The RefineBox command refines a box so its width is smaller or equal to precision. It returns a box isolating the same root as box.

The RefineListBox command refines a list of boxes so their widths are smaller or equal to precision. It returns a list of boxes isolating the same roots as l_boxes. It is more efficient than using map and RefineBox when the boxes isolate roots originating from the same regular chain. Refining a box allows one to refine instantly other boxes which share a common part.

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$

$\mathrm{with}\left(\mathrm{ChainTools}\right)\:$

$\mathrm{with}\left(\mathrm{SemiAlgebraicSetTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[y\,x\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$C≔\mathrm{Chain}\left(\left[\left(x^2-2\right)\left(x-1\right)\,\left(y-2\right)\left(y-x\right)\right]\,\mathrm{Empty}\left(R\right)\,R\right)$

$C≔\mathrm{regular\_chain}$

$L≔\mathrm{RealRootIsolate}\left(C\,R\right)$

$L≔\left[\mathrm{box}\,\mathrm{box}\,\mathrm{box}\,\mathrm{box}\,\mathrm{box}\,\mathrm{box}\right]$

$\mathrm{map}\left(\mathrm{BoxValues}\,L\,R\right)$

$\left[\left[y=\left[-\frac{46341}{32768}\,-\frac{1482909}{1048576}\right]\,x=\left[-\frac{1482911}{1048576}\,-\frac{741455}{524288}\right]\right]\,\left[y=\left[\frac{1048575}{524288}\,\frac{1048577}{524288}\right]\,x=\left[-\frac{1482911}{1048576}\,-\frac{741455}{524288}\right]\right]\,\left[y=1\,x=1\right]\,\left[y=2\,x=1\right]\,\left[y=\left[\frac{1482899}{1048576}\,\frac{741461}{524288}\right]\,x=\left[\frac{741455}{524288}\,\frac{1482911}{1048576}\right]\right]\,\left[y=\left[\frac{524285}{262144}\,\frac{524291}{262144}\right]\,x=\left[\frac{741455}{524288}\,\frac{1482911}{1048576}\right]\right]\right]$

$\mathrm{rb}≔\mathrm{RefineBox}\left(L\left[1\right]\,{10}^{-5}\,R\right)$

$\mathrm{rb}≔\mathrm{box}$

$\mathrm{BoxValues}\left(\mathrm{rb}\,R\right)$

$\left[y=\left[-\frac{46341}{32768}\,-\frac{1482909}{1048576}\right]\,x=\left[-\frac{1482911}{1048576}\,-\frac{741455}{524288}\right]\right]$

$\mathrm{rlb}≔\mathrm{RefineListBox}\left(L\,{10}^{-4}\,R\right)$

$\mathrm{rlb}≔\left[\mathrm{box}\,\mathrm{box}\,\mathrm{box}\,\mathrm{box}\,\mathrm{box}\,\mathrm{box}\right]$

$\mathrm{map}\left(\mathrm{BoxValues}\,\mathrm{rlb}\,R\right)$

$\left[\left[y=\left[\frac{524285}{262144}\,\frac{524291}{262144}\right]\,x=\left[\frac{741455}{524288}\,\frac{1482911}{1048576}\right]\right]\,\left[y=\left[\frac{1482899}{1048576}\,\frac{741461}{524288}\right]\,x=\left[\frac{741455}{524288}\,\frac{1482911}{1048576}\right]\right]\,\left[y=2\,x=1\right]\,\left[y=1\,x=1\right]\,\left[y=\left[\frac{1048575}{524288}\,\frac{1048577}{524288}\right]\,x=\left[-\frac{1482911}{1048576}\,-\frac{741455}{524288}\right]\right]\,\left[y=\left[-\frac{46341}{32768}\,-\frac{1482909}{1048576}\right]\,x=\left[-\frac{1482911}{1048576}\,-\frac{741455}{524288}\right]\right]\right]$

$\mathrm{rb}≔\mathrm{RefineBox}\left(L\left[1\right]\,\frac{1}{20}\,R\right)$

$\mathrm{rb}≔\mathrm{box}$

$\mathrm{BoxValues}\left(\mathrm{rb}\,R\right)$

$\left[y=\left[-\frac{46341}{32768}\,-\frac{1482909}{1048576}\right]\,x=\left[-\frac{1482911}{1048576}\,-\frac{741455}{524288}\right]\right]$

$\mathrm{rlb}≔\mathrm{RefineListBox}\left(L\,\frac{1}{20}\,R\right)$

$\mathrm{rlb}≔\left[\mathrm{box}\,\mathrm{box}\,\mathrm{box}\,\mathrm{box}\,\mathrm{box}\,\mathrm{box}\right]$

$\mathrm{map}\left(\mathrm{BoxValues}\,\mathrm{rlb}\,R\right)$

$\left[\left[y=\left[\frac{524285}{262144}\,\frac{524291}{262144}\right]\,x=\left[\frac{741455}{524288}\,\frac{1482911}{1048576}\right]\right]\,\left[y=\left[\frac{1482899}{1048576}\,\frac{741461}{524288}\right]\,x=\left[\frac{741455}{524288}\,\frac{1482911}{1048576}\right]\right]\,\left[y=2\,x=1\right]\,\left[y=1\,x=1\right]\,\left[y=\left[\frac{1048575}{524288}\,\frac{1048577}{524288}\right]\,x=\left[-\frac{1482911}{1048576}\,-\frac{741455}{524288}\right]\right]\,\left[y=\left[-\frac{46341}{32768}\,-\frac{1482909}{1048576}\right]\,x=\left[-\frac{1482911}{1048576}\,-\frac{741455}{524288}\right]\right]\right]$