RefineBox - Maple Help

RegularChains[SemiAlgebraicSetTools]

 RefineBox
 refine a box
 RefineListBox
 refine a list of boxes

 Calling Sequence RefineBox(box, precision, R) RefineListBox(l_boxes, precision, R)

Parameters

 R - polynomial ring box - box isolating a root precision - positive numeric constant l_boxes - list of boxes isolating roots

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ChainTools}\right):$
 > $\mathrm{with}\left(\mathrm{SemiAlgebraicSetTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[y,x\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $C≔\mathrm{Chain}\left(\left[\left({x}^{2}-2\right)\left(x-1\right),\left(y-2\right)\left(y-x\right)\right],\mathrm{Empty}\left(R\right),R\right)$
 ${C}{≔}{\mathrm{regular_chain}}$ (2)
 > $L≔\mathrm{RealRootIsolate}\left(C,R\right)$
 ${L}{≔}\left[{\mathrm{box}}{,}{\mathrm{box}}{,}{\mathrm{box}}{,}{\mathrm{box}}{,}{\mathrm{box}}{,}{\mathrm{box}}\right]$ (3)
 > $\mathrm{map}\left(\mathrm{BoxValues},L,R\right)$
 $\left[\left[{y}{=}\left[{-}\frac{{46341}}{{32768}}{,}{-}\frac{{1482909}}{{1048576}}\right]{,}{x}{=}\left[{-}\frac{{1482911}}{{1048576}}{,}{-}\frac{{741455}}{{524288}}\right]\right]{,}\left[{y}{=}\left[\frac{{1048575}}{{524288}}{,}\frac{{1048577}}{{524288}}\right]{,}{x}{=}\left[{-}\frac{{1482911}}{{1048576}}{,}{-}\frac{{741455}}{{524288}}\right]\right]{,}\left[{y}{=}{1}{,}{x}{=}{1}\right]{,}\left[{y}{=}{2}{,}{x}{=}{1}\right]{,}\left[{y}{=}\left[\frac{{1482899}}{{1048576}}{,}\frac{{741461}}{{524288}}\right]{,}{x}{=}\left[\frac{{741455}}{{524288}}{,}\frac{{1482911}}{{1048576}}\right]\right]{,}\left[{y}{=}\left[\frac{{524285}}{{262144}}{,}\frac{{524291}}{{262144}}\right]{,}{x}{=}\left[\frac{{741455}}{{524288}}{,}\frac{{1482911}}{{1048576}}\right]\right]\right]$ (4)

Refine the first box:

 > $\mathrm{rb}≔\mathrm{RefineBox}\left(L\left[1\right],{10}^{-5},R\right)$
 ${\mathrm{rb}}{≔}{\mathrm{box}}$ (5)
 > $\mathrm{BoxValues}\left(\mathrm{rb},R\right)$
 $\left[{y}{=}\left[{-}\frac{{46341}}{{32768}}{,}{-}\frac{{1482909}}{{1048576}}\right]{,}{x}{=}\left[{-}\frac{{1482911}}{{1048576}}{,}{-}\frac{{741455}}{{524288}}\right]\right]$ (6)
 > $\mathrm{rlb}≔\mathrm{RefineListBox}\left(L,{10}^{-4},R\right)$
 ${\mathrm{rlb}}{≔}\left[{\mathrm{box}}{,}{\mathrm{box}}{,}{\mathrm{box}}{,}{\mathrm{box}}{,}{\mathrm{box}}{,}{\mathrm{box}}\right]$ (7)
 > $\mathrm{map}\left(\mathrm{BoxValues},\mathrm{rlb},R\right)$
 $\left[\left[{y}{=}\left[\frac{{524285}}{{262144}}{,}\frac{{524291}}{{262144}}\right]{,}{x}{=}\left[\frac{{741455}}{{524288}}{,}\frac{{1482911}}{{1048576}}\right]\right]{,}\left[{y}{=}\left[\frac{{1482899}}{{1048576}}{,}\frac{{741461}}{{524288}}\right]{,}{x}{=}\left[\frac{{741455}}{{524288}}{,}\frac{{1482911}}{{1048576}}\right]\right]{,}\left[{y}{=}{2}{,}{x}{=}{1}\right]{,}\left[{y}{=}{1}{,}{x}{=}{1}\right]{,}\left[{y}{=}\left[\frac{{1048575}}{{524288}}{,}\frac{{1048577}}{{524288}}\right]{,}{x}{=}\left[{-}\frac{{1482911}}{{1048576}}{,}{-}\frac{{741455}}{{524288}}\right]\right]{,}\left[{y}{=}\left[{-}\frac{{46341}}{{32768}}{,}{-}\frac{{1482909}}{{1048576}}\right]{,}{x}{=}\left[{-}\frac{{1482911}}{{1048576}}{,}{-}\frac{{741455}}{{524288}}\right]\right]\right]$ (8)
 > $\mathrm{rb}≔\mathrm{RefineBox}\left(L\left[1\right],\frac{1}{20},R\right)$
 ${\mathrm{rb}}{≔}{\mathrm{box}}$ (9)
 > $\mathrm{BoxValues}\left(\mathrm{rb},R\right)$
 $\left[{y}{=}\left[{-}\frac{{46341}}{{32768}}{,}{-}\frac{{1482909}}{{1048576}}\right]{,}{x}{=}\left[{-}\frac{{1482911}}{{1048576}}{,}{-}\frac{{741455}}{{524288}}\right]\right]$ (10)
 > $\mathrm{rlb}≔\mathrm{RefineListBox}\left(L,\frac{1}{20},R\right)$
 ${\mathrm{rlb}}{≔}\left[{\mathrm{box}}{,}{\mathrm{box}}{,}{\mathrm{box}}{,}{\mathrm{box}}{,}{\mathrm{box}}{,}{\mathrm{box}}\right]$ (11)
 > $\mathrm{map}\left(\mathrm{BoxValues},\mathrm{rlb},R\right)$
 $\left[\left[{y}{=}\left[\frac{{524285}}{{262144}}{,}\frac{{524291}}{{262144}}\right]{,}{x}{=}\left[\frac{{741455}}{{524288}}{,}\frac{{1482911}}{{1048576}}\right]\right]{,}\left[{y}{=}\left[\frac{{1482899}}{{1048576}}{,}\frac{{741461}}{{524288}}\right]{,}{x}{=}\left[\frac{{741455}}{{524288}}{,}\frac{{1482911}}{{1048576}}\right]\right]{,}\left[{y}{=}{2}{,}{x}{=}{1}\right]{,}\left[{y}{=}{1}{,}{x}{=}{1}\right]{,}\left[{y}{=}\left[\frac{{1048575}}{{524288}}{,}\frac{{1048577}}{{524288}}\right]{,}{x}{=}\left[{-}\frac{{1482911}}{{1048576}}{,}{-}\frac{{741455}}{{524288}}\right]\right]{,}\left[{y}{=}\left[{-}\frac{{46341}}{{32768}}{,}{-}\frac{{1482909}}{{1048576}}\right]{,}{x}{=}\left[{-}\frac{{1482911}}{{1048576}}{,}{-}\frac{{741455}}{{524288}}\right]\right]\right]$ (12)