Next: Conclusions Up: Foundations of Computer Algebra: Introduction Previous: Challenges and Achievements
For more information about AXIOM please visit:
AXIOM SESSIONscale=2.5]
Starts dribbling to session1.out (Mon Sep 15 12:01:24 2003)
--------------------------------------------
-- There are different types of integers --
--------------------------------------------
123
(1) 123
Type: PositiveInteger
Time: 0 sec
0
(2) 0
Type: NonNegativeInteger
Time: 0 sec
-45
(3) - 45
Type: Integer
Time: 0 sec
------------------------------------------
-- There is a type inference mechanism --
------------------------------------------
123 + (-45)
(4) 78
Type: PositiveInteger
Time: 0 sec
123 + (-456)
(5) - 333
Type: Integer
Time: 0 sec
355/113
355
(6) ---
113
Type: Fraction Integer
Time: 0 sec
355/113.0
(7) 3.1415929203 539823009
Type: Float
Time: 0.01 (IN) + 0.01 (OT) + 0.01 (GC) = 0.03 sec
-------------------------------
-- Coercions are also needed --
-------------------------------
(355/113) :: Float
(8) 3.1415929203 539823009
Type: Float
Time: 0 sec
(355/113) :: DoubleFloat
(9) 3.1415929203539825
Type: DoubleFloat
Time: 0 sec
-- Type inference again
n: Integer := 12
(10) 12
Type: Integer
Time: 0 sec
a: Integer := 3
(11) 3
Type: Integer
Time: 0 sec
a^n
(12) 531441
Type: Fraction Integer
Time: 0 sec
n := -n
(13) - 12
Type: Integer
Time: 0.01 (OT) = 0.01 sec
a^n
1
(14) ------
531441
Type: Fraction Integer
Time: 0 sec
inv(a^n)
(15) 531441
Type: Fraction Integer
Time: 0 sec
f := n * 1.0
(16) - 12.0
Type: Float
Time: 0.02 (OT) = 0.02 sec
a^f
(17) 0.0000018816 7642315892 07457
Type: Float
Time: 0.01 (IN) + 0.01 (OT) = 0.02 sec
-------------------------------------------
-- Types have an influence on the result --
-------------------------------------------
gcd (123,45)
(18) 3
Type: PositiveInteger
Time: 0 sec
gcd (123,45/1)
(19) 1
Type: Fraction Integer
Time: 0.03 (IN) + 0.01 (EV) + 0.01 (OT) = 0.05 sec
x
(20) x
Type: Variable x
Time: 0 sec
x + 1
(21) x + 1
Type: Polynomial Integer
Time: 0.01 (IN) + 0.01 (GC) = 0.02 sec
x + 1/2
1
(22) x + -
2
Type: Polynomial Fraction Integer
Time: 0.04 (IN) + 0.01 (OT) = 0.05 sec
p := (2*x + 2) * (x + 2)
2
(23) 2x + 6x + 4
Type: Polynomial Integer
Time: 0.01 (IN) = 0.01 sec
q := (x + 1) * (x + 3) * 2
2
(24) 2x + 8x + 6
Type: Polynomial Integer
Time: 0.01 (OT) = 0.01 sec
gcd(p,q)
(25) 2x + 2
Type: Polynomial Integer
Time: 0.01 (OT) + 0.03 (GC) = 0.04 sec
q := (x + 1) * (x + 3) * 2/1
2
(26) 2x + 8x + 6
Type: Polynomial Fraction Integer
Time: 0.01 (IN) + 0.01 (EV) + 0.01 (OT) + 0.01 (GC) = 0.04 sec
gcd (p,q)
(27) x + 1
Type: Polynomial Fraction Integer
Time: 0.02 (IN) + 0.01 (EV) = 0.03 sec
----------------------
-- Specifying types --
----------------------
Z := Integer
(28) Integer
Type: Domain
Time: 0 sec
a: Z := 12
(29) 12
Type: Integer
Time: 0 sec
b: Z := 45
(30) 45
Type: Integer
Time: 0 sec
gcd(a,b)
(31) 3
Type: PositiveInteger
Time: 0 sec
Q := Fraction Integer
(32) Fraction Integer
Type: Domain
Time: 0 sec
r: Q := a
(33) 12
Type: Fraction Integer
Time: 0 sec
s: Q := b
(34) 45
Type: Fraction Integer
Time: 0 sec
gcd(r,s)
(35) 1
Type: Fraction Integer
Time: 0 sec
gcd(12,45)$Q
(36) 1
Type: Fraction Integer
Time: 0 sec
gcd(12,45)$Z
(37) 3
Type: Integer
Time: 0 sec
a := inv(a)
Cannot convert right-hand side of assignment
1
--
12
to an object of the type Integer of the left-hand side.
--------------------
-- Browsing types --
--------------------
)sh Z
Integer is a domain constructor.
Abbreviation for Integer is INT
This constructor is exposed in this frame.
Issue )edit /usr/local/share/axiom/axiom2.3//mnt/linuxglibc2.1//../../src/algebra/integer.spad to see algebra source code for INT
------------------------------- Operations --------------------------------
?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> %
?*? : (%,%) -> % ?**? : (%,PositiveInteger) -> %
?+? : (%,%) -> % ?-? : (%,%) -> %
-? : % -> % ?<? : (%,%) -> Boolean
?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean
?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean
D : (%,NonNegativeInteger) -> % D : % -> %
OMwrite : (%,Boolean) -> String OMwrite : % -> String
1 : () -> % 0 : () -> %
?^? : (%,PositiveInteger) -> % abs : % -> %
addmod : (%,%,%) -> % associates? : (%,%) -> Boolean
base : () -> % binomial : (%,%) -> %
bit? : (%,%) -> Boolean coerce : % -> OutputForm
coerce : Integer -> % coerce : % -> %
convert : % -> DoubleFloat convert : % -> Float
convert : % -> InputForm convert : % -> Integer
convert : % -> Pattern Integer convert : % -> String
copy : % -> % dec : % -> %
differentiate : % -> % even? : % -> Boolean
factor : % -> Factored % factorial : % -> %
gcd : List % -> % gcd : (%,%) -> %
hash : % -> SingleInteger hash : % -> %
inc : % -> % init : () -> %
invmod : (%,%) -> % latex : % -> String
lcm : List % -> % lcm : (%,%) -> %
length : % -> % mask : % -> %
max : (%,%) -> % min : (%,%) -> %
mulmod : (%,%,%) -> % negative? : % -> Boolean
odd? : % -> Boolean one? : % -> Boolean
permutation : (%,%) -> % positive? : % -> Boolean
positiveRemainder : (%,%) -> % powmod : (%,%,%) -> %
prime? : % -> Boolean ?quo? : (%,%) -> %
random : % -> % random : () -> %
rational : % -> Fraction Integer rational? : % -> Boolean
recip : % -> Union(%,"failed") ?rem? : (%,%) -> %
retract : % -> Integer sample : () -> %
shift : (%,%) -> % sign : % -> Integer
sizeLess? : (%,%) -> Boolean squareFree : % -> Factored %
squareFreePart : % -> % submod : (%,%,%) -> %
symmetricRemainder : (%,%) -> % unit? : % -> Boolean
unitCanonical : % -> % zero? : % -> Boolean
?~=? : (%,%) -> Boolean
?*? : (NonNegativeInteger,%) -> %
?**? : (%,NonNegativeInteger) -> %
OMwrite : (OpenMathDevice,%,Boolean) -> Void
OMwrite : (OpenMathDevice,%) -> Void
?^? : (%,NonNegativeInteger) -> %
characteristic : () -> NonNegativeInteger
differentiate : (%,NonNegativeInteger) -> %
divide : (%,%) -> Record(quotient: %,remainder: %)
euclideanSize : % -> NonNegativeInteger
expressIdealMember : (List %,%) -> Union(List %,"failed")
exquo : (%,%) -> Union(%,"failed")
extendedEuclidean : (%,%) -> Record(coef1: %,coef2: %,generator: %)
extendedEuclidean : (%,%,%) -> Union(Record(coef1: %,coef2: %),"failed")
gcdPolynomial : (SparseUnivariatePolynomial %,SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %
multiEuclidean : (List %,%) -> Union(List %,"failed")
nextItem : % -> Union(%,"failed")
patternMatch : (%,Pattern Integer,PatternMatchResult(Integer,%)) -> PatternMatchResult(Integer,%)
principalIdeal : List % -> Record(coef: List %,generator: %)
rationalIfCan : % -> Union(Fraction Integer,"failed")
reducedSystem : Matrix % -> Matrix Integer
reducedSystem : (Matrix %,Vector %) -> Record(mat: Matrix Integer,vec: Vector Integer)
retractIfCan : % -> Union(Integer,"failed")
subtractIfCan : (%,%) -> Union(%,"failed")
unitNormal : % -> Record(unit: %,canonical: %,associate: %)
)sh Q
Fraction Integer is a domain constructor.
Abbreviation for Fraction is FRAC
This constructor is exposed in this frame.
Issue )edit /usr/local/share/axiom/axiom2.3//mnt/linuxglibc2.1//../../src/algebra/fraction.spad to see algebra source code for FRAC
------------------------------- Operations --------------------------------
?*? : (Fraction Integer,%) -> % ?*? : (Integer,%) -> %
?*? : (PositiveInteger,%) -> % ?*? : (%,Fraction Integer) -> %
?*? : (%,Integer) -> % ?*? : (%,%) -> %
?**? : (%,Integer) -> % ?**? : (%,PositiveInteger) -> %
?+? : (%,%) -> % ?-? : (%,%) -> %
-? : % -> % ?/? : (Integer,Integer) -> %
?/? : (%,%) -> % ?<? : (%,%) -> Boolean
?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean
?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean
D : (%,List Symbol) -> % D : (%,NonNegativeInteger) -> %
D : (%,Symbol) -> % D : % -> %
OMwrite : (%,Boolean) -> String OMwrite : % -> String
1 : () -> % 0 : () -> %
?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> %
abs : % -> % associates? : (%,%) -> Boolean
ceiling : % -> Integer coerce : % -> OutputForm
coerce : Fraction Integer -> % coerce : Integer -> %
coerce : Symbol -> % coerce : % -> %
convert : % -> DoubleFloat convert : % -> Float
convert : % -> InputForm convert : % -> Pattern Float
convert : % -> Pattern Integer denom : % -> Integer
denominator : % -> % differentiate : (%,Symbol) -> %
differentiate : % -> % ?.? : (%,Integer) -> %
eval : (%,Equation Integer) -> % eval : (%,Integer,Integer) -> %
eval : (%,Symbol,Integer) -> % factor : % -> Factored %
floor : % -> Integer fractionPart : % -> %
gcd : List % -> % gcd : (%,%) -> %
hash : % -> SingleInteger init : () -> %
inv : % -> % latex : % -> String
lcm : List % -> % lcm : (%,%) -> %
max : (%,%) -> % min : (%,%) -> %
negative? : % -> Boolean numer : % -> Integer
numerator : % -> % one? : % -> Boolean
positive? : % -> Boolean prime? : % -> Boolean
?quo? : (%,%) -> % random : () -> %
recip : % -> Union(%,"failed") ?rem? : (%,%) -> %
retract : % -> Fraction Integer retract : % -> Integer
retract : % -> Symbol sample : () -> %
sign : % -> Integer sizeLess? : (%,%) -> Boolean
squareFree : % -> Factored % squareFreePart : % -> %
unit? : % -> Boolean unitCanonical : % -> %
wholePart : % -> Integer zero? : % -> Boolean
?~=? : (%,%) -> Boolean
?*? : (NonNegativeInteger,%) -> %
?**? : (%,NonNegativeInteger) -> %
D : (%,List Symbol,List NonNegativeInteger) -> %
D : (%,(Integer -> Integer),NonNegativeInteger) -> %
D : (%,(Integer -> Integer)) -> %
D : (%,Symbol,NonNegativeInteger) -> %
OMwrite : (OpenMathDevice,%,Boolean) -> Void
OMwrite : (OpenMathDevice,%) -> Void
?^? : (%,NonNegativeInteger) -> %
characteristic : () -> NonNegativeInteger
charthRoot : % -> Union(%,"failed")
conditionP : Matrix % -> Union(Vector %,"failed")
differentiate : (%,List Symbol,List NonNegativeInteger) -> %
differentiate : (%,List Symbol) -> %
differentiate : (%,(Integer -> Integer),NonNegativeInteger) -> %
differentiate : (%,(Integer -> Integer)) -> %
differentiate : (%,NonNegativeInteger) -> %
differentiate : (%,Symbol,NonNegativeInteger) -> %
divide : (%,%) -> Record(quotient: %,remainder: %)
euclideanSize : % -> NonNegativeInteger
eval : (%,List Equation Integer) -> %
eval : (%,List Integer,List Integer) -> %
eval : (%,List Symbol,List Integer) -> %
expressIdealMember : (List %,%) -> Union(List %,"failed")
exquo : (%,%) -> Union(%,"failed")
extendedEuclidean : (%,%) -> Record(coef1: %,coef2: %,generator: %)
extendedEuclidean : (%,%,%) -> Union(Record(coef1: %,coef2: %),"failed")
factorPolynomial : SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial %
factorSquareFreePolynomial : SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial %
gcdPolynomial : (SparseUnivariatePolynomial %,SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %
map : ((Integer -> Integer),%) -> %
multiEuclidean : (List %,%) -> Union(List %,"failed")
nextItem : % -> Union(%,"failed")
patternMatch : (%,Pattern Float,PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%)
patternMatch : (%,Pattern Integer,PatternMatchResult(Integer,%)) -> PatternMatchResult(Integer,%)
principalIdeal : List % -> Record(coef: List %,generator: %)
reducedSystem : Matrix % -> Matrix Integer
reducedSystem : (Matrix %,Vector %) -> Record(mat: Matrix Integer,vec: Vector Integer)
retractIfCan : % -> Union(Fraction Integer,"failed")
retractIfCan : % -> Union(Integer,"failed")
retractIfCan : % -> Union(Symbol,"failed")
solveLinearPolynomialEquation : (List SparseUnivariatePolynomial %,SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %,"failed")
squareFreePolynomial : SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial %
subtractIfCan : (%,%) -> Union(%,"failed")
unitNormal : % -> Record(unit: %,canonical: %,associate: %)
Q has Field
(38) true
Type: Boolean
Time: 0 sec
Z has Field
(39) false
Type: Boolean
Time: 0 sec
)sh Field
Field is a category constructor
Abbreviation for Field is FIELD
This constructor is exposed in this frame.
Issue )edit /usr/local/share/axiom/axiom2.3//mnt/linuxglibc2.1//../../src/algebra/catdef.spad to see algebra source code for FIELD
------------------------------- Operations --------------------------------
?*? : (Fraction Integer,%) -> % ?*? : (%,Fraction Integer) -> %
?*? : (%,%) -> % ?*? : (Integer,%) -> %
?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> %
?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> %
?-? : (%,%) -> % -? : % -> %
?/? : (%,%) -> % ?=? : (%,%) -> Boolean
1 : () -> % 0 : () -> %
?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> %
associates? : (%,%) -> Boolean coerce : Fraction Integer -> %
coerce : % -> % coerce : Integer -> %
coerce : % -> OutputForm factor : % -> Factored %
gcd : List % -> % gcd : (%,%) -> %
hash : % -> SingleInteger inv : % -> %
latex : % -> String lcm : List % -> %
lcm : (%,%) -> % one? : % -> Boolean
prime? : % -> Boolean ?quo? : (%,%) -> %
recip : % -> Union(%,"failed") ?rem? : (%,%) -> %
sample : () -> % sizeLess? : (%,%) -> Boolean
squareFree : % -> Factored % squareFreePart : % -> %
unit? : % -> Boolean unitCanonical : % -> %
zero? : % -> Boolean ?~=? : (%,%) -> Boolean
?*? : (NonNegativeInteger,%) -> %
?**? : (%,NonNegativeInteger) -> %
?^? : (%,NonNegativeInteger) -> %
characteristic : () -> NonNegativeInteger
divide : (%,%) -> Record(quotient: %,remainder: %)
euclideanSize : % -> NonNegativeInteger
expressIdealMember : (List %,%) -> Union(List %,"failed")
exquo : (%,%) -> Union(%,"failed")
extendedEuclidean : (%,%,%) -> Union(Record(coef1: %,coef2: %),"failed")
extendedEuclidean : (%,%) -> Record(coef1: %,coef2: %,generator: %)
gcdPolynomial : (SparseUnivariatePolynomial %,SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %
multiEuclidean : (List %,%) -> Union(List %,"failed")
principalIdeal : List % -> Record(coef: List %,generator: %)
subtractIfCan : (%,%) -> Union(%,"failed")
unitNormal : % -> Record(unit: %,canonical: %,associate: %)
Z has Ring
(40) true
Type: Boolean
Time: 0 sec
)sh Ring
Ring is a category constructor
Abbreviation for Ring is RING
This constructor is exposed in this frame.
Issue )edit /usr/local/share/axiom/axiom2.3//mnt/linuxglibc2.1//../../src/algebra/catdef.spad to see algebra source code for RING
------------------------------- Operations --------------------------------
?*? : (%,%) -> % ?*? : (Integer,%) -> %
?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> %
?+? : (%,%) -> % ?-? : (%,%) -> %
-? : % -> % ?=? : (%,%) -> Boolean
1 : () -> % 0 : () -> %
?^? : (%,PositiveInteger) -> % coerce : Integer -> %
coerce : % -> OutputForm hash : % -> SingleInteger
latex : % -> String one? : % -> Boolean
recip : % -> Union(%,"failed") sample : () -> %
zero? : % -> Boolean ?~=? : (%,%) -> Boolean
?*? : (NonNegativeInteger,%) -> %
?**? : (%,NonNegativeInteger) -> %
?^? : (%,NonNegativeInteger) -> %
characteristic : () -> NonNegativeInteger
subtractIfCan : (%,%) -> Union(%,"failed")
Z has OrderedRing
(41) true
Type: Boolean
Time: 0 sec
)sh OrderedRing
OrderedRing is a category constructor
Abbreviation for OrderedRing is ORDRING
This constructor is exposed in this frame.
Issue )edit /usr/local/share/axiom/axiom2.3//mnt/linuxglibc2.1//../../src/algebra/catdef.spad to see algebra source code for ORDRING
------------------------------- Operations --------------------------------
?*? : (%,%) -> % ?*? : (Integer,%) -> %
?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> %
?+? : (%,%) -> % -? : % -> %
?-? : (%,%) -> % ?<? : (%,%) -> Boolean
?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean
?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean
1 : () -> % 0 : () -> %
?^? : (%,PositiveInteger) -> % abs : % -> %
coerce : Integer -> % coerce : % -> OutputForm
hash : % -> SingleInteger latex : % -> String
max : (%,%) -> % min : (%,%) -> %
negative? : % -> Boolean one? : % -> Boolean
positive? : % -> Boolean recip : % -> Union(%,"failed")
sample : () -> % sign : % -> Integer
zero? : % -> Boolean ?~=? : (%,%) -> Boolean
?*? : (NonNegativeInteger,%) -> %
?**? : (%,NonNegativeInteger) -> %
?^? : (%,NonNegativeInteger) -> %
characteristic : () -> NonNegativeInteger
subtractIfCan : (%,%) -> Union(%,"failed")
Q has OrderedRing
(42) true
Type: Boolean
Time: 0 sec
Z has GcdDomain
(43) true
Type: Boolean
Time: 0.01 (OT) = 0.01 sec
Q has GcdDomain
(44) true
Type: Boolean
Time: 0 sec
Z has EuclideanDomain
(45) true
Type: Boolean
Time: 0.01 (OT) = 0.01 sec
Q has EuclideanDomain
(46) true
Type: Boolean
Time: 0 sec
--------------------------
-- Programming in AXIOM --
--------------------------
f(u,v) == gcd(u,v)
Type: Void
Time: 0 sec
f(a,b)
Compiling function f with type (Integer,Integer) -> Integer
(48) 3
Type: PositiveInteger
Time: 0 sec
f(r,s)
Compiling function f with type (Fraction Integer,Fraction Integer)
-> Fraction Integer
(49) 1
Type: Fraction Integer
Time: 0 sec
g(u,v) == while u ~= v repeat if u < v then v := v - u else u := u - v
Type: Void
Time: 0 sec
h(u,v) ==
g(u,v)
u
Type: Void
Time: 0 sec
h(12,45)
Compiling function g with type (PositiveInteger,PositiveInteger) ->
Void
Compiling function h with type (PositiveInteger,PositiveInteger) ->
PositiveInteger
(52) 12
Type: PositiveInteger
Time: 0.03 (IN) = 0.03 sec
h(u,v) ==
while u ~= v repeat if u < v then v := v - u else u := u - v
u
Compiled code for h has been cleared.
1 old definition(s) deleted for function or rule h
Type: Void
Time: 0 sec
h(12,45)
Compiling function h with type (PositiveInteger,PositiveInteger) ->
Integer
+++ |*2;h;1;Session1| redefined
(54) 3
Type: PositiveInteger
Time: 0.01 (OT) = 0.01 sec
-------------------------------
-- Functions returning types --
-------------------------------
3/4
3
(55) -
4
Type: Fraction Integer
Time: 0 sec
(x + 1)/(x + 2)
x + 1
(56) -----
x + 2
Type: Fraction Polynomial Integer
Time: 0.01 (OT) = 0.01 sec
)sh Fraction
Fraction S: IntegralDomain is a domain constructor
Abbreviation for Fraction is FRAC
This constructor is exposed in this frame.
Issue )edit /usr/local/share/axiom/axiom2.3//mnt/linuxglibc2.1//../../src/algebra/fraction.spad to see algebra source code for FRAC
------------------------------- Operations --------------------------------
?*? : (%,S) -> % ?*? : (S,%) -> %
?*? : (Fraction Integer,%) -> % ?*? : (%,Fraction Integer) -> %
?*? : (%,%) -> % ?*? : (Integer,%) -> %
?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> %
?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> %
?-? : (%,%) -> % -? : % -> %
?/? : (S,S) -> % ?/? : (%,%) -> %
?=? : (%,%) -> Boolean D : (%,(S -> S)) -> %
D : % -> % if S has DIFRING 1 : () -> %
0 : () -> % ?^? : (%,Integer) -> %
?^? : (%,PositiveInteger) -> % associates? : (%,%) -> Boolean
ceiling : % -> S if S has INS coerce : S -> %
coerce : Fraction Integer -> % coerce : % -> %
coerce : Integer -> % coerce : % -> OutputForm
denom : % -> S denominator : % -> %
factor : % -> Factored % floor : % -> S if S has INS
gcd : List % -> % gcd : (%,%) -> %
hash : % -> SingleInteger init : () -> % if S has STEP
inv : % -> % latex : % -> String
lcm : List % -> % lcm : (%,%) -> %
map : ((S -> S),%) -> % numer : % -> S
numerator : % -> % one? : % -> Boolean
prime? : % -> Boolean ?quo? : (%,%) -> %
random : () -> % if S has INS recip : % -> Union(%,"failed")
?rem? : (%,%) -> % retract : % -> S
sample : () -> % sizeLess? : (%,%) -> Boolean
squareFree : % -> Factored % squareFreePart : % -> %
unit? : % -> Boolean unitCanonical : % -> %
zero? : % -> Boolean ?~=? : (%,%) -> Boolean
?*? : (NonNegativeInteger,%) -> %
?**? : (%,NonNegativeInteger) -> %
?<? : (%,%) -> Boolean if S has ORDSET
?<=? : (%,%) -> Boolean if S has ORDSET
?>? : (%,%) -> Boolean if S has ORDSET
?>=? : (%,%) -> Boolean if S has ORDSET
D : (%,(S -> S),NonNegativeInteger) -> %
D : (%,List Symbol,List NonNegativeInteger) -> % if S has PDRING SYMBOL
D : (%,Symbol,NonNegativeInteger) -> % if S has PDRING SYMBOL
D : (%,List Symbol) -> % if S has PDRING SYMBOL
D : (%,Symbol) -> % if S has PDRING SYMBOL
D : (%,NonNegativeInteger) -> % if S has DIFRING
OMwrite : (OpenMathDevice,%,Boolean) -> Void if S has INS and S has OM
OMwrite : (OpenMathDevice,%) -> Void if S has INS and S has OM
OMwrite : (%,Boolean) -> String if S has INS and S has OM
OMwrite : % -> String if S has INS and S has OM
?^? : (%,NonNegativeInteger) -> %
abs : % -> % if S has OINTDOM
characteristic : () -> NonNegativeInteger
charthRoot : % -> Union(%,"failed") if $ has CHARNZ and S has PFECAT or S has CHARNZ
coerce : Symbol -> % if S has RETRACT SYMBOL
conditionP : Matrix % -> Union(Vector %,"failed") if $ has CHARNZ and S has PFECAT
convert : % -> DoubleFloat if S has REAL
convert : % -> Float if S has REAL
convert : % -> InputForm if S has KONVERT INFORM
convert : % -> Pattern Float if S has KONVERT PATTERN FLOAT
convert : % -> Pattern Integer if S has KONVERT PATTERN INT
differentiate : (%,(S -> S)) -> %
differentiate : (%,(S -> S),NonNegativeInteger) -> %
differentiate : (%,List Symbol,List NonNegativeInteger) -> % if S has PDRING SYMBOL
differentiate : (%,Symbol,NonNegativeInteger) -> % if S has PDRING SYMBOL
differentiate : (%,List Symbol) -> % if S has PDRING SYMBOL
differentiate : (%,Symbol) -> % if S has PDRING SYMBOL
differentiate : (%,NonNegativeInteger) -> % if S has DIFRING
differentiate : % -> % if S has DIFRING
divide : (%,%) -> Record(quotient: %,remainder: %)
?.? : (%,S) -> % if S has ELTAB(S,S)
euclideanSize : % -> NonNegativeInteger
eval : (%,Symbol,S) -> % if S has IEVALAB(SYMBOL,S)
eval : (%,List Symbol,List S) -> % if S has IEVALAB(SYMBOL,S)
eval : (%,List Equation S) -> % if S has EVALAB S
eval : (%,Equation S) -> % if S has EVALAB S
eval : (%,S,S) -> % if S has EVALAB S
eval : (%,List S,List S) -> % if S has EVALAB S
expressIdealMember : (List %,%) -> Union(List %,"failed")
exquo : (%,%) -> Union(%,"failed")
extendedEuclidean : (%,%,%) -> Union(Record(coef1: %,coef2: %),"failed")
extendedEuclidean : (%,%) -> Record(coef1: %,coef2: %,generator: %)
factorPolynomial : SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if S has PFECAT
factorSquareFreePolynomial : SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if S has PFECAT
fractionPart : % -> % if S has EUCDOM
gcdPolynomial : (SparseUnivariatePolynomial %,SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %
max : (%,%) -> % if S has ORDSET
min : (%,%) -> % if S has ORDSET
multiEuclidean : (List %,%) -> Union(List %,"failed")
negative? : % -> Boolean if S has OINTDOM
nextItem : % -> Union(%,"failed") if S has STEP
patternMatch : (%,Pattern Float,PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) if S has PATMAB FLOAT
patternMatch : (%,Pattern Integer,PatternMatchResult(Integer,%)) -> PatternMatchResult(Integer,%) if S has PATMAB INT
positive? : % -> Boolean if S has OINTDOM
principalIdeal : List % -> Record(coef: List %,generator: %)
reducedSystem : Matrix % -> Matrix S
reducedSystem : (Matrix %,Vector %) -> Record(mat: Matrix S,vec: Vector S)
reducedSystem : (Matrix %,Vector %) -> Record(mat: Matrix Integer,vec: Vector Integer) if S has LINEXP INT
reducedSystem : Matrix % -> Matrix Integer if S has LINEXP INT
retract : % -> Integer if S has RETRACT INT
retract : % -> Fraction Integer if S has RETRACT INT
retract : % -> Symbol if S has RETRACT SYMBOL
retractIfCan : % -> Union(Integer,"failed") if S has RETRACT INT
retractIfCan : % -> Union(Fraction Integer,"failed") if S has RETRACT INT
retractIfCan : % -> Union(Symbol,"failed") if S has RETRACT SYMBOL
retractIfCan : % -> Union(S,"failed")
sign : % -> Integer if S has OINTDOM
solveLinearPolynomialEquation : (List SparseUnivariatePolynomial %,SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %,"failed") if S has PFECAT
squareFreePolynomial : SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if S has PFECAT
subtractIfCan : (%,%) -> Union(%,"failed")
unitNormal : % -> Record(unit: %,canonical: %,associate: %)
wholePart : % -> S if S has EUCDOM
)sh Polynomial
Polynomial R: Ring is a domain constructor
Abbreviation for Polynomial is POLY
This constructor is exposed in this frame.
Issue )edit /usr/local/share/axiom/axiom2.3//mnt/linuxglibc2.1//../../src/algebra/multpoly.spad to see algebra source code for POLY
------------------------------- Operations --------------------------------
?*? : (%,R) -> % ?*? : (R,%) -> %
?*? : (%,%) -> % ?*? : (Integer,%) -> %
?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> %
?+? : (%,%) -> % ?-? : (%,%) -> %
-? : % -> % ?=? : (%,%) -> Boolean
D : (%,List Symbol) -> % D : (%,Symbol) -> %
1 : () -> % 0 : () -> %
?^? : (%,PositiveInteger) -> % coefficients : % -> List R
coerce : Symbol -> % coerce : R -> %
coerce : Integer -> % coerce : % -> OutputForm
differentiate : (%,Symbol) -> % eval : (%,Symbol,%) -> %
eval : (%,Symbol,R) -> % eval : (%,List %,List %) -> %
eval : (%,%,%) -> % eval : (%,Equation %) -> %
eval : (%,List Equation %) -> % ground : % -> R
ground? : % -> Boolean hash : % -> SingleInteger
latex : % -> String leadingCoefficient : % -> R
leadingMonomial : % -> % map : ((R -> R),%) -> %
monomial? : % -> Boolean monomials : % -> List %
one? : % -> Boolean primitiveMonomials : % -> List %
recip : % -> Union(%,"failed") reductum : % -> %
retract : % -> Symbol retract : % -> R
sample : () -> % variables : % -> List Symbol
zero? : % -> Boolean ?~=? : (%,%) -> Boolean
?*? : (Fraction Integer,%) -> % if R has ALGEBRA FRAC INT
?*? : (%,Fraction Integer) -> % if R has ALGEBRA FRAC INT
?*? : (NonNegativeInteger,%) -> %
?**? : (%,NonNegativeInteger) -> %
?/? : (%,R) -> % if R has FIELD
?<? : (%,%) -> Boolean if R has ORDSET
?<=? : (%,%) -> Boolean if R has ORDSET
?>? : (%,%) -> Boolean if R has ORDSET
?>=? : (%,%) -> Boolean if R has ORDSET
D : (%,List Symbol,List NonNegativeInteger) -> %
D : (%,Symbol,NonNegativeInteger) -> %
?^? : (%,NonNegativeInteger) -> %
associates? : (%,%) -> Boolean if R has INTDOM
binomThmExpt : (%,%,NonNegativeInteger) -> % if R has COMRING
characteristic : () -> NonNegativeInteger
charthRoot : % -> Union(%,"failed") if $ has CHARNZ and R has PFECAT or R has CHARNZ
coefficient : (%,List Symbol,List NonNegativeInteger) -> %
coefficient : (%,Symbol,NonNegativeInteger) -> %
coefficient : (%,IndexedExponents Symbol) -> R
coerce : % -> % if R has INTDOM
coerce : Fraction Integer -> % if R has ALGEBRA FRAC INT or R has RETRACT FRAC INT
conditionP : Matrix % -> Union(Vector %,"failed") if $ has CHARNZ and R has PFECAT
content : (%,Symbol) -> % if R has GCDDOM
content : % -> R if R has GCDDOM
convert : % -> InputForm if Symbol has KONVERT INFORM and R has KONVERT INFORM
convert : % -> Pattern Integer if Symbol has KONVERT PATTERN INT and R has KONVERT PATTERN INT
convert : % -> Pattern Float if Symbol has KONVERT PATTERN FLOAT and R has KONVERT PATTERN FLOAT
degree : (%,List Symbol) -> List NonNegativeInteger
degree : (%,Symbol) -> NonNegativeInteger
degree : % -> IndexedExponents Symbol
differentiate : (%,List Symbol,List NonNegativeInteger) -> %
differentiate : (%,Symbol,NonNegativeInteger) -> %
differentiate : (%,List Symbol) -> %
discriminant : (%,Symbol) -> % if R has COMRING
eval : (%,List Symbol,List %) -> %
eval : (%,List Symbol,List R) -> %
exquo : (%,%) -> Union(%,"failed") if R has INTDOM
exquo : (%,R) -> Union(%,"failed") if R has INTDOM
factor : % -> Factored % if R has PFECAT
factorPolynomial : SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PFECAT
factorSquareFreePolynomial : SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PFECAT
gcd : (%,%) -> % if R has GCDDOM
gcd : List % -> % if R has GCDDOM
gcdPolynomial : (SparseUnivariatePolynomial %,SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial % if R has GCDDOM
integrate : (%,Symbol) -> % if R has ALGEBRA FRAC INT
isExpt : % -> Union(Record(var: Symbol,exponent: NonNegativeInteger),"failed")
isPlus : % -> Union(List %,"failed")
isTimes : % -> Union(List %,"failed")
lcm : (%,%) -> % if R has GCDDOM
lcm : List % -> % if R has GCDDOM
mainVariable : % -> Union(Symbol,"failed")
mapExponents : ((IndexedExponents Symbol -> IndexedExponents Symbol),%) -> %
max : (%,%) -> % if R has ORDSET
min : (%,%) -> % if R has ORDSET
minimumDegree : (%,List Symbol) -> List NonNegativeInteger
minimumDegree : (%,Symbol) -> NonNegativeInteger
minimumDegree : % -> IndexedExponents Symbol
monicDivide : (%,%,Symbol) -> Record(quotient: %,remainder: %)
monomial : (%,List Symbol,List NonNegativeInteger) -> %
monomial : (%,Symbol,NonNegativeInteger) -> %
monomial : (R,IndexedExponents Symbol) -> %
multivariate : (SparseUnivariatePolynomial %,Symbol) -> %
multivariate : (SparseUnivariatePolynomial R,Symbol) -> %
numberOfMonomials : % -> NonNegativeInteger
patternMatch : (%,Pattern Integer,PatternMatchResult(Integer,%)) -> PatternMatchResult(Integer,%) if Symbol has PATMAB INT and R has PATMAB INT
patternMatch : (%,Pattern Float,PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) if Symbol has PATMAB FLOAT and R has PATMAB FLOAT
pomopo! : (%,R,IndexedExponents Symbol,%) -> %
prime? : % -> Boolean if R has PFECAT
primitivePart : (%,Symbol) -> % if R has GCDDOM
primitivePart : % -> % if R has GCDDOM
reducedSystem : Matrix % -> Matrix R
reducedSystem : (Matrix %,Vector %) -> Record(mat: Matrix R,vec: Vector R)
reducedSystem : (Matrix %,Vector %) -> Record(mat: Matrix Integer,vec: Vector Integer) if R has LINEXP INT
reducedSystem : Matrix % -> Matrix Integer if R has LINEXP INT
resultant : (%,%,Symbol) -> % if R has COMRING
retract : % -> Integer if R has RETRACT INT
retract : % -> Fraction Integer if R has RETRACT FRAC INT
retractIfCan : % -> Union(Symbol,"failed")
retractIfCan : % -> Union(Integer,"failed") if R has RETRACT INT
retractIfCan : % -> Union(Fraction Integer,"failed") if R has RETRACT FRAC INT
retractIfCan : % -> Union(R,"failed")
solveLinearPolynomialEquation : (List SparseUnivariatePolynomial %,SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %,"failed") if R has PFECAT
squareFree : % -> Factored % if R has GCDDOM
squareFreePart : % -> % if R has GCDDOM
squareFreePolynomial : SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PFECAT
subtractIfCan : (%,%) -> Union(%,"failed")
totalDegree : (%,List Symbol) -> NonNegativeInteger
totalDegree : % -> NonNegativeInteger
unit? : % -> Boolean if R has INTDOM
unitCanonical : % -> % if R has INTDOM
unitNormal : % -> Record(unit: %,canonical: %,associate: %) if R has INTDOM
univariate : % -> SparseUnivariatePolynomial R
univariate : (%,Symbol) -> SparseUnivariatePolynomial %
P := Polynomial Q
(57) Polynomial Fraction Integer
Type: Domain
Time: 0 sec
P has EuclideanDomain
(58) false
Type: Boolean
Time: 0.01 (IN) = 0.01 sec
(x + 1)/(y + 2)
x + 1
(59) -----
y + 2
Type: Fraction Polynomial Integer
Time: 0.01 (IN) = 0.01 sec
U := UnivariatePolynomial(x,Q)
(60) UnivariatePolynomial(x,Fraction Integer)
Type: Domain
Time: 0 sec
U has EuclideanDomain
(61) true
Type: Boolean
Time: 0 sec
p1: P := 6 * (x + 1) * (x + 2)
2
(62) 6x + 18x + 12
Type: Polynomial Fraction Integer
Time: 0 sec
p2: P := 9 * (x + 1) * ( x + 3)
2
(63) 9x + 36x + 27
Type: Polynomial Fraction Integer
Time: 0.01 (IN) = 0.01 sec
gcd(p1,p2)
(64) x + 1
Type: Polynomial Fraction Integer
Time: 0.01 (EV) = 0.01 sec
u1: U := p1
2
(65) 6x + 18x + 12
Type: UnivariatePolynomial(x,Fraction Integer)
Time: 0.03 (IN) = 0.03 sec
u2: U := p2
2
(66) 9x + 36x + 27
Type: UnivariatePolynomial(x,Fraction Integer)
Time: 0 sec
gcd(u1,u2)
(67) x + 1
Type: UnivariatePolynomial(x,Fraction Integer)
Time: 0.01 (IN) = 0.01 sec
gcd( (y^3-1) * p1, (y^2 -1) * p2)
(68) (x + 1)y - x - 1
Type: Polynomial Fraction Integer
Time: 0.02 (EV) = 0.02 sec
Session1 (69) -> )spool
Finished dribbling to session1.out.