
for (lb=2, la-=2, a-=2, b--; la; lb++, la-=2, a-=2, b--) {
/* quotient approch, peut tre trop grand d'une ou deux units */
if ((ndouble)a[lb] >= (ndouble)b[lb-1]) v = (BASE_2)+(BASE_2-1);
else {
x = (ndouble)a[lb-1] + ((ndouble)a[lb] << HW);
v = x/(ndouble)b[lb-1];
}
/* a <- a - v*b - v^2, b <- b + 2*v */
b[0] = v;
for (x=0, z=0, i=0; i<lb; i++) {
x += v*(ndouble)b[i];
z += (ndouble)a[i] - (x & ((BASE_2)+(BASE_2-1)));
a[i] = z;
x >>= HW; z >>= HW;
}
z += (ndouble)a[lb] - x;
a[lb] = z;
b[0] = v << 1;
if (v & (BASE_2)) b[1]++;
/* corrige le quotient et le reste si < 0 */
while(a[lb]) {
xn(dec1)(b,lb);
xn(inc)(a,lb+1,b,lb);
b[0]--;
}
}
#else /* pas de doubles */
chiffre u,v,x,y;
long i,lb;
/* avance a et b aux chiffres de tte */
a += la-2; b += la/2 - 1;
/* b[0] <- 2*floor(sqrt(a[0]+BASE*a[1])), a <- a - b^2/4 */
for (u = 0, v = BASE_2; v; v >>= 1) {
u += v;
xn(sqr_0)(u,&x,&y);
if ((a[1] < y) || ((a[1] == y) && (a[0] < x))) u -= v;
}
a[0] -= u*u;
a[1] = 0;
b[0] = u << 1;
/* calcule les chiffres suivants par divisions */
for (lb=2, la-=2, a-=2, b--; la; lb++, la-=2, a-=2, b--) {
/* quotient approch, peut tre trop grand d'une ou deux units */
if (a[lb] >= b[lb-1]) v = (BASE_2)+(BASE_2-1);
else xn(div_0)(a[lb-1],a[lb],b[lb-1],&v,&u);
/* a <- a - v*b - v^2, b <- b + 2*v */
b[0] = v;
for (u=0, i=0; i<lb; i++) {
xn(mul_0)(v,b[i],&x,&y);
x += u; u = y + (x < u) + (a[i] < x);
a[i] -= x;
}
a[lb] -= u;
b[0] = v << 1;
if (v & (BASE_2)) b[1]++;
/* corrige le quotient et le reste si < 0 */
while(a[lb]) {
xn(dec1)(b,lb);
xn(inc)(a,lb+1,b,lb);
b[0]--;
}
}
#endif /* ndouble */
}
#endif /* assembly_sn_sqrt_n2 */
/* +------------+
| Contrle |
+------------+ */
#ifdef debug_sqrt_n2
void xn(sqrt_n2_buggy)(chiffre *a, long la, chiffre *b);
void xn(sqrt_n2)(chiffre *a, long la, chiffre *b) {
long lb = la/2;
chiffre *x,*y, r;
/* vrifie que la est pair > 0 et BASE/16 <= a[la-1] < BASE/4 */
if ((la%2) || (la < 2))
xn(internal_error)("error, sqrt_n2 is called with la odd or la < 2",0);
if ((a[la-1] < (chiffre)(BASE_2/8)) || (a[la-1] >= (chiffre)(BASE_2/2)))
xn(internal_error)("error, sqrt_n2 is called without BASE/16 <= msb(la) < BASE/4",1,a,la);
/* calcule la racine carre douteuse */
x = xn(alloc_tmp)(2*la); y = x + la;
xn(move)(a,la,x);
xn(sqrt_n2_buggy)(a,la,b);
/* vrifie que a_entre = a_sortie + (b/2)^2 et a_sortie <= b */
xn(toomsqr)(b,lb,y);
r = xn(shift_down)(y,la,y,2);
if (r == 0) r = xn(inc)(y,la,a,lb);
if (r == 0) r = xn(cmp)(x,la,y,la);
if (r == 0) r = (xn(cmp)(a,lb,b,lb) > 0);
if (r) xn(internal_error)("error in sqrt_n2", 3,x,la,b,lb,a,lb);
xn(free_tmp)(x);
}
#endif /* debug_sqrt_n2 */
|
