RegularChains[SemiAlgebraicSetTools] - Maple Help

Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : SemiAlgebraicSetTools Subpackage : RegularChains/SemiAlgebraicSetTools/RefineBox

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)
 > $C≔\mathrm{Chain}\left(\left[{x}^{2}-2,\left(y-1\right)\left(y-x\right)\right],\mathrm{Empty}\left(R\right),R\right)$
 ${C}{:=}{\mathrm{regular_chain}}$ (2)
 > $L≔\mathrm{RealRootIsolate}\left(C,R\right)$
 ${L}{:=}\left[{\mathrm{box}}{,}{\mathrm{box}}{,}{\mathrm{box}}{,}{\mathrm{box}}\right]$ (3)

Refine the first box:

 > $\mathrm{rl}≔\mathrm{RefineBox}\left({L}_{1},{10}^{-10},R\right)$
 ${\mathrm{rl}}{:=}{\mathrm{box}}$ (4)
 > $\mathrm{BoxValues}\left(\mathrm{rl},R\right)$
 $\left[{y}{=}{1}{,}{x}{=}\left[\frac{{24296003999}}{{17179869184}}{,}\frac{{759250125}}{{536870912}}\right]\right]$ (5)
 > $\mathrm{evalf}\left(\right)$
 $\left[{y}{=}{1.}{,}{x}{=}\left[{1.414213562}{,}{1.414213562}\right]\right]$ (6)
 > $L≔\mathrm{RealRootIsolate}\left(C,R\right)$
 ${L}{:=}\left[{\mathrm{box}}{,}{\mathrm{box}}{,}{\mathrm{box}}{,}{\mathrm{box}}\right]$ (7)

Refine all boxes at the same time:

 > $\mathrm{rlb}≔\mathrm{RefineListBox}\left(L,{10}^{-10},R\right)$
 ${\mathrm{rlb}}{:=}\left[{\mathrm{box}}{,}{\mathrm{box}}{,}{\mathrm{box}}{,}{\mathrm{box}}\right]$ (8)
 > $\mathrm{map}\left(\mathrm{BoxValues},\mathrm{rlb},R\right)$
 $\left[\left[{y}{=}\left[{-}\frac{{759250125}}{{536870912}}{,}{-}\frac{{24296003999}}{{17179869184}}\right]{,}{x}{=}\left[{-}\frac{{194368031999}}{{137438953472}}{,}{-}\frac{{97184015999}}{{68719476736}}\right]\right]{,}\left[{y}{=}\left[\frac{{24296003999}}{{17179869184}}{,}\frac{{759250125}}{{536870912}}\right]{,}{x}{=}\left[\frac{{97184015999}}{{68719476736}}{,}\frac{{194368031999}}{{137438953472}}\right]\right]{,}\left[{y}{=}{1}{,}{x}{=}\left[{-}\frac{{194368031999}}{{137438953472}}{,}{-}\frac{{97184015999}}{{68719476736}}\right]\right]{,}\left[{y}{=}{1}{,}{x}{=}\left[\frac{{97184015999}}{{68719476736}}{,}\frac{{194368031999}}{{137438953472}}\right]\right]\right]$ (9)
 > $\mathrm{evalf}\left(\right)$
 $\left[\left[{y}{=}\left[{-}{1.414213562}{,}{-}{1.414213562}\right]{,}{x}{=}\left[{-}{1.414213562}{,}{-}{1.414213562}\right]\right]{,}\left[{y}{=}\left[{1.414213562}{,}{1.414213562}\right]{,}{x}{=}\left[{1.414213562}{,}{1.414213562}\right]\right]{,}\left[{y}{=}{1.}{,}{x}{=}\left[{-}{1.414213562}{,}{-}{1.414213562}\right]\right]{,}\left[{y}{=}{1.}{,}{x}{=}\left[{1.414213562}{,}{1.414213562}\right]\right]\right]$ (10)