/*
* find log(N)
*
* if places < 360
* solve with cubically convergent algorithm above
*
* else
*
* let 'X' be 'close' to the solution (we use ~110 decimal places)
*
* let Y = N * exp(-X) - 1
*
* then
*
* log(N) = X + log(1 + Y)
*
* since 'Y' will be small, we can use the efficient log_near_1 algorithm.
*
*/
void M_log_basic_iteration(M_APM rr, int places, M_APM nn)
{
M_APM tmp0, tmp1, tmp2, tmpX;
if (places < 360)
{
M_log_solve_cubic(rr, places, nn);
}
else
{
tmp0 = M_get_stack_var();
tmp1 = M_get_stack_var();
tmp2 = M_get_stack_var();
tmpX = M_get_stack_var();
M_log_solve_cubic(tmpX, 110, nn);
m_apm_negate(tmp0, tmpX);
m_apm_exp(tmp1, (places + 8), tmp0);
m_apm_multiply(tmp2, tmp1, nn);
m_apm_subtract(tmp1, tmp2, MM_One);
M_log_near_1(tmp0, (places - 104), tmp1);
m_apm_add(tmp1, tmpX, tmp0);
m_apm_round(rr, places, tmp1);
M_restore_stack(4);
}
}
/*
* return the nearest integer <= input
*/
void m_apm_floor(M_APM bb, M_APM aa)
{
M_APM mtmp;
m_apm_copy(bb, aa);
if (m_apm_is_integer(bb)) /* if integer, we're done */
return;
if (bb->m_apm_exponent <= 0) /* if |bb| < 1, result is -1 or 0 */
{
if (bb->m_apm_sign < 0)
m_apm_negate(bb, MM_One);
else
M_set_to_zero(bb);
return;
}
if (bb->m_apm_sign < 0)
{
mtmp = M_get_stack_var();
m_apm_negate(mtmp, bb);
mtmp->m_apm_datalength = mtmp->m_apm_exponent;
M_apm_normalize(mtmp);
m_apm_add(bb, mtmp, MM_One);
bb->m_apm_sign = -1;
M_restore_stack(1);
}
else
{
bb->m_apm_datalength = bb->m_apm_exponent;
M_apm_normalize(bb);
}
}
void m_apm_sqrt(M_APM rr, int places, M_APM aa)
{
M_APM last_x, guess, tmpN, tmp7, tmp8, tmp9;
int ii, bflag, nexp, tolerance, dplaces;
if (aa->m_apm_sign <= 0)
{
if (aa->m_apm_sign == -1)
{
M_apm_log_error_msg(M_APM_RETURN, "\'m_apm_sqrt\', Negative argument");
}
M_set_to_zero(rr);
return;
}
last_x = M_get_stack_var();
guess = M_get_stack_var();
tmpN = M_get_stack_var();
tmp7 = M_get_stack_var();
tmp8 = M_get_stack_var();
tmp9 = M_get_stack_var();
m_apm_copy(tmpN, aa);
/*
normalize the input number (make the exponent near 0) so
the 'guess' function will not over/under flow on large
magnitude exponents.
*/
nexp = aa->m_apm_exponent / 2;
tmpN->m_apm_exponent -= 2 * nexp;
M_get_sqrt_guess(guess, tmpN); /* actually gets 1/sqrt guess */
tolerance = places + 4;
dplaces = places + 16;
bflag = FALSE;
m_apm_negate(last_x, MM_Ten);
/* Use the following iteration to calculate 1 / sqrt(N) :
X = 0.5 * X * [ 3 - N * X^2 ]
n+1
*/
ii = 0;
while (TRUE)
{
m_apm_multiply(tmp9, tmpN, guess);
m_apm_multiply(tmp8, tmp9, guess);
m_apm_round(tmp7, dplaces, tmp8);
m_apm_subtract(tmp9, MM_Three, tmp7);
m_apm_multiply(tmp8, tmp9, guess);
m_apm_multiply(tmp9, tmp8, MM_0_5);
if (bflag)
break;
m_apm_round(guess, dplaces, tmp9);
/* force at least 2 iterations so 'last_x' has valid data */
if (ii != 0)
{
m_apm_subtract(tmp7, guess, last_x);
if (tmp7->m_apm_sign == 0)
break;
/*
* if we are within a factor of 4 on the error term,
* we will be accurate enough after the *next* iteration
* is complete. (note that the sign of the exponent on
* the error term will be a negative number).
*/
if ((-4 * tmp7->m_apm_exponent) > tolerance)
bflag = TRUE;
}
m_apm_copy(last_x, guess);
ii++;
}
/*
* multiply by the starting number to get the final
* sqrt and then adjust the exponent since we found
* the sqrt of the normalized number.
*/
m_apm_multiply(tmp8, tmp9, tmpN);
m_apm_round(rr, places, tmp8);
rr->m_apm_exponent += nexp;
M_restore_stack(6);
}
void m_apm_reciprocal(M_APM rr, int places, M_APM aa)
{
M_APM last_x, guess, tmpN, tmp1, tmp2;
int ii, bflag, dplaces, nexp, tolerance;
if (aa->m_apm_sign == 0)
{
M_apm_log_error_msg(M_APM_RETURN,
"Warning! ... \'m_apm_reciprocal\', Input = 0");
M_set_to_zero(rr);
return;
}
last_x = M_get_stack_var();
guess = M_get_stack_var();
tmpN = M_get_stack_var();
tmp1 = M_get_stack_var();
tmp2 = M_get_stack_var();
m_apm_absolute_value(tmpN, aa);
/*
normalize the input number (make the exponent 0) so
the 'guess' below will not over/under flow on large
magnitude exponents.
*/
nexp = aa->m_apm_exponent;
tmpN->m_apm_exponent -= nexp;
m_apm_set_double(guess, (1.0 / m_apm_get_double(tmpN)));
tolerance = places + 4;
dplaces = places + 16;
bflag = FALSE;
m_apm_negate(last_x, MM_Ten);
/* Use the following iteration to calculate the reciprocal :
X = X * [ 2 - N * X ]
n+1
*/
ii = 0;
while (TRUE)
{
m_apm_multiply(tmp1, tmpN, guess);
m_apm_subtract(tmp2, MM_Two, tmp1);
m_apm_multiply(tmp1, tmp2, guess);
if (bflag)
break;
m_apm_round(guess, dplaces, tmp1);
/* force at least 2 iterations so 'last_x' has valid data */
if (ii != 0)
{
m_apm_subtract(tmp2, guess, last_x);
if (tmp2->m_apm_sign == 0)
break;
/*
* if we are within a factor of 4 on the error term,
* we will be accurate enough after the *next* iteration
* is complete.
*/
if ((-4 * tmp2->m_apm_exponent) > tolerance)
bflag = TRUE;
}
m_apm_copy(last_x, guess);
ii++;
}
m_apm_round(rr, places, tmp1);
rr->m_apm_exponent -= nexp;
rr->m_apm_sign = aa->m_apm_sign;
M_restore_stack(5);
}
void m_apm_negate_mt(M_APM d, M_APM s)
{
m_apm_enter();
m_apm_negate(d,s);
m_apm_leave();
}
MAPM MAPM::neg(void) const
{
MAPM ret;
m_apm_negate(ret.val(),cval());
return ret;
}
/*
* From Knuth, The Art of Computer Programming:
*
* This is the binary GCD algorithm as described
* in the book (Algorithm B)
*/
void m_apm_gcd(M_APM r, M_APM u, M_APM v)
{
M_APM tmpM, tmpN, tmpT, tmpU, tmpV;
int kk, kr, mm;
long pow_2;
/* 'is_integer' will return 0 || 1 */
if ((m_apm_is_integer(u) + m_apm_is_integer(v)) != 2)
{
M_apm_log_error_msg(M_APM_RETURN,
"Warning! \'m_apm_gcd\', Non-integer input");
M_set_to_zero(r);
return;
}
if (u->m_apm_sign == 0)
{
m_apm_absolute_value(r, v);
return;
}
if (v->m_apm_sign == 0)
{
m_apm_absolute_value(r, u);
return;
}
tmpM = M_get_stack_var();
tmpN = M_get_stack_var();
tmpT = M_get_stack_var();
tmpU = M_get_stack_var();
tmpV = M_get_stack_var();
m_apm_absolute_value(tmpU, u);
m_apm_absolute_value(tmpV, v);
/* Step B1 */
kk = 0;
while (TRUE)
{
mm = 1;
if (m_apm_is_odd(tmpU))
break;
mm = 0;
if (m_apm_is_odd(tmpV))
break;
m_apm_multiply(tmpN, MM_0_5, tmpU);
m_apm_copy(tmpU, tmpN);
m_apm_multiply(tmpN, MM_0_5, tmpV);
m_apm_copy(tmpV, tmpN);
kk++;
}
/* Step B2 */
if (mm)
{
m_apm_negate(tmpT, tmpV);
goto B4;
}
m_apm_copy(tmpT, tmpU);
/* Step: */
B3:
m_apm_multiply(tmpN, MM_0_5, tmpT);
m_apm_copy(tmpT, tmpN);
/* Step: */
B4:
if (m_apm_is_even(tmpT))
goto B3;
/* Step B5 */
if (tmpT->m_apm_sign == 1)
m_apm_copy(tmpU, tmpT);
else
m_apm_negate(tmpV, tmpT);
/* Step B6 */
m_apm_subtract(tmpT, tmpU, tmpV);
if (tmpT->m_apm_sign != 0)
goto B3;
/*
* result = U * 2 ^ kk
*/
if (kk == 0)
m_apm_copy(r, tmpU);
else
{
if (kk == 1)
m_apm_multiply(r, tmpU, MM_Two);
if (kk == 2)
m_apm_multiply(r, tmpU, MM_Four);
if (kk >= 3)
{
mm = kk / 28;
kr = kk % 28;
pow_2 = 1L << kr;
if (mm == 0)
{
m_apm_set_long(tmpN, pow_2);
m_apm_multiply(r, tmpU, tmpN);
}
else
{
m_apm_copy(tmpN, MM_One);
m_apm_set_long(tmpM, 0x10000000L); /* 2 ^ 28 */
while (TRUE)
{
m_apm_multiply(tmpT, tmpN, tmpM);
m_apm_copy(tmpN, tmpT);
if (--mm == 0)
break;
}
if (kr == 0)
{
m_apm_multiply(r, tmpU, tmpN);
}
else
{
m_apm_set_long(tmpM, pow_2);
m_apm_multiply(tmpT, tmpN, tmpM);
m_apm_multiply(r, tmpU, tmpT);
}
}
}
}
M_restore_stack(5);
}
