mdw@git.distorted.org.uk Git - catacomb/blob - math/mp-sqrt.c

   1 /* -*-c-*-
   2  *
   3  * Compute integer square roots
   4  *
   5  * (c) 2000 Straylight/Edgeware
   6  */
   7
   8 /*----- Licensing notice --------------------------------------------------*
   9  *
  10  * This file is part of Catacomb.
  11  *
  12  * Catacomb is free software; you can redistribute it and/or modify
  13  * it under the terms of the GNU Library General Public License as
  14  * published by the Free Software Foundation; either version 2 of the
  15  * License, or (at your option) any later version.
  16  *
  17  * Catacomb is distributed in the hope that it will be useful,
  18  * but WITHOUT ANY WARRANTY; without even the implied warranty of
  19  * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  20  * GNU Library General Public License for more details.
  21  *
  22  * You should have received a copy of the GNU Library General Public
  23  * License along with Catacomb; if not, write to the Free
  24  * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
  25  * MA 02111-1307, USA.
  26  */
  27
  28 /*----- Header files ------------------------------------------------------*/
  29
  30 #include "mp.h"
  31
  32 /*----- Main code ---------------------------------------------------------*/
  33
  34 /* --- @mp_sqrt@ --- *
  35  *
  36  * Arguments:   @mp *d@ = pointer to destination integer
  37  *              @mp *a@ = (nonnegative) integer to take square root of
  38  *
  39  * Returns:     The largest integer %$x$% such that %$x^2 \le a$%.
  40  *
  41  * Use:         Computes integer square roots.
  42  *
  43  *              The current implementation isn't very good: it uses the
  44  *              Newton-Raphson method to find an approximation to %$a$%.  If
  45  *              there's any demand for a better version, I'll write one.
  46  */
  47
  48 mp *mp_sqrt(mp *d, mp *a)
  49 {
  50   unsigned long z;
  51   mp *q = MP_NEW, *r = MP_NEW;
  52
  53   /* --- Sanity preservation --- */
  54
  55   assert(!MP_NEGP(a));
  56
  57   /* --- Deal with trivial cases --- */
  58
  59   MP_SHRINK(a);
  60   if (MP_ZEROP(a)) {
  61     mp_drop(d);
  62     return (MP_ZERO);
  63   }
  64
  65   /* --- Find an initial guess of about the right size --- */
  66
  67   z = mp_bits(a);
  68   z >>= 1;
  69   mp_copy(a);
  70   d = mp_lsr(d, a, z);
  71
  72   /* --- Main approximation --- *
  73    *
  74    * The Newton--Raphson method finds approximate zeroes of a function by
  75    * starting with a guess and repeatedly refining the guess by approximating
  76    * the function near the guess by its tangent at the guess
  77    * %$x$%-coordinate, using where the tangent cuts the %$x$%-axis as the new
  78    * guess.
  79    *
  80    * Given a function %$f(x)$% and a guess %$x_i$%, the tangent has the
  81    * equation %$y = f(x_i) + f'(x_i) (x - x_i)$%, which is zero when
  82    *
  83    *    %$\displaystyle x = x_i - \frac{f(x_i)}{f'(x_i)}
  84    *
  85    * We set %$f(x) = x^2 - a$%, so our recurrence will be
  86    *
  87    *    %$\displaystyle x_{i+1} = x_i - \frac{x_i^2 - a}{2 x_i}$%
  88    *
  89    * It's possible to simplify this, but it's useful to see %$q = x_i^2 - a$%
  90    * so that we know when to stop.  We want the largest integer not larger
  91    * than the true square root.  If %$q > 0$% then %$x_i$% is definitely too
  92    * large, and we should decrease it by at least one even if the adjustment
  93    * term %$(x_i^2 - a)/2 x$% is less than one.
  94    *
  95    * Suppose, then, that %$q \le 0$%.  Then %$(x_i + 1)^2 - a = {}$%
  96    * $%x_i^2 + 2 x_i + 1 - a = q + 2 x_i + 1$%.  Hence, if %$q \ge -2 x_i$%
  97    * then %$x_i + 1$% is an overestimate and we should settle where we are.
  98    * Otherwise, %$x_i + 1$% is an underestimate -- but, in this case the
  99    * adjustment will always be at least one.
 100    */
 101
 102   for (;;) {
 103     q = mp_sqr(q, d);
 104     q = mp_sub(q, q, a);
 105     if (MP_ZEROP(q))
 106       break;
 107     if (MP_NEGP(q)) {
 108       r = mp_lsl(r, d, 1);
 109       r->f |= MP_NEG;
 110       if (MP_CMP(q, >=, r))
 111         break;
 112     }
 113     mp_div(&r, &q, q, d);
 114     r = mp_lsr(r, r, 1);
 115     if (r->v == r->vl)
 116       d = mp_sub(d, d, MP_ONE);
 117     else
 118       d = mp_sub(d, d, r);
 119   }
 120
 121   /* --- Finished, at last --- */
 122
 123   mp_drop(a);
 124   mp_drop(q);
 125   mp_drop(r);
 126   return (d);
 127 }
 128
 129 /* --- @mp_squarep@ --- *
 130  *
 131  * Arguments:   @mp *n@ = an integer
 132  *
 133  * Returns:     Nonzero if and only if @n@ is a perfect square, i.e.,
 134  *              %$n = a^2$% for some rational integer %$a$%.
 135  */
 136
 137 int mp_squarep(mp *n)
 138 {
 139   mp *t = MP_NEW;
 140   int rc;
 141
 142   if (MP_NEGP(n)) return (0);
 143   t = mp_sqrt(t, n); t = mp_sqr(t, t);
 144   rc = MP_EQ(t, n); mp_drop(t); return (rc);
 145 }
 146
 147 /*----- Test rig ----------------------------------------------------------*/
 148
 149 #ifdef TEST_RIG
 150
 151 #include <mLib/testrig.h>
 152
 153 static int verify(dstr *v)
 154 {
 155   mp *a = *(mp **)v[0].buf;
 156   mp *qq = *(mp **)v[1].buf;
 157   mp *q = mp_sqrt(MP_NEW, a);
 158   int ok = 1;
 159
 160   if (!MP_EQ(q, qq)) {
 161     ok = 0;
 162     fputs("\n*** sqrt failed", stderr);
 163     fputs("\n*** a      = ", stderr); mp_writefile(a, stderr, 10);
 164     fputs("\n*** result = ", stderr); mp_writefile(q, stderr, 10);
 165     fputs("\n*** expect = ", stderr); mp_writefile(qq, stderr, 10);
 166     fputc('\n', stderr);
 167   }
 168
 169   mp_drop(a);
 170   mp_drop(q);
 171   mp_drop(qq);
 172   assert(mparena_count(MPARENA_GLOBAL) == 0);
 173
 174   return (ok);
 175 }
 176
 177 static test_chunk tests[] = {
 178   { "sqrt", verify, { &type_mp, &type_mp, 0 } },
 179   { 0, 0, { 0 } },
 180 };
 181
 182 int main(int argc, char *argv[])
 183 {
 184   sub_init();
 185   test_run(argc, argv, tests, SRCDIR "/t/mp");
 186   return (0);
 187 }
 188
 189 #endif
 190
 191 /*----- That's all, folks -------------------------------------------------*/