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-- This software was written by Renaud Rioboo (Computer Algebra group of
-- Laboratoire d'Informatique de Paris 6) and is the property of university
-- Paris 6.
-----------------------------------------------------------------------------
)abbrev package POLUTIL RealPolynomialUtilitiesPackage
++ Author: Renaud Rioboo
++ Date Created: summer 1992
++ Basic Functions: provides polynomial utilities
++ Related Constructors: RealClosure,
++ Date Last Updated: July 2004
++ Also See:
++ AMS Classifications:
++ Keywords: Sturm sequences
++ References:
++ Description:
++ \axiomType{RealPolynomialUtilitiesPackage} provides common functions used
++ by interval coding.
RealPolynomialUtilitiesPackage(TheField,ThePols) : PUB == PRIV where
TheField : Field
ThePols : UnivariatePolynomialCategory(TheField)
Z ==> Integer
N ==> NonNegativeInteger
P ==> ThePols
PUB == with
sylvesterSequence : (ThePols,ThePols) -> List ThePols
++ \axiom{sylvesterSequence(p,q)} is the negated remainder sequence
++ of p and q divided by the last computed term
sturmSequence : ThePols -> List ThePols
++ \axiom{sturmSequence(p) = sylvesterSequence(p,p')}
if TheField has OrderedRing then
boundOfCauchy : ThePols -> TheField
++ \axiom{boundOfCauchy(p)} bounds the roots of p
sturmVariationsOf : List TheField -> N
++ \axiom{sturmVariationsOf(l)} is the number of sign variations
++ in the list of numbers l,
++ note that the first term counts as a sign
lazyVariations : (List(TheField), Z, Z) -> N
++ \axiom{lazyVariations(l,s1,sn)} is the number of sign variations
++ in the list of non null numbers [s1::l]@sn,
PRIV == add
sturmSequence(p) ==
sylvesterSequence(p,differentiate(p))
sylvesterSequence(p1,p2) ==
res : List(ThePols) := [p1]
while (p2 ~= 0) repeat
res := cons(p2 , res)
(p1 , p2) := (p2 , -(p1 rem p2))
if positive? degree(p1)
then
p1 := unitCanonical(p1)
res := [ term quo p1 for term in res ]
reverse! res
if TheField has OrderedRing
then
boundOfCauchy(p) ==
c :TheField := inv(leadingCoefficient(p))
l := [ c*term for term in rest(coefficients(p))]
null(l) => 1
1 + ("max" / [ abs(t) for t in l ])
-- sturmVariationsOf(l) ==
-- res : N := 0
-- lsg := sign(first(l))
-- for term in l repeat
-- if ^( (sg := sign(term) ) = 0 ) then
-- if (sg ~= lsg) then res := res + 1
-- lsg := sg
-- res
sturmVariationsOf(l) ==
null(l) => error "POLUTIL: sturmVariationsOf: empty list !"
l1 := first(l)
-- first 0 counts as a sign
ll : List(TheField) := []
for term in rest(l) repeat
-- zeros don't count
if not(zero?(term)) then ll := cons(term,ll)
-- if l1 is not zero then ll = reverse(l)
null(ll) => error "POLUTIL: sturmVariationsOf: Bad sequence"
ln := first(ll)
ll := reverse(rest(ll))
-- if l1 is not zero then first(l) = first(ll)
-- if l1 is zero then first zero should count as a sign
zero?(l1) => 1 + lazyVariations(rest(ll),sign(first(ll)),sign(ln))
lazyVariations(ll, sign(l1), sign(ln))
lazyVariations(l,sl,sh) ==
zero?(sl) or zero?(sh) => error "POLUTIL: lazyVariations: zero sign!"
null(l) =>
if sl = sh then 0 else 1
null(rest(l)) =>
if zero?(first(l))
then error "POLUTIL: lazyVariations: zero sign!"
else
if sl = sh
then
if (sl = sign(first(l)))
then 0
else 2
-- in this case we save one test
else 1
s := sign(l.2)
lazyVariations([first(l)],sl,s) +
lazyVariations(rest(rest(l)),s,sh)
)abbrev category RRCC RealRootCharacterizationCategory
++ Author: Renaud Rioboo
++ Date Created: summer 1992
++ Date Last Updated: January 2004
++ Basic Functions: provides operations with generic real roots of
++ polynomials
++ Related Constructors: RealClosure, RightOpenIntervalRootCharacterization
++ Also See:
++ AMS Classifications:
++ Keywords: Real Algebraic Numbers
++ References:
++ Description:
++ \axiomType{RealRootCharacterizationCategory} provides common acces
++ functions for all real root codings.
RealRootCharacterizationCategory(TheField, ThePols ) : Category == PUB where
TheField : Join(OrderedRing, Field)
ThePols : UnivariatePolynomialCategory(TheField)
Z ==> Integer
N ==> PositiveInteger
PUB ==>
SetCategory with
sign: ( ThePols, $ ) -> Z
++ \axiom{sign(pol,aRoot)} gives the sign of \axiom{pol}
++ interpreted as \axiom{aRoot}
zero? : ( ThePols, $ ) -> Boolean
++ \axiom{zero?(pol,aRoot)} answers if \axiom{pol}
++ interpreted as \axiom{aRoot} is \axiom{0}
negative?: ( ThePols, $ ) -> Boolean
++ \axiom{negative?(pol,aRoot)} answers if \axiom{pol}
++ interpreted as \axiom{aRoot} is negative
positive?: ( ThePols, $ ) -> Boolean
++ \axiom{positive?(pol,aRoot)} answers if \axiom{pol}
++ interpreted as \axiom{aRoot} is positive
recip: ( ThePols, $ ) -> Union(ThePols,"failed")
++ \axiom{recip(pol,aRoot)} tries to inverse \axiom{pol}
++ interpreted as \axiom{aRoot}
definingPolynomial: $ -> ThePols
++ \axiom{definingPolynomial(aRoot)} gives a polynomial
++ such that \axiom{definingPolynomial(aRoot).aRoot = 0}
allRootsOf: ThePols -> List $
++ \axiom{allRootsOf(pol)} creates all the roots of \axiom{pol}
++ in the Real Closure, assumed in order.
rootOf: ( ThePols, N ) -> Union($,"failed")
++ \axiom{rootOf(pol,n)} gives the nth root for the order of the
++ Real Closure
approximate : (ThePols,$,TheField) -> TheField
++ \axiom{approximate(term,root,prec)} gives an approximation
++ of \axiom{term} over \axiom{root} with precision \axiom{prec}
relativeApprox : (ThePols,$,TheField) -> TheField
++ \axiom{approximate(term,root,prec)} gives an approximation
++ of \axiom{term} over \axiom{root} with precision \axiom{prec}
add
zero?(toTest, rootChar) ==
sign(toTest, rootChar) = 0
negative?(toTest, rootChar) ==
negative? sign(toTest, rootChar)
positive?(toTest, rootChar) ==
positive? sign(toTest, rootChar)
rootOf(pol,n) ==
liste:List($):= allRootsOf(pol)
# liste > n => "failed"
liste.n
recip(toInv,rootChar) ==
degree(toInv) = 0 =>
res := recip(leadingCoefficient(toInv))
if (res case "failed") then "failed" else (res::TheField::ThePols)
defPol := definingPolynomial(rootChar)
d := principalIdeal([defPol,toInv])
zero?(d.generator,rootChar) => "failed"
if (degree(d.generator) ~= 0 )
then
defPol := (defPol exquo (d.generator))::ThePols
d := principalIdeal([defPol,toInv])
d.coef.2
)abbrev category RCFIELD RealClosedField
++ Author: Renaud Rioboo
++ Date Created: may 1993
++ Date Last Updated: January 2004
++ Basic Functions: provides computations with generic real roots of
++ polynomials
++ Related Constructors: SimpleOrderedAlgebraicExtension, RealClosure
++ Also See:
++ AMS Classifications:
++ Keywords: Real Algebraic Numbers
++ References:
++ Description:
++ \axiomType{RealClosedField} provides common acces
++ functions for all real closed fields.
RealClosedField : Category == PUB where
E ==> OutputForm
SUP ==> SparseUnivariatePolynomial
OFIELD ==> Join(OrderedRing,Field)
PME ==> SUP($)
N ==> NonNegativeInteger
PI ==> PositiveInteger
RN ==> Fraction(Integer)
Z ==> Integer
POLY ==> Polynomial
PACK ==> SparseUnivariatePolynomialFunctions2
PUB == Join(CharacteristicZero,
OrderedRing,
CommutativeRing,
Field,
FullyRetractableTo(Fraction(Integer)),
Algebra Integer,
Algebra(Fraction(Integer)),
RadicalCategory) with
mainForm : $ -> Union(E,"failed")
++ \axiom{mainForm(x)} is the main algebraic quantity name of
++ \axiom{x}
mainDefiningPolynomial : $ -> Union(PME,"failed")
++ \axiom{mainDefiningPolynomial(x)} is the defining
++ polynomial for the main algebraic quantity of \axiom{x}
mainValue : $ -> Union(PME,"failed")
++ \axiom{mainValue(x)} is the expression of \axiom{x} in terms
++ of \axiom{SparseUnivariatePolynomial($)}
rootOf: (PME,PI,E) -> Union($,"failed")
++ \axiom{rootOf(pol,n,name)} creates the nth root for the order
++ of \axiom{pol} and names it \axiom{name}
rootOf: (PME,PI) -> Union($,"failed")
++ \axiom{rootOf(pol,n)} creates the nth root for the order
++ of \axiom{pol} and gives it unique name
allRootsOf: PME -> List $
++ \axiom{allRootsOf(pol)} creates all the roots
++ of \axiom{pol} naming each uniquely
allRootsOf: (SUP(RN)) -> List $
++ \axiom{allRootsOf(pol)} creates all the roots
++ of \axiom{pol} naming each uniquely
allRootsOf: (SUP(Z)) -> List $
++ \axiom{allRootsOf(pol)} creates all the roots
++ of \axiom{pol} naming each uniquely
allRootsOf: (POLY($)) -> List $
++ \axiom{allRootsOf(pol)} creates all the roots
++ of \axiom{pol} naming each uniquely
allRootsOf: (POLY(RN)) -> List $
++ \axiom{allRootsOf(pol)} creates all the roots
++ of \axiom{pol} naming each uniquely
allRootsOf: (POLY(Z)) -> List $
++ \axiom{allRootsOf(pol)} creates all the roots
++ of \axiom{pol} naming each uniquely
sqrt: ($,PI) -> $
++ \axiom{sqrt(x,n)} is \axiom{x ** (1/n)}
sqrt: $ -> $
++ \axiom{sqrt(x)} is \axiom{x ** (1/2)}
sqrt: RN -> $
++ \axiom{sqrt(x)} is \axiom{x ** (1/2)}
sqrt: Z -> $
++ \axiom{sqrt(x)} is \axiom{x ** (1/2)}
rename! : ($,E) -> $
++ \axiom{rename!(x,name)} changes the way \axiom{x} is printed
rename : ($,E) -> $
++ \axiom{rename(x,name)} gives a new number that prints as name
approximate: ($,$) -> RN
++ \axiom{approximate(n,p)} gives an approximation of \axiom{n}
++ that has precision \axiom{p}
add
sqrt(a:$):$ == sqrt(a,2)
sqrt(a:RN):$ == sqrt(a::$,2)
sqrt(a:Z):$ == sqrt(a::$,2)
characteristic == 0
rootOf(pol,n,o) ==
r := rootOf(pol,n)
r case "failed" => "failed"
rename!(r,o)
rootOf(pol,n) ==
liste:List($):= allRootsOf(pol)
# liste > n => "failed"
liste.n
sqrt(x,n) ==
n = 1 => x
zero?(x) => 0
one?(x) => 1
if odd?(n)
then
r := rootOf(monomial(1,n) - (x :: PME), 1)
else
r := rootOf(monomial(1,n) - (x :: PME), 2)
r case "failed" => error "no roots"
n = 2 => rename(r,root(x::E)$E)
rename(r,root(x :: E, n :: E)$E)
(x : $) ** (rn : RN) == sqrt(x**numer(rn),denom(rn)::PI)
nthRoot(x, n) ==
zero?(n) => x
negative?(n) => inv(sqrt(x,(-n) :: PI))
sqrt(x,n :: PI)
allRootsOf(p:SUP(RN)) == allRootsOf(map(#1 :: $ ,p)$PACK(RN,$))
allRootsOf(p:SUP(Z)) == allRootsOf(map(#1 :: $ ,p)$PACK(Z,$))
allRootsOf(p:POLY($)) == allRootsOf(univariate(p))
allRootsOf(p:POLY(RN)) == allRootsOf(univariate(p))
allRootsOf(p:POLY(Z)) == allRootsOf(univariate(p))
)abbrev domain ROIRC RightOpenIntervalRootCharacterization
++ Author: Renaud Rioboo
++ Date Created: summer 1992
++ Date Last Updated: January 2004
++ Basic Functions: provides computations with real roots of olynomials
++ Related Constructors: RealRootCharacterizationCategory, RealClosure
++ Also See:
++ AMS Classifications:
++ Keywords: Real Algebraic Numbers
++ References:
++ Description:
++ \axiomType{RightOpenIntervalRootCharacterization} provides work with
++ interval root coding.
RightOpenIntervalRootCharacterization(TheField,ThePolDom) : PUB == PRIV where
TheField : Join(OrderedRing,Field)
ThePolDom : UnivariatePolynomialCategory(TheField)
Z ==> Integer
P ==> ThePolDom
N ==> NonNegativeInteger
B ==> Boolean
UTIL ==> RealPolynomialUtilitiesPackage(TheField,ThePolDom)
RRCC ==> RealRootCharacterizationCategory
O ==> OutputForm
TwoPoints ==> Record(low:TheField , high:TheField)
PUB == RealRootCharacterizationCategory(TheField, ThePolDom) with
left : $ -> TheField
++ \axiom{left(rootChar)} is the left bound of the isolating
++ interval
right : $ -> TheField
++ \axiom{right(rootChar)} is the right bound of the isolating
++ interval
size : $ -> TheField
++ The size of the isolating interval
middle : $ -> TheField
++ \axiom{middle(rootChar)} is the middle of the isolating
++ interval
refine : $ -> $
++ \axiom{refine(rootChar)} shrinks isolating interval around
++ \axiom{rootChar}
mightHaveRoots : (P,$) -> B
++ \axiom{mightHaveRoots(p,r)} is false if \axiom{p.r} is not 0
relativeApprox : (P,$,TheField) -> TheField
++ \axiom{relativeApprox(exp,c,p) = a} is relatively close to exp
++ as a polynomial in c ip to precision p
PRIV == add
-- local functions
makeChar: (TheField,TheField,ThePolDom) -> $
refine! : $ -> $
sturmIsolate : (List(P), TheField, TheField,N,N) -> List TwoPoints
isolate : List(P) -> List TwoPoints
rootBound : P -> TheField
-- varStar : P -> N
linearRecip : ( P , $) -> Union(P, "failed")
linearZero? : (TheField,$) -> B
linearSign : (P,$) -> Z
sturmNthRoot : (List(P), TheField, TheField,N,N,N) -> Union(TwoPoints,"failed")
addOne : P -> P
minus : P -> P
translate : (P,TheField) -> P
dilate : (P,TheField) -> P
invert : P -> P
evalOne : P -> TheField
hasVarsl: List(TheField) -> B
hasVars: P -> B
-- Representation
Rep:= Record(low:TheField,high:TheField,defPol:ThePolDom)
-- and now the code !
size(rootCode) ==
rootCode.high - rootCode.low
relativeApprox(pval,rootCode,prec) ==
-- beurk !
dPol := rootCode.defPol
degree(dPol) = 1 =>
c := -coefficient(dPol,0)/leadingCoefficient(dPol)
pval.c
pval := pval rem dPol
degree(pval) = 0 => leadingCoefficient(pval)
zero?(pval,rootCode) => 0
while mightHaveRoots(pval,rootCode) repeat
rootCode := refine(rootCode)
dpval := differentiate(pval)
degree(dpval) = 0 =>
l := left(rootCode)
r := right(rootCode)
a := pval.l
b := pval.r
while ( abs(2*(a-b)/(a+b)) > prec ) repeat
rootCode := refine(rootCode)
l := left(rootCode)
r := right(rootCode)
a := pval.l
b := pval.r
(a+b)/(2::TheField)
zero?(dpval,rootCode) =>
relativeApprox(pval,
[left(rootCode),
right(rootCode),
gcd(dpval,rootCode.defPol)]$Rep,
prec)
while mightHaveRoots(dpval,rootCode) repeat
rootCode := refine(rootCode)
l := left(rootCode)
r := right(rootCode)
a := pval.l
b := pval.r
while ( abs(2*(a-b)/(a+b)) > prec ) repeat
rootCode := refine(rootCode)
l := left(rootCode)
r := right(rootCode)
a := pval.l
b := pval.r
(a+b)/(2::TheField)
approximate(pval,rootCode,prec) ==
-- glurp
dPol := rootCode.defPol
degree(dPol) = 1 =>
c := -coefficient(dPol,0)/leadingCoefficient(dPol)
pval.c
pval := pval rem dPol
degree(pval) = 0 => leadingCoefficient(pval)
dpval := differentiate(pval)
a : TheField
b : TheField
degree(dpval) = 0 =>
l := left(rootCode)
r := right(rootCode)
while ( abs((a := pval.l) - (b := pval.r)) > prec ) repeat
rootCode := refine(rootCode)
l := left(rootCode)
r := right(rootCode)
(a+b)/(2::TheField)
zero?(dpval,rootCode) =>
approximate(pval,
[left(rootCode),
right(rootCode),
gcd(dpval,rootCode.defPol)]$Rep,
prec)
while mightHaveRoots(dpval,rootCode) repeat
rootCode := refine(rootCode)
l := left(rootCode)
r := right(rootCode)
while ( abs((a := pval.l) - (b := pval.r)) > prec ) repeat
rootCode := refine(rootCode)
l := left(rootCode)
r := right(rootCode)
(a+b)/(2::TheField)
addOne(p) == p.(monomial(1,1)+(1::P))
minus(p) == p.(monomial(-1,1))
translate(p,a) == p.(monomial(1,1)+(a::P))
dilate(p,a) == p.(monomial(a,1))
evalOne(p) == "+" / coefficients(p)
invert(p) ==
d := degree(p)
mapExponents((d-#1)::N, p)
rootBound(p) ==
res : TheField := 1
raw :TheField := 1+boundOfCauchy(p)$UTIL
while (res < raw) repeat
res := 2*(res)
res
sturmNthRoot(lp,l,r,vl,vr,n) ==
nv := (vl - vr)::N
nv < n => "failed"
((nv = 1) and (n = 1)) => [l,r]
int := (l+r)/(2::TheField)
lt:List(TheField):=[]
for t in lp repeat
lt := cons(t.int , lt)
vi := sturmVariationsOf(reverse! lt)$UTIL
o :Z := n - vl + vi
if positive? o
then
sturmNthRoot(lp,int,r,vi,vr,o::N)
else
sturmNthRoot(lp,l,int,vl,vi,n)
sturmIsolate(lp,l,r,vl,vr) ==
r <= l => error "ROIRC: sturmIsolate: bad bounds"
n := (vl - vr)::N
zero?(n) => []
one?(n) => [[l,r]]
int := (l+r)/(2::TheField)
vi := sturmVariationsOf( [t.int for t in lp ] )$UTIL
append(sturmIsolate(lp,l,int,vl,vi),sturmIsolate(lp,int,r,vi,vr))
isolate(lp) ==
b := rootBound(first(lp))
l1,l2 : List(TheField)
(l1,l2) := ([] , [])
for t in reverse(lp) repeat
if odd?(degree(t))
then
(l1,l2):= (cons(-leadingCoefficient(t),l1),
cons(leadingCoefficient(t),l2))
else
(l1,l2):= (cons(leadingCoefficient(t),l1),
cons(leadingCoefficient(t),l2))
sturmIsolate(lp,
-b,
b,
sturmVariationsOf(l1)$UTIL,
sturmVariationsOf(l2)$UTIL)
rootOf(pol,n) ==
ls := sturmSequence(pol)$UTIL
pol := unitCanonical(first(ls)) -- this one is SqFR
degree(pol) = 0 => "failed"
numberOfMonomials(pol) = 1 => ([0,1,monomial(1,1)]$Rep)::$
b := rootBound(pol)
l1,l2 : List(TheField)
(l1,l2) := ([] , [])
for t in reverse(ls) repeat
if odd?(degree(t))
then
(l1,l2):= (cons(leadingCoefficient(t),l1),
cons(-leadingCoefficient(t),l2))
else
(l1,l2):= (cons(leadingCoefficient(t),l1),
cons(leadingCoefficient(t),l2))
res := sturmNthRoot(ls,
-b,
b,
sturmVariationsOf(l2)$UTIL,
sturmVariationsOf(l1)$UTIL,
n)
res case "failed" => "failed"
makeChar(res.low,res.high,pol)
allRootsOf(pol) ==
ls := sturmSequence(unitCanonical pol)$UTIL
pol := unitCanonical(first(ls)) -- this one is SqFR
degree(pol) = 0 => []
numberOfMonomials(pol) = 1 => [[0,1,monomial(1,1)]$Rep]
[ makeChar(term.low,term.high,pol) for term in isolate(ls) ]
hasVarsl(l:List(TheField)) ==
null(l) => false
f := sign(first(l))
for term in rest(l) repeat
if negative?(f*term) then return(true)
false
hasVars(p:P) ==
zero?(p) => error "ROIRC: hasVars: null polynonial"
zero?(coefficient(p,0)) => true
hasVarsl(coefficients(p))
mightHaveRoots(p,rootChar) ==
a := rootChar.low
q := translate(p,a)
not(hasVars(q)) => false
-- varStar(q) = 0 => false
a := (rootChar.high) - a
q := dilate(q,a)
sign(coefficient(q,0))*sign(evalOne(q)) <= 0 => true
q := minus(addOne(q))
not(hasVars(q)) => false
-- varStar(q) = 0 => false
q := invert(q)
hasVars(addOne(q))
-- ^(varStar(addOne(q)) = 0)
coerce(rootChar:$):O ==
commaSeparate([ hconcat("[" :: O , (rootChar.low)::O),
hconcat((rootChar.high)::O,"[" ::O ) ])
c1 = c2 ==
mM := max(c1.low,c2.low)
Mm := min(c1.high,c2.high)
mM >= Mm => false
rr : ThePolDom := gcd(c1.defPol,c2.defPol)
degree(rr) = 0 => false
sign(rr.mM) * sign(rr.Mm) <= 0
makeChar(left,right,pol) ==
res :$ := [left,right,leadingMonomial(pol)+reductum(pol)]$Rep -- safe copy
while zero?(pol.(res.high)) repeat refine!(res)
while negative?(res.high * res.low) repeat refine!(res)
zero?(pol.(res.low)) => [res.low,res.high,monomial(1,1)-(res.low)::P]
res
definingPolynomial(rootChar) == rootChar.defPol
linearRecip(toTest,rootChar) ==
c := - inv(leadingCoefficient(toTest)) * coefficient(toTest,0)
r := recip(rootChar.defPol.c)
if (r case "failed")
then
if (c - rootChar.low) * (c - rootChar.high) <= 0
then
"failed"
else
newPol := (rootChar.defPol exquo toTest)::P
((1$ThePolDom - inv(newPol.c)*newPol) exquo toTest)::P
else
((1$ThePolDom - (r::TheField)*rootChar.defPol) exquo toTest)::P
recip(toTest,rootChar) ==
degree(toTest) = 0 or degree(rootChar.defPol) <= degree(toTest) =>
error "IRC: recip: Not reduced"
degree(rootChar.defPol) = 1 =>
error "IRC: recip: Linear Defining Polynomial"
degree(toTest) = 1 =>
linearRecip(toTest, rootChar)
d := extendedEuclidean((rootChar.defPol),toTest)
(degree(d.generator) = 0 ) =>
d.coef2
d.generator := unitCanonical(d.generator)
(d.generator.(rootChar.low) *
d.generator.(rootChar.high)<= 0) => "failed"
newPol := (rootChar.defPol exquo (d.generator))::P
degree(newPol) = 1 =>
c := - inv(leadingCoefficient(newPol)) * coefficient(newPol,0)
inv(toTest.c)::P
degree(toTest) = 1 =>
c := - coefficient(toTest,0)/ leadingCoefficient(toTest)
((1$ThePolDom - inv(newPol.(c))*newPol) exquo toTest)::P
d := extendedEuclidean(newPol,toTest)
d.coef2
linearSign(toTest,rootChar) ==
c := - inv(leadingCoefficient(toTest)) * coefficient(toTest,0)
ev := sign(rootChar.defPol.c)
if zero?(ev)
then
if (c - rootChar.low) * (c - rootChar.high) <= 0
then
0
else
sign(toTest.(rootChar.high))
else
if (ev*sign(rootChar.defPol.(rootChar.high)) <= 0 )
then
sign(toTest.(rootChar.high))
else
sign(toTest.(rootChar.low))
sign(toTest,rootChar) ==
degree(toTest) = 0 or degree(rootChar.defPol) <= degree(toTest) =>
error "IRC: sign: Not reduced"
degree(rootChar.defPol) = 1 =>
error "IRC: sign: Linear Defining Polynomial"
degree(toTest) = 1 =>
linearSign(toTest, rootChar)
s := sign(leadingCoefficient(toTest))
toTest := monomial(1,degree(toTest))+
inv(leadingCoefficient(toTest))*reductum(toTest)
delta := gcd(toTest,rootChar.defPol)
newChar := [rootChar.low,rootChar.high,rootChar.defPol]$Rep
if positive? degree(delta)
then
if sign(delta.(rootChar.low) * delta.(rootChar.high)) <= 0
then
return(0)
else
newChar.defPol := (newChar.defPol exquo delta) :: P
toTest := toTest rem (newChar.defPol)
degree(toTest) = 0 => s * sign(leadingCoefficient(toTest))
degree(toTest) = 1 => s * linearSign(toTest, newChar)
while mightHaveRoots(toTest,newChar) repeat
newChar := refine(newChar)
s*sign(toTest.(newChar.low))
linearZero?(c,rootChar) ==
zero?((rootChar.defPol).c) and
(c - rootChar.low) * (c - rootChar.high) <= 0
zero?(toTest,rootChar) ==
degree(toTest) = 0 or degree(rootChar.defPol) <= degree(toTest) =>
error "IRC: zero?: Not reduced"
degree(rootChar.defPol) = 1 =>
error "IRC: zero?: Linear Defining Polynomial"
degree(toTest) = 1 =>
linearZero?(- inv(leadingCoefficient(toTest)) * coefficient(toTest,0),
rootChar)
toTest := monomial(1,degree(toTest))+
inv(leadingCoefficient(toTest))*reductum(toTest)
delta := gcd(toTest,rootChar.defPol)
degree(delta) = 0 => false
sign(delta.(rootChar.low) * delta.(rootChar.high)) <= 0
refine!(rootChar) ==
-- this is not a safe function, it can work with badly created object
-- we do not assume (rootChar.defPol).(rootChar.high) <> 0
int := middle(rootChar)
s1 := sign((rootChar.defPol).(rootChar.low))
zero?(s1) =>
rootChar.high := int
rootChar.defPol := monomial(1,1) - (rootChar.low)::P
rootChar
s2 := sign((rootChar.defPol).int)
zero?(s2) =>
rootChar.low := int
rootChar.defPol := monomial(1,1) - int::P
rootChar
if negative?(s1*s2)
then
rootChar.high := int
else
rootChar.low := int
rootChar
refine(rootChar) ==
-- we assume (rootChar.defPol).(rootChar.high) <> 0
int := middle(rootChar)
s:= (rootChar.defPol).int * (rootChar.defPol).(rootChar.high)
zero?(s) => [int,rootChar.high,monomial(1,1)-int::P]
if negative? s
then
[int,rootChar.high,rootChar.defPol]
else
[rootChar.low,int,rootChar.defPol]
left(rootChar) == rootChar.low
right(rootChar) == rootChar.high
middle(rootChar) == (rootChar.low + rootChar.high)/(2::TheField)
-- varStar(p) == -- if 0 no roots in [0,:infty[
-- res : N := 0
-- lsg := sign(coefficient(p,0))
-- l := [ sign(i) for i in reverse!(coefficients(p))]
-- for sg in l repeat
-- if (sg ~= lsg) then res := res + 1
-- lsg := sg
-- res
)abbrev domain RECLOS RealClosure
++ Author: Renaud Rioboo
++ Date Created: summer 1988
++ Date Last Updated: January 2004
++ Basic Functions: provides computations in an ordered real closure
++ Related Constructors: RightOpenIntervalRootCharacterization
++ Also See:
++ AMS Classifications:
++ Keywords: Real Algebraic Numbers
++ References:
++ Description:
++ This domain implements the real closure of an ordered field.
++ Note:
++ The code here is generic i.e. it does not depend of the way the operations
++ are done. The two macros PME and SEG should be passed as functorial
++ arguments to the domain. It does not help much to write a category
++ since non trivial methods cannot be placed there either.
++
RealClosure(TheField): PUB == PRIV where
TheField : Join(OrderedRing, Field, RealConstant)
-- ThePols : UnivariatePolynomialCategory($)
-- PME ==> ThePols
-- TheCharDom : RealRootCharacterizationCategory($, ThePols )
-- SEG ==> TheCharDom
-- this does not work yet
E ==> OutputForm
Z ==> Integer
SE ==> Symbol
B ==> Boolean
SUP ==> SparseUnivariatePolynomial($)
N ==> PositiveInteger
RN ==> Fraction Z
LF ==> ListFunctions2($,N)
-- *****************************************************************
-- *****************************************************************
-- PUT YOUR OWN PREFERENCE HERE
-- *****************************************************************
-- *****************************************************************
PME ==> SparseUnivariatePolynomial($)
SEG ==> RightOpenIntervalRootCharacterization($,PME)
-- *****************************************************************
-- *****************************************************************
PUB == Join(RealClosedField,
FullyRetractableTo TheField,
Algebra TheField) with
algebraicOf : (SEG,E) -> $
++ \axiom{algebraicOf(char)} is the external number
mainCharacterization : $ -> Union(SEG,"failed")
++ \axiom{mainCharacterization(x)} is the main algebraic
++ quantity of \axiom{x} (\axiom{SEG})
relativeApprox : ($,$) -> RN
++ \axiom{relativeApprox(n,p)} gives a relative
++ approximation of \axiom{n}
++ that has precision \axiom{p}
PRIV == add
-- local functions
lessAlgebraic : $ -> $
newElementIfneeded : (SEG,E) -> $
-- Representation
Rec := Record(seg: SEG, val:PME, outForm:E, order:N)
Rep := Union(TheField,Rec)
-- global (mutable) variables
orderOfCreation : N := 1$N
-- it is internally used to sort the algebraic levels
instanceName : Symbol := new()$Symbol
-- this used to print the results, thus different instanciations
-- use different names
-- now the code
relativeApprox(nbe,prec) ==
nbe case TheField => retract(nbe)
appr := relativeApprox(nbe.val, nbe.seg, prec)
-- now appr has the good exact precision but is $
relativeApprox(appr,prec)
approximate(nbe,prec) ==
abs(nbe) < prec => 0
nbe case TheField => retract(nbe)
appr := approximate(nbe.val, nbe.seg, prec)
-- now appr has the good exact precision but is $
approximate(appr,prec)
newElementIfneeded(s,o) ==
p := definingPolynomial(s)
degree(p) = 1 =>
- coefficient(p,0) / leadingCoefficient(p)
res := [s, monomial(1,1), o, orderOfCreation ]$Rec
orderOfCreation := orderOfCreation + 1
res :: $
algebraicOf(s,o) ==
pol := definingPolynomial(s)
degree(pol) = 1 =>
-coefficient(pol,0) / leadingCoefficient(pol)
res := [s, monomial(1,1), o, orderOfCreation ]$Rec
orderOfCreation := orderOfCreation + 1
res :: $
rename!(x,o) ==
x.outForm := o
x
rename(x,o) ==
[x.seg, x.val, o, x.order]$Rec
rootOf(pol,n) ==
degree(pol) = 0 => "failed"
degree(pol) = 1 =>
if n=1
then
-coefficient(pol,0) / leadingCoefficient(pol)
else
"failed"
r := rootOf(pol,n)$SEG
r case "failed" => "failed"
o := hconcat(instanceName :: E , orderOfCreation :: E)$E
algebraicOf(r,o)
allRootsOf(pol:SUP):List($) ==
degree(pol)=0 => []
degree(pol)=1 => [-coefficient(pol,0) / leadingCoefficient(pol)]
liste := allRootsOf(pol)$SEG
res : List $ := []
for term in liste repeat
o := hconcat(instanceName :: E , orderOfCreation :: E)$E
res := cons(algebraicOf(term,o), res)
reverse! res
coerce(x:$):$ ==
x case TheField => x
[x.seg,x.val rem$PME definingPolynomial(x.seg),x.outForm,x.order]$Rec
positive?(x) ==
x case TheField => positive?(x)$TheField
positive?(x.val,x.seg)$SEG
negative?(x) ==
x case TheField => negative?(x)$TheField
negative?(x.val,x.seg)$SEG
abs(x) == sign(x)*x
sign(x) ==
x case TheField => sign(x)$TheField
sign(x.val,x.seg)$SEG
x < y == positive?(y-x)
x = y == zero?(x-y)
mainCharacterization(x) ==
x case TheField => "failed"
x.seg
mainDefiningPolynomial(x) ==
x case TheField => "failed"
definingPolynomial x.seg
mainForm(x) ==
x case TheField => "failed"
x.outForm
mainValue(x) ==
x case TheField => "failed"
x.val
coerce(x:$):E ==
x case TheField => x::TheField :: E
xx:$ := coerce(x)
outputForm(univariate(xx.val),x.outForm)$SUP
inv(x) ==
(res:= recip x) case "failed" => error "Division by 0"
res :: $
recip(x) ==
x case TheField =>
if ((r := recip(x)$TheField) case TheField)
then r::$
else "failed"
if ((r := recip(x.val,x.seg)$SEG) case "failed")
then "failed"
else lessAlgebraic([x.seg,r::PME,x.outForm,x.order]$Rec)
(n:Z * x:$):$ ==
x case TheField => n *$TheField x
zero?(n) => 0
one?(n) => x
[x.seg,map(n * #1, x.val),x.outForm,x.order]$Rec
(rn:TheField * x:$):$ ==
x case TheField => rn *$TheField x
zero?(rn) => 0
one?(rn) => x
[x.seg,map(rn * #1, x.val),x.outForm,x.order]$Rec
(x:$ * y:$):$ ==
(x case TheField) and (y case TheField) => x *$TheField y
(x case TheField) => x::TheField * y
-- x is no longer TheField
(y case TheField) => y::TheField * x
-- now both are algebraic
y.order > x.order =>
[y.seg,map(x * #1 , y.val),y.outForm,y.order]$Rec
x.order > y.order =>
[x.seg,map( #1 * y , x.val),x.outForm,x.order]$Rec
-- now x.exp = y.exp
-- we will multiply the polynomials and then reduce
-- however wee need to call lessAlgebraic
lessAlgebraic([x.seg,
(x.val * y.val) rem definingPolynomial(x.seg),
x.outForm,
x.order]$Rec)
nonNull(r:Rec):$ ==
degree(r.val)=0 => leadingCoefficient(r.val)
numberOfMonomials(r.val) = 1 => r
zero?(r.val,r.seg)$SEG => 0
r
-- zero?(x) ==
-- x case TheField => zero?(x)$TheField
-- zero?(x.val,x.seg)$SEG
zero?(x) ==
x case TheField => zero?(x)$TheField
false
x + y ==
(x case TheField) and (y case TheField) => x +$TheField y
(x case TheField) =>
if zero?(x)
then
y
else
nonNull([y.seg,x::PME+(y.val),y.outForm,y.order]$Rec)
-- x is no longer TheField
(y case TheField) =>
if zero?(y)
then
x
else
nonNull([x.seg,(x.val)+y::PME,x.outForm,x.order]$Rec)
-- now both are algebraic
y.order > x.order =>
nonNull([y.seg,x::PME+y.val,y.outForm,y.order]$Rec)
x.order > y.order =>
nonNull([x.seg,(x.val)+y::PME,x.outForm,x.order]$Rec)
-- now x.exp = y.exp
-- we simply add polynomials (since degree cannot increase)
-- however wee need to call lessAlgebraic
nonNull([x.seg,x.val + y.val,x.outForm,x.order])
-x ==
x case TheField => -$TheField (x::TheField)
[x.seg,-$PME x.val,x.outForm,x.order]$Rec
retractIfCan(x:$):Union(TheField,"failed") ==
x case TheField => x
o := x.order
res := lessAlgebraic x
res case TheField => res
o = res.order => "failed"
retractIfCan res
retract(x:$):TheField ==
x case TheField => x
o := x.order
res := lessAlgebraic x
res case TheField => res
o = res.order => error "Can't retract"
retract res
lessAlgebraic(x) ==
x case TheField => x
degree(x.val) = 0 => leadingCoefficient(x.val)
def := definingPolynomial(x.seg)
degree(def) = 1 =>
x.val.(- coefficient(def,0) / leadingCoefficient(def))
x
0 == (0$TheField) :: $
1 == (1$TheField) :: $
coerce(rn:TheField):$ == rn :: $
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