static int
get_approximated_bits (mps_approximation * appr)
{
rdpe_t module;
mpc_rmod (module, appr->mvalue);
return (rdpe_log (module) - rdpe_log (appr->drad)) / LOG2 - 1;
}
static rdpe_t *
evaluate_root_conditioning (mps_context * ctx, mps_polynomial * p, mps_approximation ** appr, int n)
{
int i;
rdpe_t * root_conditioning = rdpe_valloc (n);
for (i = 0; i < n; i++)
{
mpc_t value;
rdpe_t error, module;
mpc_init2 (value, appr[i]->wp);
mps_polynomial_meval (ctx, p, appr[i]->mvalue, value, error);
mpc_rmod (module, value);
/* Get the relative error of this evaluation */
if (!rdpe_eq_zero (module))
rdpe_div_eq (error, module);
else
rdpe_set_d (error, DBL_EPSILON * p->degree);
/* log2(error) + wp - log(n) is a good estimate of log(k) */
rdpe_set_d (root_conditioning[i], rdpe_log (error) / LOG2 + appr[i]->wp - log2 (n));
rdpe_exp_eq (root_conditioning[i]);
rdpe_div_eq (root_conditioning[i], appr[i]->drad);
mpc_clear (value);
}
return root_conditioning;
}
/**
* @brief Improve all the approximations up to prec_out digits.
*
* For each approximation compute the value of sigma such that, given some
* approximations \f$x_j\f$ of the roots, \f$r_j\f$ the values of the
* inclusion radii and \f$d_i\f$ the number of correct digits:
* \f[
* e_j < e_0 * \sigma^{2^j} \qquad \sigma=\frac{k}{k-1}=\frac{1}{1-t} \qquad k=\frac{1}{t}
* \f]
* and
* \f[
* t = \min_j |z_i-z_j|-r_j
* \f]
* Then compute the number of digits needed for the j-th
* iteration i.e., if \f$cond\f$ is the conditioning of the root:
* \f[
* d_j = \log(\frac{e_j}{|x|}) + cond
* \f]
* where
* \f[
* \log(\frac{e_j}{|x|}) = (f+g){2j} \qquad
* cond = \log(\frac{rad}{\epsilon})
* \f]
* and
* \f[
* cond \approx \lVert p \rVert (1+ \frac{|x_i|}{a_n \prod_{j \neq i} |x_i-x_j|}
* \f]
* and
* \f[
* cond \approx \frac{r_i}{\epsilon |x_i|}
* \f]
* for user-defined polynomials.
*
* <code>s->mpwp</code> denotes the number of bits of the current working
* precision.
*
* @param ctx The mps_context associated with the computation.
*/
MPS_PRIVATE void
mps_improve (mps_context * ctx)
{
int i;
long int current_precision = 0L;
int approximated_roots = 0;
mps_polynomial * p = ctx->active_poly;
rdpe_t * root_conditioning = NULL;
ctx->operation = MPS_OPERATION_REFINEMENT;
/* We need to be able to evaluate the Newton correction in a point
* in order to perform the refinement. This is not necessary true
* for custom polynomial types, so add a check in here */
if (p->mnewton == NULL && p->density != MPS_DENSITY_USER)
return;
/* Set lastphase to mp */
ctx->lastphase = mp_phase;
/* Determine the conditioning of the roots */
root_conditioning = evaluate_root_conditioning (ctx, p, ctx->root, ctx->n);
/* We adopt the strategy of various iterations refinements on
* the approximations by setting the precision of the input
* polynomial in an increasing sequence. */
/* We start by determining the minimum precision at which we can
* extract some information. */
current_precision = LONG_MAX;
for (i = 0; i < ctx->n; i++)
{
if (ctx->root[i]->wp < current_precision)
current_precision = ctx->root[i]->wp;
if (MPS_ROOT_STATUS_IS_APPROXIMATED (ctx->root[i]->status) ||
ctx->root[i]->inclusion == MPS_ROOT_INCLUSION_OUT)
approximated_roots++;
}
/* Start by iterating on the roots that are not approximated, and
* continue until we get all of them. */
while (approximated_roots < ctx->n)
{
mps_polynomial_raise_data (ctx, p, current_precision);
MPS_DEBUG (ctx, "Step of improvement");
for (i = 0; i < ctx->n; i++)
if (ctx->root[i]->status == MPS_ROOT_STATUS_ISOLATED &&
ctx->root[i]->inclusion != MPS_ROOT_INCLUSION_OUT)
{
/* Evaluate the necessary precision to iterate on this root.
* If the the current polynomial precision is enough, iterate on it.
* Otherwise, let it for the next round. */
long int necessary_precision = get_approximated_bits (ctx->root[i]) + log2 (ctx->n) +
rdpe_log (root_conditioning[i]) / LOG2;
if (necessary_precision < current_precision)
{
__improve_root_data * data = mps_new (__improve_root_data);
data->ctx = ctx;
data->p = p;
data->root = ctx->root[i];
data->precision = current_precision;
mps_thread_pool_assign (ctx, NULL, improve_root_wrapper, data);
}
}
mps_thread_pool_wait (ctx, ctx->pool);
for (i = 0; i < ctx->n; i++)
if (!MPS_ROOT_STATUS_IS_APPROXIMATED (ctx->root[i]->status) &&
get_approximated_bits (ctx->root[i]) >= ctx->output_config->prec)
{
ctx->root[i]->status = MPS_ROOT_STATUS_APPROXIMATED;
approximated_roots++;
if (ctx->debug_level & MPS_DEBUG_IMPROVEMENT)
MPS_DEBUG (ctx, "Approximated roots = %d", approximated_roots);
}
/* Increase precision to reach the desired number of approximated roots */
current_precision = 2 * current_precision;
/* Check if we have gone too far with the precision, and we have gone over
* the maximum precision allowed for this polynomial. */
if (current_precision > p->prec && p->prec != 0)
{
ctx->over_max = true;
goto cleanup;
}
/* Increase data prec max that will be useful to the end user to know
* the precision needed to hold these approximations. */
ctx->data_prec_max.value = current_precision;
if (ctx->debug_level & MPS_DEBUG_IMPROVEMENT)
MPS_DEBUG (ctx, "Increasing precision to %ld", current_precision);
}
cleanup:
free (root_conditioning);
}
/**
* @brief Check consistency of data and makes some basic adjustments.
*
* This routine check, for example, if there are zero roots in the polynomial
* (i.e. no costant term) and deflates the polynomial if necessary (shifting
* the coefficients).
*
* It sets the value of the parameter <code>which_case</code> to <code>'f'</code>
* if a floating point phase is enough, or to <code>'d'</code> if
* a <code>dpe</code> phase is needed.
*
* @param s The <code>mps_context</code> associated with the current computation.
* @param which_case the address of the variable which_case;
*/
MPS_PRIVATE void
mps_check_data (mps_context * s, char *which_case)
{
rdpe_t min_coeff, max_coeff, tmp;
mps_monomial_poly *p = NULL;
int i;
/* case of user-defined polynomial */
if (! MPS_IS_MONOMIAL_POLY (s->active_poly))
{
if (s->output_config->multiplicity)
mps_error (s,
"Multiplicity detection not yet implemented for user polynomial");
if (s->output_config->root_properties)
mps_error (s,
"Real/imaginary detection not yet implemented for user polynomial");
*which_case = 'd';
return;
}
else
p = MPS_MONOMIAL_POLY (s->active_poly);
/* Check consistency of input */
if (rdpe_eq (p->dap[s->n], rdpe_zero))
{
mps_warn (s, "The leading coefficient is zero");
do
(s->n)--;
while (rdpe_eq (p->dap[s->n], rdpe_zero));
MPS_POLYNOMIAL (p)->degree = s->n;
}
/* Compute min_coeff */
if (rdpe_lt (p->dap[0], p->dap[s->n]))
rdpe_set (min_coeff, p->dap[0]);
else
rdpe_set (min_coeff, p->dap[s->n]);
/* Compute max_coeff and its logarithm */
rdpe_set (max_coeff, p->dap[0]);
for (i = 1; i <= s->n; i++)
if (rdpe_lt (max_coeff, p->dap[i]))
rdpe_set (max_coeff, p->dap[i]);
s->lmax_coeff = rdpe_log (max_coeff);
/* Multiplicity and sep */
if (s->output_config->multiplicity)
{
if (MPS_STRUCTURE_IS_INTEGER (s->active_poly->structure))
{
mps_compute_sep (s);
}
else if (MPS_STRUCTURE_IS_RATIONAL (s->active_poly->structure))
{
mps_warn (s, "The multiplicity option has not been yet implemented");
s->sep = 0.0;
}
else
{
mps_warn (s, "The input polynomial has neither integer nor rational");
mps_warn (s, " coefficients: unable to compute multiplicities");
s->sep = 0.0;
}
}
/* Real/Imaginary detection */
if (s->output_config->root_properties ||
s->output_config->search_set == MPS_SEARCH_SET_REAL ||
s->output_config->search_set == MPS_SEARCH_SET_IMAG)
{
if (MPS_STRUCTURE_IS_INTEGER (s->active_poly->structure))
{
mps_compute_sep (s);
}
else if (MPS_STRUCTURE_IS_RATIONAL (s->active_poly->structure))
{
mps_error (s,
"The real/imaginary option has not been yet implemented for rational input");
return;
}
else
{
mps_error (s, "The input polynomial has neither integer nor rational "
"coefficients: unable to perform real/imaginary options");
return;
}
}
/* Select cases (dpe or floating point)
* First normalize the polynomial (only the float version) */
rdpe_div (tmp, max_coeff, min_coeff);
rdpe_mul_eq_d (tmp, (double)(s->n + 1));
rdpe_mul_eq (tmp, rdpe_mind);
rdpe_div_eq (tmp, rdpe_maxd);
if (rdpe_lt (tmp, rdpe_one))
{
mpc_t m_min_coeff;
cdpe_t c_min_coeff;
/* if (n+1)*max_coeff/min_coeff < dhuge/dtiny - float case */
*which_case = 'f';
rdpe_mul_eq (min_coeff, max_coeff);
rdpe_mul (tmp, rdpe_mind, rdpe_maxd);
rdpe_div (min_coeff, tmp, min_coeff);
rdpe_sqrt_eq (min_coeff);
rdpe_set (cdpe_Re (c_min_coeff), min_coeff);
rdpe_set (cdpe_Im (c_min_coeff), rdpe_zero);
mpc_init2 (m_min_coeff, mpc_get_prec (p->mfpc[0]));
mpc_set_cdpe (m_min_coeff, c_min_coeff);
/* min_coeff = sqrt(dhuge*dtiny/(min_coeff*max_coeff))
* NOTE: This is enabled for floating point polynomials only
* for the moment, but it may work nicely also for other representations. */
{
for (i = 0; i <= s->n; i++)
{
/* Multiply the MP leading coefficient */
mpc_mul_eq (p->mfpc[i], m_min_coeff);
rdpe_mul (tmp, p->dap[i], min_coeff);
rdpe_set (p->dap[i], tmp);
p->fap[i] = rdpe_get_d (tmp);
mpc_get_cdpe (p->dpc[i], p->mfpc[i]);
cdpe_get_x (p->fpc[i], p->dpc[i]);
}
}
mpc_clear (m_min_coeff);
}
else
*which_case = 'd';
}
