/*
* http://en.wikipedia.org/wiki/Pell's_equation
*
* We find the continued fraction expansion of sqrt(d), and
* expand out the convergents as h/k. At some point, the
* solution x=h, y=k will satisfy the equation and this is
* guaranteed to be minimal in x.
*
* To find the continued fraction expansion of sqrt(d):
* http://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Continued_fraction_expansion
*
* To find the convergents:
* http://www.proofwiki.org/wiki/Definition:Continued_Fraction/Numerators_and_Denominators
* (our h, k are their p, q)
*/
static void minx(int d, mpz_t x) {
int a0 = sqrt(d);
int mn = 0, dn = 1, an = a0;
int m1, d1, a1;
mpz_t h, k; // h_n, k_n
mpz_t h1, k1; // h_{n-1}, k_{n-1}
mpz_t h2, k2; // h_{n-2}, k_{n-2}
mpz_t hsq, ksq; // h^2, k^2
mpz_init(h); mpz_init(k);
mpz_init(h1); mpz_init(k1);
mpz_init(h2); mpz_init(k2);
mpz_init(hsq); mpz_init(ksq);
mpz_set_ui(h1, 1);
mpz_set_ui(k1, 0);
mpz_set_ui(h, a0);
mpz_set_ui(k, 1);
do {
// Get the next a_n in the continued fraction
m1 = dn * an - mn;
d1 = (d - m1*m1) / dn;
a1 = (a0 + m1) / d1;
mn = m1;
dn = d1;
an = a1;
// Get the next convergent
mpz_set(h2, h1);
mpz_set(k2, k1);
mpz_set(h1, h);
mpz_set(k1, k);
mpz_mul_ui(h, h1, an); mpz_add(h, h, h2); // h_n = a_n*h_{n-1} + h_{n-2}
mpz_mul_ui(k, k1, an); mpz_add(k, k, k2); // k_n = a_n*k_{n-1} + k_{n-2}
// Check if h, k is a solution
mpz_mul(hsq, h, h);
mpz_mul(ksq, k, k);
mpz_submul_ui(hsq, ksq, d);
} while (mpz_cmp_ui(hsq, 1) != 0);
mpz_set(x, h);
mpz_clear(h); mpz_clear(k);
mpz_clear(h1); mpz_clear(k1);
mpz_clear(h2); mpz_clear(k2);
mpz_clear(hsq); mpz_clear(ksq);
}
void
testmain (int argc, char **argv)
{
unsigned i;
mpz_t a, b, res, ref;
mpz_init (a);
mpz_init (b);
mpz_init_set_ui (res, 5);
mpz_init (ref);
for (i = 0; i < COUNT; i++)
{
mini_random_op3 (OP_MUL, MAXBITS, a, b, ref);
if (i & 1) {
mpz_add (ref, ref, res);
if (mpz_fits_ulong_p (b))
mpz_addmul_ui (res, a, mpz_get_ui (b));
else
mpz_addmul (res, a, b);
} else {
mpz_sub (ref, res, ref);
if (mpz_fits_ulong_p (b))
mpz_submul_ui (res, a, mpz_get_ui (b));
else
mpz_submul (res, a, b);
}
if (mpz_cmp (res, ref))
{
if (i & 1)
fprintf (stderr, "mpz_addmul failed:\n");
else
fprintf (stderr, "mpz_submul failed:\n");
dump ("a", a);
dump ("b", b);
dump ("r", res);
dump ("ref", ref);
abort ();
}
}
mpz_clear (a);
mpz_clear (b);
mpz_clear (res);
mpz_clear (ref);
}
void
check_one_ui (mpz_ptr w, mpz_ptr x, unsigned long y)
{
mpz_t want, got;
mpz_init (want);
mpz_init (got);
mpz_mul_ui (want, x, (unsigned long) y);
mpz_add (want, w, want);
mpz_set (got, w);
mpz_addmul_ui (got, x, (unsigned long) y);
MPZ_CHECK_FORMAT (got);
if (mpz_cmp (want, got) != 0)
{
printf ("mpz_addmul_ui fail\n");
fail:
mpz_trace ("w", w);
mpz_trace ("x", x);
printf ("y=0x%lX %lu\n", y, y);
mpz_trace ("want", want);
mpz_trace ("got ", got);
abort ();
}
mpz_mul_ui (want, x, y);
mpz_sub (want, w, want);
mpz_set (got, w);
mpz_submul_ui (got, x, y);
MPZ_CHECK_FORMAT (got);
if (mpz_cmp (want, got) != 0)
{
printf ("mpz_submul_ui fail\n");
goto fail;
}
mpz_clear (want);
mpz_clear (got);
}
static void eliminate_digit(unsigned int d)
{
mpz_submul_ui(accum, denom, d);
mpz_mul_ui(accum, accum, 10);
mpz_mul_ui(numer, numer, 10);
}
void eliminate_digit(ui d) {
mpz_submul_ui(acc, den, d);
mpz_mul_ui(acc, acc, 10);
mpz_mul_ui(num, num, 10);
}
Integer& Integer::maxpyin(Integer& res, const Integer& a, const uint64_t x)
{
if (isZero(a) || isZero(x)) return res;
mpz_submul_ui( (mpz_ptr)&res.gmp_rep, (mpz_srcptr)&a.gmp_rep, x);
return res;
}
