Refining of isolated approximations by means of Newton method. More...

#include <mps/mps.h>

Functions
MPS_BEGIN_DECLS void	mps_improve (mps_context *s)
	Improve all the approximations up to prec_out digits.

Detailed Description

Refining of isolated approximations by means of Newton method.

Function Documentation

◆ mps_improve()

MPS_BEGIN_DECLS void mps_improve ( mps_context * ctx )

Improve all the approximations up to prec_out digits.

For each approximation compute the value of sigma such that, given some approximations of the roots, the values of the inclusion radii and the number of correct digits:

$\‍[ e_j < e_0 * \sigma^{2^j} \qquad \sigma=\frac{k}{k-1}=\frac{1}{1-t} \qquad k=\frac{1}{t} \‍]$

and

$\‍[ t = \min_j |z_i-z_j|-r_j \‍]$

Then compute the number of digits needed for the j-th iteration i.e., if is the conditioning of the root:

$\‍[ d_j = \log(\frac{e_j}{|x|}) + cond \‍]$

where

$\‍[ \log(\frac{e_j}{|x|}) = (f+g){2j} \qquad cond = \log(\frac{rad}{\epsilon}) \‍]$

and

$\‍[ cond \approx \lVert p \rVert (1+ \frac{|x_i|}{a_n \prod_{j \neq i} |x_i-x_j|} \‍]$

and

$\‍[ cond \approx \frac{r_i}{\epsilon |x_i|} \‍]$

for user-defined polynomials.

s->mpwp denotes the number of bits of the current working precision.

Parameters

ctx	The mps_context associated with the computation.

Functions

Detailed Description

Function Documentation

◆ mps_improve()