mps_boolean
mps_quadratic_poly_meval (mps_context * ctx, mps_polynomial * p, mpc_t x, mpc_t value, rdpe_t error)
{
int i;
int m = (int) (log (p->degree + 1.0) / LOG2);
rdpe_t ax, rtmp;
mpc_t tmp;
long int wp = mpc_get_prec (x);
/* Correct the working precision in case of a limited
precision polynomial (quite unlikely
* in the quadratic case, but still. */
if (p->prec > 0 && p->prec < wp)
wp = p->prec;
if ((1 << m) <= p->degree)
m++;
mpc_init2 (tmp, wp);
mpc_rmod (ax, x);
mpc_set_ui (value, 1U, 0U);
mpc_rmod (error, value);
for (i = 1; i <= m; i++)
{
mpc_sqr (tmp, value);
mpc_mul (value, x, tmp);
mpc_add_eq_ui (value, 1U, 0U);
rdpe_mul_eq (error, ax);
mpc_rmod (rtmp, value);
rdpe_add_eq (error, rtmp);
}
rdpe_set_2dl (rtmp, 1.0, -wp);
rdpe_mul_eq (error, rtmp);
mpc_clear (tmp);
return true;
}
static void
reuse_bug (void)
{
mpc_t z1;
/* reuse bug found by Paul Zimmermann 20081021 */
mpc_init2 (z1, 2);
/* RE (z1^2) overflows, IM(z^2) = -0 */
mpfr_set_str (mpc_realref (z1), "0.11", 2, MPFR_RNDN);
mpfr_mul_2si (mpc_realref (z1), mpc_realref (z1), mpfr_get_emax (), MPFR_RNDN);
mpfr_set_ui (mpc_imagref (z1), 0, MPFR_RNDN);
mpc_conj (z1, z1, MPC_RNDNN);
mpc_sqr (z1, z1, MPC_RNDNN);
if (!mpfr_inf_p (mpc_realref (z1)) || mpfr_signbit (mpc_realref (z1))
||!mpfr_zero_p (mpc_imagref (z1)) || !mpfr_signbit (mpc_imagref (z1)))
{
printf ("Error: Regression, bug 20081021 reproduced\n");
MPC_OUT (z1);
exit (1);
}
mpc_clear (z1);
}
int
mpc_asin (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd)
{
mpfr_prec_t p, p_re, p_im, incr_p = 0;
mpfr_rnd_t rnd_re, rnd_im;
mpc_t z1;
int inex;
/* special values */
if (mpfr_nan_p (mpc_realref (op)) || mpfr_nan_p (mpc_imagref (op)))
{
if (mpfr_inf_p (mpc_realref (op)) || mpfr_inf_p (mpc_imagref (op)))
{
mpfr_set_nan (mpc_realref (rop));
mpfr_set_inf (mpc_imagref (rop), mpfr_signbit (mpc_imagref (op)) ? -1 : +1);
}
else if (mpfr_zero_p (mpc_realref (op)))
{
mpfr_set (mpc_realref (rop), mpc_realref (op), GMP_RNDN);
mpfr_set_nan (mpc_imagref (rop));
}
else
{
mpfr_set_nan (mpc_realref (rop));
mpfr_set_nan (mpc_imagref (rop));
}
return 0;
}
if (mpfr_inf_p (mpc_realref (op)) || mpfr_inf_p (mpc_imagref (op)))
{
int inex_re;
if (mpfr_inf_p (mpc_realref (op)))
{
int inf_im = mpfr_inf_p (mpc_imagref (op));
inex_re = set_pi_over_2 (mpc_realref (rop),
(mpfr_signbit (mpc_realref (op)) ? -1 : 1), MPC_RND_RE (rnd));
mpfr_set_inf (mpc_imagref (rop), (mpfr_signbit (mpc_imagref (op)) ? -1 : 1));
if (inf_im)
mpfr_div_2ui (mpc_realref (rop), mpc_realref (rop), 1, GMP_RNDN);
}
else
{
mpfr_set_zero (mpc_realref (rop), (mpfr_signbit (mpc_realref (op)) ? -1 : 1));
inex_re = 0;
mpfr_set_inf (mpc_imagref (rop), (mpfr_signbit (mpc_imagref (op)) ? -1 : 1));
}
return MPC_INEX (inex_re, 0);
}
/* pure real argument */
if (mpfr_zero_p (mpc_imagref (op)))
{
int inex_re;
int inex_im;
int s_im;
s_im = mpfr_signbit (mpc_imagref (op));
if (mpfr_cmp_ui (mpc_realref (op), 1) > 0)
{
if (s_im)
inex_im = -mpfr_acosh (mpc_imagref (rop), mpc_realref (op),
INV_RND (MPC_RND_IM (rnd)));
else
inex_im = mpfr_acosh (mpc_imagref (rop), mpc_realref (op),
MPC_RND_IM (rnd));
inex_re = set_pi_over_2 (mpc_realref (rop),
(mpfr_signbit (mpc_realref (op)) ? -1 : 1), MPC_RND_RE (rnd));
if (s_im)
mpc_conj (rop, rop, MPC_RNDNN);
}
else if (mpfr_cmp_si (mpc_realref (op), -1) < 0)
{
mpfr_t minus_op_re;
minus_op_re[0] = mpc_realref (op)[0];
MPFR_CHANGE_SIGN (minus_op_re);
if (s_im)
inex_im = -mpfr_acosh (mpc_imagref (rop), minus_op_re,
INV_RND (MPC_RND_IM (rnd)));
else
inex_im = mpfr_acosh (mpc_imagref (rop), minus_op_re,
MPC_RND_IM (rnd));
inex_re = set_pi_over_2 (mpc_realref (rop),
(mpfr_signbit (mpc_realref (op)) ? -1 : 1), MPC_RND_RE (rnd));
if (s_im)
mpc_conj (rop, rop, MPC_RNDNN);
}
else
{
inex_im = mpfr_set_ui (mpc_imagref (rop), 0, MPC_RND_IM (rnd));
if (s_im)
mpfr_neg (mpc_imagref (rop), mpc_imagref (rop), GMP_RNDN);
inex_re = mpfr_asin (mpc_realref (rop), mpc_realref (op), MPC_RND_RE (rnd));
}
return MPC_INEX (inex_re, inex_im);
}
/* pure imaginary argument */
if (mpfr_zero_p (mpc_realref (op)))
{
int inex_im;
int s;
s = mpfr_signbit (mpc_realref (op));
mpfr_set_ui (mpc_realref (rop), 0, GMP_RNDN);
if (s)
mpfr_neg (mpc_realref (rop), mpc_realref (rop), GMP_RNDN);
inex_im = mpfr_asinh (mpc_imagref (rop), mpc_imagref (op), MPC_RND_IM (rnd));
return MPC_INEX (0, inex_im);
}
/* regular complex: asin(z) = -i*log(i*z+sqrt(1-z^2)) */
p_re = mpfr_get_prec (mpc_realref(rop));
p_im = mpfr_get_prec (mpc_imagref(rop));
rnd_re = MPC_RND_RE(rnd);
rnd_im = MPC_RND_IM(rnd);
p = p_re >= p_im ? p_re : p_im;
mpc_init2 (z1, p);
while (1)
{
mpfr_exp_t ex, ey, err;
p += mpc_ceil_log2 (p) + 3 + incr_p; /* incr_p is zero initially */
incr_p = p / 2;
mpfr_set_prec (mpc_realref(z1), p);
mpfr_set_prec (mpc_imagref(z1), p);
/* z1 <- z^2 */
mpc_sqr (z1, op, MPC_RNDNN);
/* err(x) <= 1/2 ulp(x), err(y) <= 1/2 ulp(y) */
/* z1 <- 1-z1 */
ex = mpfr_get_exp (mpc_realref(z1));
mpfr_ui_sub (mpc_realref(z1), 1, mpc_realref(z1), GMP_RNDN);
mpfr_neg (mpc_imagref(z1), mpc_imagref(z1), GMP_RNDN);
ex = ex - mpfr_get_exp (mpc_realref(z1));
ex = (ex <= 0) ? 0 : ex;
/* err(x) <= 2^ex * ulp(x) */
ex = ex + mpfr_get_exp (mpc_realref(z1)) - p;
/* err(x) <= 2^ex */
ey = mpfr_get_exp (mpc_imagref(z1)) - p - 1;
/* err(y) <= 2^ey */
ex = (ex >= ey) ? ex : ey; /* err(x), err(y) <= 2^ex, i.e., the norm
of the error is bounded by |h|<=2^(ex+1/2) */
/* z1 <- sqrt(z1): if z1 = z + h, then sqrt(z1) = sqrt(z) + h/2/sqrt(t) */
ey = mpfr_get_exp (mpc_realref(z1)) >= mpfr_get_exp (mpc_imagref(z1))
? mpfr_get_exp (mpc_realref(z1)) : mpfr_get_exp (mpc_imagref(z1));
/* we have |z1| >= 2^(ey-1) thus 1/|z1| <= 2^(1-ey) */
mpc_sqrt (z1, z1, MPC_RNDNN);
ex = (2 * ex + 1) - 2 - (ey - 1); /* |h^2/4/|t| <= 2^ex */
ex = (ex + 1) / 2; /* ceil(ex/2) */
/* express ex in terms of ulp(z1) */
ey = mpfr_get_exp (mpc_realref(z1)) <= mpfr_get_exp (mpc_imagref(z1))
? mpfr_get_exp (mpc_realref(z1)) : mpfr_get_exp (mpc_imagref(z1));
ex = ex - ey + p;
/* take into account the rounding error in the mpc_sqrt call */
err = (ex <= 0) ? 1 : ex + 1;
/* err(x) <= 2^err * ulp(x), err(y) <= 2^err * ulp(y) */
/* z1 <- i*z + z1 */
ex = mpfr_get_exp (mpc_realref(z1));
ey = mpfr_get_exp (mpc_imagref(z1));
mpfr_sub (mpc_realref(z1), mpc_realref(z1), mpc_imagref(op), GMP_RNDN);
mpfr_add (mpc_imagref(z1), mpc_imagref(z1), mpc_realref(op), GMP_RNDN);
if (mpfr_cmp_ui (mpc_realref(z1), 0) == 0 || mpfr_cmp_ui (mpc_imagref(z1), 0) == 0)
continue;
ex -= mpfr_get_exp (mpc_realref(z1)); /* cancellation in x */
ey -= mpfr_get_exp (mpc_imagref(z1)); /* cancellation in y */
ex = (ex >= ey) ? ex : ey; /* maximum cancellation */
err += ex;
err = (err <= 0) ? 1 : err + 1; /* rounding error in sub/add */
/* z1 <- log(z1): if z1 = z + h, then log(z1) = log(z) + h/t with
|t| >= min(|z1|,|z|) */
ex = mpfr_get_exp (mpc_realref(z1));
ey = mpfr_get_exp (mpc_imagref(z1));
ex = (ex >= ey) ? ex : ey;
err += ex - p; /* revert to absolute error <= 2^err */
mpc_log (z1, z1, GMP_RNDN);
err -= ex - 1; /* 1/|t| <= 1/|z| <= 2^(1-ex) */
/* express err in terms of ulp(z1) */
ey = mpfr_get_exp (mpc_realref(z1)) <= mpfr_get_exp (mpc_imagref(z1))
? mpfr_get_exp (mpc_realref(z1)) : mpfr_get_exp (mpc_imagref(z1));
err = err - ey + p;
/* take into account the rounding error in the mpc_log call */
err = (err <= 0) ? 1 : err + 1;
/* z1 <- -i*z1 */
mpfr_swap (mpc_realref(z1), mpc_imagref(z1));
mpfr_neg (mpc_imagref(z1), mpc_imagref(z1), GMP_RNDN);
if (mpfr_can_round (mpc_realref(z1), p - err, GMP_RNDN, GMP_RNDZ,
p_re + (rnd_re == GMP_RNDN)) &&
mpfr_can_round (mpc_imagref(z1), p - err, GMP_RNDN, GMP_RNDZ,
p_im + (rnd_im == GMP_RNDN)))
break;
}
inex = mpc_set (rop, z1, rnd);
mpc_clear (z1);
return inex;
}
/******************************************************
* SUBROUTINE MNEWTON_USR *
*******************************************************
multiprecision computation
******************************************************/
void
mnewton_usr (mpc_t x, rdpe_t rad, mpc_t corr, mps_boolean * again)
{
int i, m;
rdpe_t ap, ax, eps, temp, apeps, atmp;
cdpe_t ctmp;
tmpc_t p, pp, pt, tmp;
tmpc_init2 (p, mpwp);
tmpc_init2 (pp, mpwp);
tmpc_init2 (pt, mpwp);
tmpc_init2 (tmp, mpwp);
m = (int) (log (n + 1.0) / LOG2);
if ((1 << m) <= n)
m++;
rdpe_set (eps, mp_epsilon);
rdpe_mul_eq_d (eps, (double) 4 * n);
mpc_get_cdpe (ctmp, x);
cdpe_mod (ax, ctmp);
mpc_set_ui (p, 1, 0);
mpc_set_ui (pp, 0, 0);
rdpe_set (ap, rdpe_one);
for (i = 1; i <= m; i++)
{
mpc_sqr (tmp, p);
mpc_mul (pt, x, tmp);
mpc_add_eq_ui (pt, 1, 0);
mpc_mul_eq (pp, x);
mpc_mul_eq (pp, p);
mpc_mul_eq_ui (pp, 2);
mpc_add_eq (pp, tmp);
mpc_set (p, pt);
rdpe_mul_eq (ap, ax);
mpc_get_cdpe (ctmp, p);
cdpe_mod (atmp, ctmp);
rdpe_add_eq (ap, atmp);
}
rdpe_mul_eq (ap, ax);
mpc_div (corr, p, pp);
mpc_get_cdpe (ctmp, p);
cdpe_mod (temp, ctmp);
rdpe_mul (apeps, ap, eps);
rdpe_mul_eq_d (apeps, 3.0);
*again = rdpe_gt (temp, apeps);
rdpe_add (rad, temp, apeps);
rdpe_mul_eq_d (rad, (double) n);
mpc_get_cdpe (ctmp, pp);
cdpe_mod (temp, ctmp);
rdpe_div_eq (rad, temp);
if (rdpe_eq (rad, rdpe_zero))
rdpe_mul (rad, ax, eps);
tmpc_clear (tmp);
tmpc_clear (pt);
tmpc_clear (pp);
tmpc_clear (p);
}
static void
cmpsqr (mpc_srcptr x, mpc_rnd_t rnd)
/* computes the square of x with the specific function or by simple */
/* multiplication using the rounding mode rnd and compares the results */
/* and return values. */
/* In our current test suite, the real and imaginary parts of x have */
/* the same precision, and we use this precision also for the result. */
/* Furthermore, we check whether computing the square in the same */
/* place yields the same result. */
/* We also compute the result with four times the precision and check */
/* whether the rounding is correct. Error reports in this part of the */
/* algorithm might still be wrong, though, since there are two */
/* consecutive roundings. */
{
mpc_t z, t, u;
int inexact_z, inexact_t;
mpc_init2 (z, MPC_MAX_PREC (x));
mpc_init2 (t, MPC_MAX_PREC (x));
mpc_init2 (u, 4 * MPC_MAX_PREC (x));
inexact_z = mpc_sqr (z, x, rnd);
inexact_t = mpc_mul (t, x, x, rnd);
if (mpc_cmp (z, t))
{
fprintf (stderr, "sqr and mul differ for rnd=(%s,%s) \nx=",
mpfr_print_rnd_mode(MPC_RND_RE(rnd)),
mpfr_print_rnd_mode(MPC_RND_IM(rnd)));
mpc_out_str (stderr, 2, 0, x, MPC_RNDNN);
fprintf (stderr, "\nmpc_sqr gives ");
mpc_out_str (stderr, 2, 0, z, MPC_RNDNN);
fprintf (stderr, "\nmpc_mul gives ");
mpc_out_str (stderr, 2, 0, t, MPC_RNDNN);
fprintf (stderr, "\n");
exit (1);
}
if (inexact_z != inexact_t)
{
fprintf (stderr, "The return values of sqr and mul differ for rnd=(%s,%s) \nx= ",
mpfr_print_rnd_mode(MPC_RND_RE(rnd)),
mpfr_print_rnd_mode(MPC_RND_IM(rnd)));
mpc_out_str (stderr, 2, 0, x, MPC_RNDNN);
fprintf (stderr, "\nx^2=");
mpc_out_str (stderr, 2, 0, z, MPC_RNDNN);
fprintf (stderr, "\nmpc_sqr gives %i", inexact_z);
fprintf (stderr, "\nmpc_mul gives %i", inexact_t);
fprintf (stderr, "\n");
exit (1);
}
mpc_set (t, x, MPC_RNDNN);
inexact_t = mpc_sqr (t, t, rnd);
if (mpc_cmp (z, t))
{
fprintf (stderr, "sqr and sqr in place differ for rnd=(%s,%s) \nx=",
mpfr_print_rnd_mode(MPC_RND_RE(rnd)),
mpfr_print_rnd_mode(MPC_RND_IM(rnd)));
mpc_out_str (stderr, 2, 0, x, MPC_RNDNN);
fprintf (stderr, "\nmpc_sqr gives ");
mpc_out_str (stderr, 2, 0, z, MPC_RNDNN);
fprintf (stderr, "\nmpc_sqr in place gives ");
mpc_out_str (stderr, 2, 0, t, MPC_RNDNN);
fprintf (stderr, "\n");
exit (1);
}
if (inexact_z != inexact_t)
{
fprintf (stderr, "The return values of sqr and sqr in place differ for rnd=(%s,%s) \nx= ",
mpfr_print_rnd_mode(MPC_RND_RE(rnd)),
mpfr_print_rnd_mode(MPC_RND_IM(rnd)));
mpc_out_str (stderr, 2, 0, x, MPC_RNDNN);
fprintf (stderr, "\nx^2=");
mpc_out_str (stderr, 2, 0, z, MPC_RNDNN);
fprintf (stderr, "\nmpc_sqr gives %i", inexact_z);
fprintf (stderr, "\nmpc_sqr in place gives %i", inexact_t);
fprintf (stderr, "\n");
exit (1);
}
mpc_sqr (u, x, rnd);
mpc_set (t, u, rnd);
if (mpc_cmp (z, t))
{
fprintf (stderr, "rounding in sqr might be incorrect for rnd=(%s,%s) \nx=",
mpfr_print_rnd_mode(MPC_RND_RE(rnd)),
mpfr_print_rnd_mode(MPC_RND_IM(rnd)));
mpc_out_str (stderr, 2, 0, x, MPC_RNDNN);
fprintf (stderr, "\nmpc_sqr gives ");
mpc_out_str (stderr, 2, 0, z, MPC_RNDNN);
fprintf (stderr, "\nmpc_sqr quadruple precision gives ");
mpc_out_str (stderr, 2, 0, u, MPC_RNDNN);
fprintf (stderr, "\nand is rounded to ");
mpc_out_str (stderr, 2, 0, t, MPC_RNDNN);
fprintf (stderr, "\n");
exit (1);
}
mpc_clear (z);
mpc_clear (t);
mpc_clear (u);
}
/**
* @brief User-defined program for the computation of \f$p\f$, \f$p'\f$.
*
* @param s The current mps_context
* @param poly The mps_polynomial being solved.
* @param root The approximation whose Newton correction shall be computed.
* @param corr The output value where the newton correction will be stored.
*
* This sample computes the 'Quadratic polynomial by
* means of the relation: p=1+x*p**2, starting with p=1
*/
void
mps_quadratic_poly_mnewton (mps_context * ctx, mps_polynomial * poly,
mps_approximation * root, mpc_t corr, long int wp)
{
int i, m, n = poly->degree;
rdpe_t ap, ax, eps, temp, apeps, atmp, epsilon, drad;
cdpe_t ctmp;
mpc_t p, pp, pt, tmp, x;
mps_boolean again;
mpc_init2 (p, wp);
mpc_init2 (pp, wp);
mpc_init2 (pt, wp);
mpc_init2 (tmp, wp);
mpc_init2 (x, wp);
mps_approximation_get_mvalue (ctx, root, x);
mps_approximation_get_drad (ctx, root, drad);
again = mps_approximation_get_again (ctx, root);
rdpe_set_2dl (epsilon, 1.0, 2 - wp);
m = (int) (log (n + 1.0) / LOG2);
if ((1 << m) <= n)
m++;
rdpe_set (eps, epsilon);
rdpe_mul_eq_d (eps, (double) 4 * n);
mpc_get_cdpe (ctmp, x);
cdpe_mod (ax, ctmp);
mpc_set_ui (p, 1, 0);
mpc_set_ui (pp, 0, 0);
rdpe_set (ap, rdpe_one);
for (i = 1; i <= m; i++)
{
mpc_sqr (tmp, p);
mpc_mul (pt, x, tmp);
mpc_add_eq_ui (pt, 1, 0);
mpc_mul_eq (pp, x);
mpc_mul_eq (pp, p);
mpc_mul_eq_ui (pp, 2);
mpc_add_eq (pp, tmp);
mpc_set (p, pt);
rdpe_mul_eq (ap, ax);
mpc_get_cdpe (ctmp, p);
cdpe_mod (atmp, ctmp);
rdpe_add_eq (ap, atmp);
}
rdpe_mul_eq (ap, ax);
mpc_div (corr, p, pp);
mpc_get_cdpe (ctmp, p);
cdpe_mod (temp, ctmp);
rdpe_mul (apeps, ap, eps);
rdpe_mul_eq_d (apeps, 3.0);
mps_approximation_set_again (ctx, root, rdpe_gt (temp, apeps));
rdpe_add (drad, temp, apeps);
rdpe_mul_eq_d (drad, (double) n);
mpc_get_cdpe (ctmp, pp);
cdpe_mod (temp, ctmp);
rdpe_div_eq (drad, temp);
if (rdpe_eq (drad, rdpe_zero))
rdpe_mul (drad, ax, eps);
mps_approximation_set_drad (ctx, root, drad);
mps_approximation_set_again (ctx, root, again);
mpc_clear (tmp);
mpc_clear (pt);
mpc_clear (pp);
mpc_clear (p);
mpc_clear (x);
}
void mpcomplex::sqr3() {
mpc_sqr(mpc_val, mpc_val, default_rnd);
}
