/usr/share/axiom-20170501/src/algebra/LAUPOL.spad is in axiom-source 20170501-3.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 | )abbrev domain LAUPOL LaurentPolynomial
++ Author: Manuel Bronstein
++ Date Created: May 1988
++ Date Last Updated: 26 Apr 1990
++ Description:
++ Univariate polynomials with negative and positive exponents.
LaurentPolynomial(R, UP) : SIG == CODE where
R : IntegralDomain
UP : UnivariatePolynomialCategory R
O ==> OutputForm
B ==> Boolean
N ==> NonNegativeInteger
Z ==> Integer
RF ==> Fraction UP
SIG ==> Join(DifferentialExtension UP, IntegralDomain,
ConvertibleTo RF, FullyRetractableTo R, RetractableTo UP) with
monomial? : % -> B
++ monomial?(x) is not documented
degree : % -> Z
++ degree(x) is not documented
order : % -> Z
++ order(x) is not documented
++
++X w:SparseUnivariateLaurentSeries(Fraction(Integer),'z,0):=0
++X order(w,0)
reductum : % -> %
++ reductum(x) is not documented
leadingCoefficient : % -> R
++ leadingCoefficient(p) is not documented
trailingCoefficient : % -> R
++ trailingCoefficient(p) is not documented
coefficient : (%, Z) -> R
++ coefficient(x,n) is not documented
monomial : (R, Z) -> %
++ monomial(x,n) is not documented
if R has CharacteristicZero then CharacteristicZero
if R has CharacteristicNonZero then CharacteristicNonZero
if R has Field then
EuclideanDomain
separate : RF -> Record(polyPart:%, fracPart:RF)
++ separate(x) is not documented
CODE ==> add
Rep := Record(polypart: UP, order0: Z)
poly : % -> UP
check0 : (Z, UP) -> %
mkgpol : (Z, UP) -> %
gpol : (UP, Z) -> %
toutput: (R, Z, O) -> O
monTerm: (R, Z, O) -> O
0 == [0, 0]
1 == [1, 0]
p = q == p.order0 = q.order0 and p.polypart = q.polypart
poly p == p.polypart
order p == p.order0
gpol(p, n) == [p, n]
monomial(r, n) == check0(n, r::UP)
coerce(p:UP):% == mkgpol(0, p)
reductum p == check0(order p, reductum poly p)
n:Z * p:% == check0(order p, n * poly p)
characteristic() == characteristic()$R
coerce(n:Z):% == n::R::%
degree p == degree(poly p)::Z + order p
monomial? p == monomial? poly p
coerce(r:R):% == gpol(r::UP, 0)
convert(p:%):RF == poly(p) * (monomial(1, 1)$UP)::RF ** order p
p:% * q:% == check0(order p + order q, poly p * poly q)
- p == gpol(- poly p, order p)
check0(n, p) == (zero? p => 0; gpol(p, n))
trailingCoefficient p == coefficient(poly p, 0)
leadingCoefficient p == leadingCoefficient poly p
coerce(p:%):O ==
zero? p => 0::Z::O
l := nil()$List(O)
v := monomial(1, 1)$UP :: O
while p ^= 0 repeat
l := concat(l, toutput(leadingCoefficient p, degree p, v))
p := reductum p
reduce("+", l)
coefficient(p, n) ==
(m := n - order p) < 0 => 0
coefficient(poly p, m::N)
differentiate(p:%, derivation:UP -> UP) ==
t := monomial(1, 1)$UP
mkgpol(order(p) - 1,
derivation(poly p) * t + order(p) * poly(p) * derivation t)
monTerm(r, n, v) ==
zero? n => r::O
(n = 1) => v
v ** (n::O)
toutput(r, n, v) ==
mon := monTerm(r, n, v)
zero? n or (r = 1) => mon
r = -1 => - mon
r::O * mon
recip p ==
(q := recip poly p) case "failed" => "failed"
gpol(q::UP, - order p)
p + q ==
zero? q => p
zero? p => q
(d := order p - order q) > 0 =>
gpol(poly(p) * monomial(1, d::N) + poly q, order q)
d < 0 => gpol(poly(p) + poly(q) * monomial(1, (-d)::N), order p)
mkgpol(order p, poly(p) + poly q)
mkgpol(n, p) ==
zero? p => 0
d := order(p, monomial(1, 1)$UP)
gpol((p exquo monomial(1, d))::UP, n + d::Z)
p exquo q ==
(r := poly(p) exquo poly q) case "failed" => "failed"
check0(order p - order q, r::UP)
retractIfCan(p:%):Union(UP, "failed") ==
order(p) < 0 => error "Not retractable"
poly(p) * monomial(1, order(p)::N)$UP
retractIfCan(p:%):Union(R, "failed") ==
order(p) ^= 0 => "failed"
retractIfCan poly p
if R has Field then
gcd(p, q) == gcd(poly p, poly q)::%
separate f ==
n := order(q := denom f, monomial(1, 1))
q := (q exquo (tn := monomial(1, n)$UP))::UP
bc := extendedEuclidean(tn,q,numer f)::Record(coef1:UP,coef2:UP)
qr := divide(bc.coef1, q)
[mkgpol(-n, bc.coef2 + tn * qr.quotient), qr.remainder / q]
-- returns (z, r) s.t. p = q z + r,
-- and degree(r) < degree(q), order(r) >= min(order(p), order(q))
divide(p, q) ==
c := min(order p, order q)
qr := divide(poly(p) * monomial(1, (order p - c)::N)$UP, poly q)
[mkgpol(c - order q, qr.quotient), mkgpol(c, qr.remainder)]
euclideanSize p == degree poly p
extendedEuclidean(a, b, c) ==
(bc := extendedEuclidean(poly a, poly b, poly c)) case "failed"
=> "failed"
[mkgpol(order c - order a, bc.coef1),
mkgpol(order c - order b, bc.coef2)]
|