Section9.66

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## 9.66 RealClosure

The Real Closure 1.0 package provided by Renaud Rioboo (Renaud.Rioboo@lip6.fr) consists of different packages, categories and domains :

The package RealPolynomialUtilitiesPackage which needs a Field F and a UnivariatePolynomialCategory domain with coefficients in F. It computes some simple functions such as Sturm and Sylvester sequences (sturmSequencesturmSequenceRealPolynomialUtilitiesPackage, sylvesterSequencesylvesterSequenceRealPolynomialUtilitiesPackage).
The category RealRootCharacterizationCategory provides abstract functions to work with ``real roots'' of univariate polynomials. These resemble variables with some functionality needed to compute important operations.
The category RealClosedField provides common operations available over real closed fiels. These include finding all the roots of a univariate polynomial, taking square (and higher) roots, ...
The domain RightOpenIntervalRootCharacterization is the main code that provides the functionality of RealRootCharacterizationCategory for the case of archimedean fields. Abstract roots are encoded with a left closed right open interval containing the root together with a defining polynomial for the root.
The RealClosure domain is the end-user code. It provides usual arithmetic with real algebraic numbers, along with the functionality of a real closed field. It also provides functions to approximate a real algebraic number by an element of the base field. This approximation may either be absolute (approximateapproximateRealClosure) or relative (relativeApproxrelativeApproxRealClosure).
CAVEATS

Since real algebraic expressions are stored as depending on ``real roots'' which are managed like variables, there is an ordering on these. This ordering is dynamical in the sense that any new algebraic takes precedence over older ones. In particular every creation function raises a new ``real root''. This has the effect that when you type something like sqrt(2) + sqrt(2) you have two new variables which happen to be equal. To avoid this name the expression such as in s2 := sqrt(2) ; s2 + s2

Also note that computing times depend strongly on the ordering you implicitly provide. Please provide algebraics in the order which seems most natural to you.

LIMITATIONS

These packages use algorithms which are published in [1] and [2] which are based on field arithmetics, in particular for polynomial gcd related algorithms. This can be quite slow for high degree polynomials and subresultants methods usually work best. Beta versions of the package try to use these techniques in a better way and work significantly faster. These are mostly based on unpublished algorithms and cannot be distributed. Please contact the author if you have a particular problem to solve or want to use these versions.

Be aware that approximations behave as post-processing and that all computations are done exactly. They can thus be quite time consuming when depending on several ``real roots''.

REFERENCES

[1] R. Rioboo : Real Algebraic Closure of an ordered Field : Implementation in Axiom. In proceedings of the ISSAC'92 Conference, Berkeley 1992 pp. 206-215.

[2] Z. Ligatsikas, R. Rioboo, M. F. Roy : Generic computation of the real closure of an ordered field. In Mathematics and Computers in Simulation Volume 42, Issue 4-6, November 1996.

EXAMPLES

We shall work with the real closure of the ordered field of rational numbers.

Ran := RECLOS(FRAC INT)

RealClosureFractionInteger

Type: Domain

Some simple signs for square roots, these correspond to an extension of degree 16 of the rational numbers. Examples provided by J. Abbot.

fourSquares(a:Ran,b:Ran,c:Ran,d:Ran):Ran == sqrt(a)+sqrt(b) - sqrt(c)-sqrt(d)
Function declaration fourSquares : (RealClosure Fraction Integer,
RealClosure Fraction Integer,RealClosure Fraction Integer,
RealClosure Fraction Integer) -> RealClosure Fraction Integer has
been added to workspace.
Void

These produce values very close to zero.

squareDiff1 := fourSquares(73,548,60,586)

-586-60+548+73

Type: RealClosure Fraction Integer
recip(squareDiff1)

((54602 548+149602 73)60+49502 73 548+9900895)586+(154702 73 548+30941947)60+10238421 548+28051871 73

Type: Union(RealClosure Fraction Integer,...)
sign(squareDiff1)

1

Type: PositiveInteger
squareDiff2 := fourSquares(165,778,86,990)

-990-86+778+165

Type: RealClosure Fraction Integer
recip(squareDiff2)

((556778 778+1209010 165)86+401966 165 778+144019431)990+(1363822 165 778+488640503)86+162460913 778+352774119 165

Type: Union(RealClosure Fraction Integer,...)
sign(squareDiff2)

1

Type: PositiveInteger
squareDiff3 := fourSquares(217,708,226,692)

-692-226+708+217

Type: RealClosure Fraction Integer
recip(squareDiff3)

((-34102 708-61598 217)226-34802 217 708-13641141)692+(-60898 217 708-23869841)226-13486123 708-24359809 217

Type: Union(RealClosure Fraction Integer,...)
sign(squareDiff3)

-1

Type: Integer
squareDiff4 := fourSquares(155,836,162,820)

-820-162+836+155

Type: RealClosure Fraction Integer
recip(squareDiff4)

((-37078 836-86110 155)162-37906 155 836-13645107)820+(-85282 155 836-30699151)162-13513901 836-31384703 155

Type: Union(RealClosure Fraction Integer,...)
sign(squareDiff4)

-1

Type: Integer
squareDiff5 := fourSquares(591,772,552,818)

-818-552+772+591

Type: RealClosure Fraction Integer
recip(squareDiff5)

((70922 772+81058 591)552+68542 591 772+46297673)818+(83438 591 772+56359389)552+47657051 772+54468081 591

Type: Union(RealClosure Fraction Integer,...)
sign(squareDiff5)

1

Type: PositiveInteger
squareDiff6 := fourSquares(434,1053,412,1088)

-1088-412+1053+434

Type: RealClosure Fraction Integer
recip(squareDiff6)

((115442 1053+179818 434)412+112478 434 1053+76037291)1088+(182782 434 1053+123564147)412+77290639 1053+120391609 434

Type: Union(RealClosure Fraction Integer,...)
sign(squareDiff6)

1

Type: PositiveInteger
squareDiff7 := fourSquares(514,1049,446,1152)

-1152-446+1049+514

Type: RealClosure Fraction Integer
recip(squareDiff7)

((349522 1049+499322 514)446+325582 514 1049+239072537)1152+(523262 514 1049+384227549)446+250534873 1049+357910443 514

Type: Union(RealClosure Fraction Integer,...)
sign(squareDiff7)

1

Type: PositiveInteger
squareDiff8 := fourSquares(190,1751,208,1698)

-1698-208+1751+190

Type: RealClosure Fraction Integer
recip(squareDiff8)

((-214702 1751-651782 190)208-224642 190 1751-129571901)1698+(-641842 190 1751-370209881)208-127595865 1751-387349387 190

Type: Union(RealClosure Fraction Integer,...)
sign(squareDiff8)

-1

Type: Integer

This should give three digits of precision

relativeApprox(squareDiff8,10**(-3))::Float

-0. 2340527771 5937700123 E-10

Type: Float

The sum of these 4 roots is 0

l := allRootsOf((x**2-2)**2-2)$Ran

[%A 33,%A 34,%A 35,%A 36]

Type: List RealClosure Fraction Integer

Check that they are all roots of the same polynomial

removeDuplicates map(mainDefiningPolynomial,l)

[?4-4? 2+2]

Type: List Union(SparseUnivariatePolynomial RealClosure Fraction Integer,"failed")

We can see at a glance that they are separate roots

map(mainCharacterization,l)

[[-2,-1[,[-1, 0[,[0, 1[,[1, 2[]

Type: List Union( RightOpenIntervalRootCharacterization( RealClosure Fraction Integer, SparseUnivariatePolynomial RealClosure Fraction Integer), "failed")

Check the sum and product

[reduce(+,l),reduce(*,l)-2]

[0, 0]

Type: List RealClosure Fraction Integer

A more complicated test that involve an extension of degree 256. This is a way of checking nested radical identities.

(s2, s5, s10) := (sqrt(2)$Ran, sqrt(5)$Ran, sqrt(10)$Ran)

10

Type: RealClosure Fraction Integer
eq1:=sqrt(s10+3)*sqrt(s5+2) - sqrt(s10-3)*sqrt(s5-2) = sqrt(10*s2+10)

-10-3 5-2+10+3 5+2=10 2+10

Type: Equation RealClosure Fraction Integer
eq1::Boolean

true

Type: Boolean
eq2:=sqrt(s5+2)*sqrt(s2+1) - sqrt(s5-2)*sqrt(s2-1) = sqrt(2*s10+2)

-5-2 2-1+5+2 2+1=2 10+2

Type: Equation RealClosure Fraction Integer
eq2::Boolean

true

Type: Boolean

Some more examples from J. M. Arnaudies

s3 := sqrt(3)$Ran

3

Type: RealClosure Fraction Integer
s7:= sqrt(7)$Ran

7

Type: RealClosure Fraction Integer
e1 := sqrt(2*s7-3*s3,3)

2 7-3 3 3

Type: RealClosure Fraction Integer
e2 := sqrt(2*s7+3*s3,3)

2 7+3 3 3

Type: RealClosure Fraction Integer

This should be null

e2-e1-s3

0

Type: RealClosure Fraction Integer

A quartic polynomial

pol : UP(x,Ran) := x**4+(7/3)*x**2+30*x-(100/3)

x 4+7 3 x 2+30 x-100 3

Type: UnivariatePolynomial(x,RealClosure Fraction Integer)

Add some cubic roots

r1 := sqrt(7633)$Ran

7633

Type: RealClosure Fraction Integer
alpha := sqrt(5*r1-436,3)/3

1 3 5 7633-436 3

Type: RealClosure Fraction Integer
beta := -sqrt(5*r1+436,3)/3

-1 3 5 7633+436 3

Type: RealClosure Fraction Integer

this should be null

pol.(alpha+beta-1/3)

0

Type: RealClosure Fraction Integer

A quintic polynomial

qol : UP(x,Ran) := x**5+10*x**3+20*x+22

x 5+10 x 3+20 x+22

Type: UnivariatePolynomial(x,RealClosure Fraction Integer)

Add some cubic roots

r2 := sqrt(153)$Ran

153

Type: RealClosure Fraction Integer
alpha2 := sqrt(r2-11,5)

153-11 5

Type: RealClosure Fraction Integer
beta2 := -sqrt(r2+11,5)

-153+11 5

Type: RealClosure Fraction Integer

this should be null

qol(alpha2+beta2)

0

Type: RealClosure Fraction Integer

Finally, some examples from the book Computer Algebra by Davenport, Siret and Tournier (page 77). The last one is due to Ramanujan.

dst1:=sqrt(9+4*s2)=1+2*s2

4 2+9=2 2+1

Type: Equation RealClosure Fraction Integer
dst1::Boolean

true

Type: Boolean
s6:Ran:=sqrt 6

6

Type: RealClosure Fraction Integer
dst2:=sqrt(5+2*s6)+sqrt(5-2*s6) = 2*s3

-2 6+5+2 6+5=2 3

Type: Equation RealClosure Fraction Integer
dst2::Boolean

true

Type: Boolean
s29:Ran:=sqrt 29

29

Type: RealClosure Fraction Integer
dst4:=sqrt(16-2*s29+2*sqrt(55-10*s29)) = sqrt(22+2*s5)-sqrt(11+2*s29)+s5

2-10 29+55-2 29+16=-2 29+11+2 5+22+5

Type: Equation RealClosure Fraction Integer
dst4::Boolean

true

Type: Boolean
dst6:=sqrt((112+70*s2)+(46+34*s2)*s5) = (5+4*s2)+(3+s2)*s5

(34 2+46)5+70 2+112=(2+3)5+4 2+5

Type: Equation RealClosure Fraction Integer
dst6::Boolean

true

Type: Boolean
f3:Ran:=sqrt(3,5)

3 5

Type: RealClosure Fraction Integer
f25:Ran:=sqrt(1/25,5)

1 25 5

Type: RealClosure Fraction Integer
f32:Ran:=sqrt(32/5,5)

32 5 5

Type: RealClosure Fraction Integer
f27:Ran:=sqrt(27/5,5)

27 5 5

Type: RealClosure Fraction Integer
dst5:=sqrt((f32-f27,3)) = f25*(1+f3-f3**2)

-27 5 5+32 5 5 3=(-3 5 2+3 5+1)1 25 5

Type: Equation RealClosure Fraction Integer
dst5::Boolean

true

Type: Boolean
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