Edit detail for Cylindrical Algebraic Decomposition revision 2 of 4

On Wednesday, October 12, 2005 5:29 PM Renaud Rioboo wrote:

Dear Axiom fans,

In the last few years there have been a regain of interest in some Computer Algebra users circles about Cylindrical Algebraic Decomposition (aka CAD). As some of you may know my thesis was about CAD and I am regularly receiving queries about CAD and the Axiom implementation I wrote.

I already privately gave sources or ran particular problems for some people but I have never officially released my sources because I do not think they are at production level and many work has still to be done on them.

Recently Martin Rubey thought it could be a good idea to release them and I have of course no objection on that point nor on sharing these sources with the community. If there is some interest on it you may include it into the Axiom distribution with the same permissions than the rest of the software.

While I wait for my lab to give me the necessary permissions to be able to use Axiom and export this software you may find it's sources at the unlisted url

http://rioboo.free.fr/CadPub/

I would like to add just a few words about that work which is a straightforward implementation of the standard papers by Arnon, Collins and McCallum? which are more than 20 years old now.

My work on Cad is 15 years old and was developped for the defense of my thesis where CAD was an sample application for real algebraic numbers manipulations. By the time I wrote it I also needed some subresultant calculations which were not in Axiom (Scratchpad by that time) and the required machinery to work with real algebraic numbers.

Along the times, real algebraic numbers were included into Axiom and subresultant calculations which are used in several places of the Axiom Library were modified to all use Lionel Ducos's package which enables a performance gain. While making these modifications to Axiom I tried to keep my CAD package up-to-date with the Axiom Library and have used it as a test for other packages. It thus should compile under recent Axiom versions and should compute something that resembles a CAD.

Don't expect this package to be the absolute CAD program, I never found time to further work on it in order to include many enhancements which are present in other CAD packages. While Axiom versions evolved I had to remove what I thought were enhancements and remove some support for generalization. As of today it only includes one projection method which is the McCallum? projection and no reconstruction nor adjacency's are taken into account.

However it did reasonabily compare with more recent techniques such as Rational Univariate Representation (aka RUR) for simple cases though it could not produce results for some more complicated cases. By the time this comparison was made Axiom had some severe memory restrictions which seem to have disappear now. I thus think that many objections to using CAD could now be revised.

For hard problems this package should thus not be worse than any other CAD package except that you cannot fall into an optimzation drawback :-) There is no optimization !

I will of course try to maintain the package to the best of my ability. One thing I should do is include comments describing the different functions, but even that requires going into the code and the algorithms which are quite old for me now.

Let me know if there is some interest on it.

Best regards,

Renaud

CylindricalAlgebraicDecompositionUtilities?

)ab pack CADU CylindricalAlgebraicDecompositionUtilities

CylindricalAlgebraicDecompositionUtilities(R,P) : PUB == PRIV where

-- Tese are some standard tools which are needed to compute with univariate
-- polynomials. 
--
-- A gcd basis for a set of polynomials is a set of pairwise relatively prime 
-- polynomials which all divide the original set and whose product is the
-- same than the product of the original set.
--
-- A square free basis for a list of polynomials is just a list
-- of square free polynomials which are pairwise relatively primes and have
-- the same roots than the original polynomials.

  R : GcdDomain
  P : UnivariatePolynomialCategory(R)

  PUB == with
      squareFreeBasis : List(P) -> List(P)
        ++ 
      gcdBasis : List(P) -> List(P)
        ++ decompose a list of polynomials into pairwise relatively 
        ++ prime polynomials
      gcdBasisAdd : (P,List(P)) -> List(P)
        ++ add one polynomial to list of pairwise relatively prime polynomials

  PRIV == add

     squareFreeBasis(lpols) ==
       lpols = [] => []
       pol := first(lpols)
       sqpol := unitCanonical(squareFreePart(pol))
       gcdBasis(cons(sqpol,squareFreeBasis(rest(lpols))))

     gcdBasisAdd(p,lpols) ==
       (degree(p) = 0) => lpols
       null lpols => [unitCanonical p]
       p1 := first(lpols)
       g := gcd(p,p1)
       (degree(g) = 0) => cons(p1,gcdBasisAdd(p,rest lpols))
       p := (p exquo g)::P
       p1 := (p1 exquo g)::P
       basis := gcdBasisAdd(p,rest(lpols))
       if degree(p1) > 0 then basis := cons(p1,basis)
       gcdBasisAdd(g,basis)

     gcdBasis(lpols) ==
       (#lpols <= 1) => lpols
       basis := gcdBasis(rest lpols)
       gcdBasisAdd(first(lpols),basis)

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/4986109030324487666-25px001.spad
      using old system compiler.
   CADU abbreviates package CylindricalAlgebraicDecompositionUtilities 
------------------------------------------------------------------------
   initializing NRLIB CADU for CylindricalAlgebraicDecompositionUtilities 
   compiling into NRLIB CADU 
   compiling exported squareFreeBasis : List P -> List P
Time: 0.04 SEC.

   compiling exported gcdBasisAdd : (P,List P) -> List P
Time: 0.01 SEC.

   compiling exported gcdBasis : List P -> List P
Time: 0 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |CylindricalAlgebraicDecompositionUtilities| REDEFINED

;;;     ***       |CylindricalAlgebraicDecompositionUtilities| REDEFINED
Time: 0 SEC.

   Cumulative Statistics for Constructor CylindricalAlgebraicDecompositionUtilities
      Time: 0.05 seconds

   finalizing NRLIB CADU 
   Processing CylindricalAlgebraicDecompositionUtilities for Browser database:
--->-->CylindricalAlgebraicDecompositionUtilities(constructor): Not documented!!!!
--------(squareFreeBasis ((List P) (List P)))---------
--------(gcdBasis ((List P) (List P)))---------
--->-->CylindricalAlgebraicDecompositionUtilities((gcdBasis ((List P) (List P)))): Improper first word in comments: decompose
"decompose a list of polynomials into pairwise relatively prime polynomials"
--------(gcdBasisAdd ((List P) P (List P)))---------
--->-->CylindricalAlgebraicDecompositionUtilities((gcdBasisAdd ((List P) P (List P)))): Improper first word in comments: add
"add one polynomial to list of pairwise relatively prime polynomials"
--->-->CylindricalAlgebraicDecompositionUtilities(): Missing Description
; compiling file "/var/aw/var/LatexWiki/CADU.NRLIB/CADU.lsp" (written 11 APR 2014 06:34:38 PM):

; /var/aw/var/LatexWiki/CADU.NRLIB/CADU.fasl written
; compilation finished in 0:00:00.026
------------------------------------------------------------------------
   CylindricalAlgebraicDecompositionUtilities is now explicitly exposed
      in frame initial 
   CylindricalAlgebraicDecompositionUtilities will be automatically 
      loaded when needed from /var/aw/var/LatexWiki/CADU.NRLIB/CADU

SimpleCell?

)ab domain SCELL SimpleCell

-- This domain is made to work with one-dimensional semi-algebraic cells
-- ie either an algebraic point, either an interval. The point is specified as 
-- specification of an algebraic value.

SimpleCell(TheField,ThePols) : PUB == PRIV where
  TheField : RealClosedField
  ThePols : UnivariatePolynomialCategory(TheField)
  O           ==> OutputForm
  B           ==> Boolean
  Z           ==> Integer
  N           ==> NonNegativeInteger

  --  VARS ==> VariationsPackage(TheField,ThePols)
  VARS ==> RealPolynomialUtilitiesPackage(TheField,ThePols)
  LF ==> List(TheField)

  PUB == CoercibleTo(O) with

     allSimpleCells : (ThePols,Symbol) -> List %
     allSimpleCells : (List(ThePols),Symbol) -> List %
     hasDimension? : % -> B
     samplePoint : % -> TheField
     stablePol : % -> ThePols
     variableOf : % -> Symbol
     separe : (LF,TheField,TheField) -> LF
     pointToCell : (TheField,B,Symbol) -> %

  PRIV == add

     Rep := Record(samplePoint:TheField,
                   hasDim:B,
                   varOf:Symbol)

     samplePoint(c) == c.samplePoint

     stablePol(c) == error "Prout"

     hasDimension?(c) == c.hasDim

     variableOf(c) == c.varOf

     coerce(c:%):O ==
       o : O := ((c.varOf)::O) = ((c.samplePoint)::O)
       brace [o,(c.hasDim)::O]

     separe(liste,gauche,droite) ==
       milieu : TheField := (gauche + droite) / (2::TheField)
       liste = [] => [milieu]
       #liste = 1 => [gauche,first(liste),droite]
       nbe := first(liste)
       lg :List(TheField) := []
       ld :List(TheField) := rest(liste)
       sg := sign(milieu-nbe)
       while sg > 0 repeat
         lg := cons(nbe,lg)
         ld = [] => return(separe(reverse(lg),gauche,milieu))
         nbe := first(ld)
         sg := sign(milieu-nbe)
         ld := rest(ld)
       sg < 0 =>
         append(separe(reverse(lg),gauche,milieu),
                rest(separe(cons(nbe,ld),milieu,droite)))
       newDroite := (gauche+milieu)/(2::TheField)
       null lg =>
           newGauche := (milieu+droite)/(2::TheField)
           while newGauche >= first(ld) repeat
             newGauche := (milieu+newGauche)/(2::TheField)
           append([gauche,milieu],separe(ld,newGauche,droite))
       while newDroite <= first(lg) repeat
         newDroite := (newDroite+milieu)/(2::TheField)
       newGauche := (milieu+droite)/(2::TheField)
       null ld => append(separe(reverse(lg),gauche,newDroite),[milieu,droite])
       while newGauche >= first(ld) repeat
         newGauche := (milieu+newGauche)/(2::TheField)
       append(separe(reverse(lg),gauche,newDroite),
              cons(milieu,separe(ld,newGauche,droite)))

     pointToCell(sp,hasDim?,varName) ==
       [sp,hasDim?,varName]$Rep

     allSimpleCells(p:ThePols,var:Symbol) ==
       allSimpleCells([p],var)

     PACK ==> CylindricalAlgebraicDecompositionUtilities(TheField,ThePols)
     allSimpleCells(lp:List(ThePols),var:Symbol) ==
       lp1 := gcdBasis(lp)$PACK
       null(lp1) => [pointToCell(0,true,var)]
       b := ("max" / [ boundOfCauchy(p)$VARS for p in lp1 ])::TheField
       l := "append" / [allRootsOf(makeSUP(unitCanonical(p))) for p in lp1]
       l := sort(l)
       l1 := separe(l,-b,b)
       res : List(%) := [pointToCell(first(l1),true,var)]
       l1 := rest(l1)
       while not(null(l1)) repeat
         res := cons(pointToCell(first(l1),false,var),res)
         l1 := rest(l1)
         l1 = [] => return(error "Liste vide")
         res := cons(pointToCell(first(l1),true,var),res)
         l1 := rest(l1)
       reverse! res

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/2449457199855673372-25px002.spad
      using old system compiler.
   SCELL abbreviates domain SimpleCell 
------------------------------------------------------------------------
   initializing NRLIB SCELL for SimpleCell 
   compiling into NRLIB SCELL 
   compiling exported samplePoint : $ -> TheField
      SCELL;samplePoint;$TheField;1 is replaced by QVELTc0 
Time: 0.01 SEC.

   compiling exported stablePol : $ -> ThePols
      SCELL;stablePol;$ThePols;2 is replaced by errorProut 
Time: 0 SEC.

   compiling exported hasDimension? : $ -> Boolean
      SCELL;hasDimension?;$B;3 is replaced by QVELTc1 
Time: 0 SEC.

   compiling exported variableOf : $ -> Symbol
      SCELL;variableOf;$S;4 is replaced by QVELTc2 
Time: 0 SEC.

   compiling exported coerce : $ -> OutputForm
Time: 0 SEC.

   compiling exported separe : (List TheField,TheField,TheField) -> List TheField
Time: 0.01 SEC.

   compiling exported pointToCell : (TheField,Boolean,Symbol) -> $
      SCELL;pointToCell;TheFieldBS$;7 is replaced by VECTOR 
Time: 0 SEC.

   compiling exported allSimpleCells : (ThePols,Symbol) -> List $
Time: 0 SEC.

   processing macro definition PACK ==> CylindricalAlgebraicDecompositionUtilities(TheField,ThePols) 
   compiling exported allSimpleCells : (List ThePols,Symbol) -> List $
Time: 0.02 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |SimpleCell| REDEFINED

;;;     ***       |SimpleCell| REDEFINED
Time: 0 SEC.

   Cumulative Statistics for Constructor SimpleCell
      Time: 0.04 seconds

   finalizing NRLIB SCELL 
   Processing SimpleCell for Browser database:
--->-->SimpleCell(constructor): Not documented!!!!
--->-->SimpleCell((allSimpleCells ((List %) ThePols (Symbol)))): Not documented!!!!
--->-->SimpleCell((allSimpleCells ((List %) (List ThePols) (Symbol)))): Not documented!!!!
--->-->SimpleCell((hasDimension? ((Boolean) %))): Not documented!!!!
--->-->SimpleCell((samplePoint (TheField %))): Not documented!!!!
--->-->SimpleCell((stablePol (ThePols %))): Not documented!!!!
--->-->SimpleCell((variableOf ((Symbol) %))): Not documented!!!!
--->-->SimpleCell((separe ((List TheField) (List TheField) TheField TheField))): Not documented!!!!
--->-->SimpleCell((pointToCell (% TheField (Boolean) (Symbol)))): Not documented!!!!
--->-->SimpleCell(): Missing Description
; compiling file "/var/aw/var/LatexWiki/SCELL.NRLIB/SCELL.lsp" (written 11 APR 2014 06:34:40 PM):

; /var/aw/var/LatexWiki/SCELL.NRLIB/SCELL.fasl written
; compilation finished in 0:00:00.071
------------------------------------------------------------------------
   SimpleCell is now explicitly exposed in frame initial 
   SimpleCell will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/SCELL.NRLIB/SCELL

Cell

)ab domain CELL Cell

Cell(TheField) : PUB == PRIV where
  TheField : RealClosedField

  ThePols ==> Polynomial(TheField)

  O           ==> OutputForm
  B           ==> Boolean
  Z           ==> Integer
  N           ==> NonNegativeInteger
  BUP         ==> SparseUnivariatePolynomial(TheField)
  SCELL       ==> SimpleCell(TheField,BUP)

  PUB == CoercibleTo(O) with

     samplePoint : % -> List(TheField)
     dimension : % -> N
     hasDimension? :  (%,Symbol) -> B
     makeCell : List(SCELL) -> %
     makeCell : (SCELL,%) -> %
     mainVariableOf : % -> Symbol
     variablesOf : % -> List Symbol
     projection : % -> Union(%,"failed")

  PRIV == add

    Rep := List(SCELL)

    coerce(c:%):O == 
      paren [sc::O for sc in c]

    projection(cell) ==
      null cell => error "projection: should not appear"
      r := rest(cell)
      null r => "failed"
      r

    makeCell(l:List(SCELL)) == l

    makeCell(scell,toAdd) == cons(scell,toAdd)

    mainVariableOf(cell) == 
      null(cell) => 
        error "Should not appear"
      variableOf(first(cell))

    variablesOf(cell) ==
      null(cell) => []
      cons(mainVariableOf(cell),variablesOf(rest(cell)::%))

    dimension(cell) ==
      null(cell) => 0
      hasDimension?(first(cell)) => 1+dimension(rest(cell))
      dimension(rest(cell))

    hasDimension?(cell,var) ==
      null(cell) => 
        error "Should not appear"
      sc : SCELL := first(cell)
      v := variableOf(sc)
      v = var => hasDimension?(sc)
      v < var => false
      v > var => true
      error "Caca Prout"

    samplePoint(cell) ==
      null(cell) => []
      cons(samplePoint(first(cell)),samplePoint(rest(cell)))

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/1601075696811996402-25px003.spad
      using old system compiler.
   CELL abbreviates domain Cell 
------------------------------------------------------------------------
   initializing NRLIB CELL for Cell 
   compiling into NRLIB CELL 
Local variable Rep type redefined: (Join (ListAggregate (SimpleCell TheField (SparseUnivariatePolynomial TheField))) (CATEGORY domain (SIGNATURE nil ($)) (SIGNATURE null ((Boolean) $)) (SIGNATURE cons ($ (SimpleCell TheField (SparseUnivariatePolynomial TheField)) $)) (SIGNATURE append ($ $ $)) (IF (has (SimpleCell TheField (SparseUnivariatePolynomial TheField)) (BasicType)) (PROGN (SIGNATURE setUnion ($ $ $)) (SIGNATURE setIntersection ($ $ $)) (SIGNATURE setDifference ($ $ $))) noBranch) (IF (has (SimpleCell TheField (SparseUnivariatePolynomial TheField)) (OpenMath)) (ATTRIBUTE (OpenMath)) noBranch))) to (Join (SetCategory) (CATEGORY domain (SIGNATURE construct ((Record (: samplePoint TheField) (: hasDim (Boolean)) (: varOf (Symbol))) TheField (Boolean) (Symbol))) (SIGNATURE ~= ((Boolean) (Record (: samplePoint TheField) (: hasDim (Boolean)) (: varOf (Symbol))) (Record (: samplePoint TheField) (: hasDim (Boolean)) (: varOf (Symbol))))) (SIGNATURE coerce ((OutputForm) (Record (: samplePoint TheField) (: hasDim (Boolean)) (: varOf (Symbol))))) (SIGNATURE elt (TheField (Record (: samplePoint TheField) (: hasDim (Boolean)) (: varOf (Symbol))) samplePoint)) (SIGNATURE elt ((Boolean) (Record (: samplePoint TheField) (: hasDim (Boolean)) (: varOf (Symbol))) hasDim)) (SIGNATURE elt ((Symbol) (Record (: samplePoint TheField) (: hasDim (Boolean)) (: varOf (Symbol))) varOf)) (SIGNATURE setelt (TheField (Record (: samplePoint TheField) (: hasDim (Boolean)) (: varOf (Symbol))) samplePoint TheField)) (SIGNATURE setelt ((Boolean) (Record (: samplePoint TheField) (: hasDim (Boolean)) (: varOf (Symbol))) hasDim (Boolean))) (SIGNATURE setelt ((Symbol) (Record (: samplePoint TheField) (: hasDim (Boolean)) (: varOf (Symbol))) varOf (Symbol))) (SIGNATURE copy ((Record (: samplePoint TheField) (: hasDim (Boolean)) (: varOf (Symbol))) (Record (: samplePoint TheField) (: hasDim (Boolean)) (: varOf (Symbol)))))))
   compiling exported coerce : $ -> OutputForm
Time: 0.01 SEC.

   compiling exported projection : $ -> Union($,failed)
Time: 0 SEC.

   compiling exported makeCell : List SimpleCell(TheField,SparseUnivariatePolynomial TheField) -> $
      CELL;makeCell;L$;3 is replaced by l 
Time: 0 SEC.

   compiling exported makeCell : (SimpleCell(TheField,SparseUnivariatePolynomial TheField),$) -> $
      CELL;makeCell;Sc2$;4 is replaced by CONS 
Time: 0 SEC.

   compiling exported mainVariableOf : $ -> Symbol
Time: 0 SEC.

   compiling exported variablesOf : $ -> List Symbol
Time: 0 SEC.

   compiling exported dimension : $ -> NonNegativeInteger
Time: 0 SEC.

   compiling exported hasDimension? : ($,Symbol) -> Boolean
Time: 0 SEC.

   compiling exported samplePoint : $ -> List TheField
Time: 0 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |Cell| REDEFINED

;;;     ***       |Cell| REDEFINED
Time: 0 SEC.

   Cumulative Statistics for Constructor Cell
      Time: 0.01 seconds

   finalizing NRLIB CELL 
   Processing Cell for Browser database:
--->-->Cell(constructor): Not documented!!!!
--->-->Cell((samplePoint ((List TheField) %))): Not documented!!!!
--->-->Cell((dimension ((NonNegativeInteger) %))): Not documented!!!!
--->-->Cell((hasDimension? ((Boolean) % (Symbol)))): Not documented!!!!
--->-->Cell((makeCell (% (List (SimpleCell TheField (SparseUnivariatePolynomial TheField)))))): Not documented!!!!
--->-->Cell((makeCell (% (SimpleCell TheField (SparseUnivariatePolynomial TheField)) %))): Not documented!!!!
--->-->Cell((mainVariableOf ((Symbol) %))): Not documented!!!!
--->-->Cell((variablesOf ((List (Symbol)) %))): Not documented!!!!
--->-->Cell((projection ((Union % failed) %))): Not documented!!!!
--->-->Cell(): Missing Description
; compiling file "/var/aw/var/LatexWiki/CELL.NRLIB/CELL.lsp" (written 11 APR 2014 06:34:41 PM):

; /var/aw/var/LatexWiki/CELL.NRLIB/CELL.fasl written
; compilation finished in 0:00:00.029
------------------------------------------------------------------------
   Cell is now explicitly exposed in frame initial 
   Cell will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CELL.NRLIB/CELL

CylindricalAlgebraicDecompositionPackage?

)ab pack CAD CylindricalAlgebraicDecompositionPackage

CylindricalAlgebraicDecompositionPackage(TheField) : PUB == PRIV where

  TheField : RealClosedField

  ThePols ==> Polynomial(TheField)
  P ==> ThePols
  BUP ==> SparseUnivariatePolynomial(TheField)
  RUP ==> SparseUnivariatePolynomial(ThePols)

  Z           ==> Integer
  N           ==> NonNegativeInteger

  CELL ==> Cell(TheField)
  SCELL ==> SimpleCell(TheField,BUP)

  PUB == with

      cylindricalDecomposition: List P ->        List CELL
      cylindricalDecomposition: (List(P),List(Symbol)) ->        List CELL
      projectionSet: (List RUP)                ->    List P
      coefficientSet: RUP                ->    List P
      discriminantSet : List RUP -> List(P)
      resultantSet :  List RUP -> List P 
      principalSubResultantSet : (RUP,RUP) -> List P
      specialise : (List(ThePols),CELL) -> List(BUP)

  PRIV == add

     cylindricalDecomposition(lpols) ==
       lv : List(Symbol) := []
       for pol in lpols repeat
         ground?(pol) => "next pol"
         lv := removeDuplicates(append(variables(pol),lv))
       lv := reverse(sort(lv))
       cylindricalDecomposition(lpols,lv)

     cylindricalDecomposition(lpols,lvars) ==
       lvars = [] => error("CAD: cylindricalDecomposition: empty list of vars")
       mv := first(lvars)
       lv := rest(lvars)
       lv = [] =>
         lp1 := [ univariate(pol) for pol in lpols ]
         scells := allSimpleCells(lp1,mv)$SCELL
         [ makeCell([scell]) for scell in scells ]
       lpols1 := projectionSet [univariate(pol,mv) for pol in lpols]
       previousCad := cylindricalDecomposition(lpols1,lv)
       res : List(CELL) := []
       for cell in previousCad repeat
         lspec := specialise(lpols,cell)
         scells := allSimpleCells(lspec,mv)
         res := append(res,[makeCell(scell,cell) for scell in scells])
       res

     PACK1 ==> CylindricalAlgebraicDecompositionUtilities(ThePols,RUP)
     PACK2 ==> CylindricalAlgebraicDecompositionUtilities(TheField,BUP)

     specialise(lpols,cell) ==
       lpols = [] => error("CAD: specialise: empty list of pols")
       sp := samplePoint(cell)
       vl := variablesOf(cell)
       res : List(BUP) := []
       for pol in lpols repeat
         p1 := univariate(eval(pol,vl,sp))
--         zero?(p1) => return(error "Bad sepcialise")
         degree(p1) = 0 => "next pol"
         res := cons(p1,res)
       res

     coefficientSet(pol) ==
       res : List(ThePols) := []
       for c in coefficients(pol) repeat
         ground?(c) => return(res)
         res := cons(c,res)
       res

     SUBRES ==> SubResultantPackage(ThePols,RUP)
     discriminantSet(lpols) ==
       res : List(ThePols) := []
       for p in lpols repeat
         v := subresultantVector(p,differentiate(p))$SUBRES
         not(zero?(degree(v.0))) => return(error "Bad discriminant")
         d : ThePols :=  leadingCoefficient(v.0)
--         d := discriminant p
         zero?(d) => return(error "Non Square Free polynomial")
         if not(ground? d) then res := cons(d,res)
       res

     principalSubResultantSet(p,q) ==
        if degree(p) < degree(q)
        then (p,q) := (q,p)
        if degree(p) = degree(q)
        then (p,q) := (q,pseudoRemainder(p, q))
        v := subresultantVector(p,q)$SUBRES
        [coefficient(v.i,i) for i in 0..(((#v)-2)::N)]

     resultantSet(lpols) ==
       res : List(ThePols) := []
       laux := lpols
       for p in lpols repeat
         laux := rest(laux)
         for q in laux repeat
           r : ThePols :=  first(principalSubResultantSet(p,q))
--           r := resultant(p,q)
           zero?(r) => return(error "Non relatively prime polynomials")
           if not(ground? r) then res := cons(r,res)
       res

     projectionSet(lpols) ==
       res : List(ThePols) := []
       for p in lpols repeat
         c := content(p)
         ground?(c) => "next p"
         res := cons(c,res)
       lp1 := [primitivePart p for p in lpols]
       f : ((RUP,RUP) -> Boolean) := (x,y)+->(degree(x) <= degree(y))
       lp1 := sort(f,lp1)
       lsqfrb := squareFreeBasis(lp1)$PACK1
       lsqfrb := sort(f,lsqfrb)
       for p in lp1 repeat
         res := append(res,coefficientSet(p))
       res := append(res,discriminantSet(lsqfrb))
       append(res,resultantSet(lsqfrb))

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/4314612298553742433-25px004.spad
      using old system compiler.
   CAD abbreviates package CylindricalAlgebraicDecompositionPackage 
------------------------------------------------------------------------
   initializing NRLIB CAD for CylindricalAlgebraicDecompositionPackage 
   compiling into NRLIB CAD 
   compiling exported cylindricalDecomposition : List Polynomial TheField -> List Cell TheField
Time: 0.01 SEC.

   compiling exported cylindricalDecomposition : (List Polynomial TheField,List Symbol) -> List Cell TheField
Time: 0.02 SEC.

   processing macro definition PACK1 ==> CylindricalAlgebraicDecompositionUtilities(Polynomial TheField,SparseUnivariatePolynomial Polynomial TheField) 
   processing macro definition PACK2 ==> CylindricalAlgebraicDecompositionUtilities(TheField,SparseUnivariatePolynomial TheField) 
   compiling exported specialise : (List Polynomial TheField,Cell TheField) -> List SparseUnivariatePolynomial TheField
Time: 0.01 SEC.

   compiling exported coefficientSet : SparseUnivariatePolynomial Polynomial TheField -> List Polynomial TheField
Time: 0 SEC.

   processing macro definition SUBRES ==> SubResultantPackage(Polynomial TheField,SparseUnivariatePolynomial Polynomial TheField) 
   compiling exported discriminantSet : List SparseUnivariatePolynomial Polynomial TheField -> List Polynomial TheField
Time: 0.02 SEC.

   compiling exported principalSubResultantSet : (SparseUnivariatePolynomial Polynomial TheField,SparseUnivariatePolynomial Polynomial TheField) -> List Polynomial TheField
Time: 0.01 SEC.

   compiling exported resultantSet : List SparseUnivariatePolynomial Polynomial TheField -> List Polynomial TheField
Time: 0.01 SEC.

   compiling exported projectionSet : List SparseUnivariatePolynomial Polynomial TheField -> List Polynomial TheField
Time: 0.01 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |CylindricalAlgebraicDecompositionPackage| REDEFINED

;;;     ***       |CylindricalAlgebraicDecompositionPackage| REDEFINED
Time: 0 SEC.

   Warnings: 
      [1] cylindricalDecomposition:  lv has no value
      [2] specialise:  res has no value
      [3] discriminantSet:  res has no value
      [4] resultantSet:  res has no value
      [5] projectionSet:  res has no value

   Cumulative Statistics for Constructor CylindricalAlgebraicDecompositionPackage
      Time: 0.09 seconds

   finalizing NRLIB CAD 
   Processing CylindricalAlgebraicDecompositionPackage for Browser database:
--->-->CylindricalAlgebraicDecompositionPackage(constructor): Not documented!!!!
--->-->CylindricalAlgebraicDecompositionPackage((cylindricalDecomposition ((List (Cell TheField)) (List (Polynomial TheField))))): Not documented!!!!
--->-->CylindricalAlgebraicDecompositionPackage((cylindricalDecomposition ((List (Cell TheField)) (List (Polynomial TheField)) (List (Symbol))))): Not documented!!!!
--->-->CylindricalAlgebraicDecompositionPackage((projectionSet ((List (Polynomial TheField)) (List (SparseUnivariatePolynomial (Polynomial TheField)))))): Not documented!!!!
--->-->CylindricalAlgebraicDecompositionPackage((coefficientSet ((List (Polynomial TheField)) (SparseUnivariatePolynomial (Polynomial TheField))))): Not documented!!!!
--->-->CylindricalAlgebraicDecompositionPackage((discriminantSet ((List (Polynomial TheField)) (List (SparseUnivariatePolynomial (Polynomial TheField)))))): Not documented!!!!
--->-->CylindricalAlgebraicDecompositionPackage((resultantSet ((List (Polynomial TheField)) (List (SparseUnivariatePolynomial (Polynomial TheField)))))): Not documented!!!!
--->-->CylindricalAlgebraicDecompositionPackage((principalSubResultantSet ((List (Polynomial TheField)) (SparseUnivariatePolynomial (Polynomial TheField)) (SparseUnivariatePolynomial (Polynomial TheField))))): Not documented!!!!
--->-->CylindricalAlgebraicDecompositionPackage((specialise ((List (SparseUnivariatePolynomial TheField)) (List (Polynomial TheField)) (Cell TheField)))): Not documented!!!!
--->-->CylindricalAlgebraicDecompositionPackage(): Missing Description
; compiling file "/var/aw/var/LatexWiki/CAD.NRLIB/CAD.lsp" (written 11 APR 2014 06:34:42 PM):

; /var/aw/var/LatexWiki/CAD.NRLIB/CAD.fasl written
; compilation finished in 0:00:00.059
------------------------------------------------------------------------
   CylindricalAlgebraicDecompositionPackage is now explicitly exposed 
      in frame initial 
   CylindricalAlgebraicDecompositionPackage will be automatically 
      loaded when needed from /var/aw/var/LatexWiki/CAD.NRLIB/CAD

Test Input

)lib SCELL CELL CAD

   SimpleCell is already explicitly exposed in frame initial 
   SimpleCell will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/SCELL.NRLIB/SCELL
   Cell is already explicitly exposed in frame initial 
   Cell will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CELL.NRLIB/CELL
   CylindricalAlgebraicDecompositionPackage is already explicitly 
      exposed in frame initial 
   CylindricalAlgebraicDecompositionPackage will be automatically 
      loaded when needed from /var/aw/var/LatexWiki/CAD.NRLIB/CAD
Ran := RECLOS(FRAC INT)

(1)

Cad := CAD(Ran)

(2)

p1 : POLY(Ran) := x^2+y^2-1

(3)

p2 : POLY(Ran) := y-x^2

(4)

lpols : List(POLY(Ran)) := [p1,p2]

(5)

cad := cylindricalDecomposition(lpols)$Cad

(6)

precision(30)

(7)

ls := [ samplePoint cell for cell in cad]

(8)

lf := [ [relativeApprox(coord,2^(-precision()))::Float for coord in point] for point in ls]

(9)

lp := [ point(term::List(DoubleFloat))$Point(DoubleFloat) for term in lf ]

(10)

[ sign(eval(p1,variables(p1),samplePoint(cell))::Ran) for cell in cad ]

(11)