/usr/share/axiom-20170501/src/algebra/MODMON.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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 | )abbrev domain MODMON ModMonic
++ Author: Mark Botch
++ Description:
++ This package has not been documented
-- following line prevents caching ModMonic
)bo PUSH('ModMonic, $mutableDomains)
ModMonic(R,Rep) : SIG == CODE where
R : Ring
Rep : UnivariatePolynomialCategory(R)
SIG ==> UnivariatePolynomialCategory(R) with
setPoly : Rep -> Rep
++ setPoly(x) is not documented
modulus : -> Rep
++ modulus() is not documented
reduce : Rep -> %
++ reduce(x) is not documented
lift : % -> Rep --reduce lift = identity
++ lift(x) is not documented
coerce : Rep -> %
++ coerce(x) is not documented
Vectorise : % -> Vector(R)
++ Vectorise(x) is not documented
UnVectorise : Vector(R) -> %
++ UnVectorise(v) is not documented
An : % -> Vector(R)
++ An(x) is not documented
pow : -> PrimitiveArray(%)
++ pow() is not documented
computePowers : -> PrimitiveArray(%)
++ computePowers() is not documented
if R has FiniteFieldCategory then
frobenius : % -> %
++ frobenius(x) is not documented
if R has Finite then Finite
CODE ==> add
m:Rep := monomial(1,1)$Rep --| degree(m) > 0 and LeadingCoef(m) = R$1
d := degree(m)$Rep
d1 := (d-1):NonNegativeInteger
twod := 2*d1+1
frobenius?:Boolean := R has FiniteFieldCategory
--VectorRep:= DirectProduct(d:NonNegativeInteger,R)
x,y: %
p: Rep
d,n: Integer
e,k1,k2: NonNegativeInteger
c: R
--vect: Vector(R)
power:PrimitiveArray(%)
frobeniusPower:PrimitiveArray(%)
computeFrobeniusPowers : () -> PrimitiveArray(%)
power := new(0,0)
frobeniusPower := new(0,0)
setPoly (mon : Rep) ==
mon =$Rep m => mon
oldm := m
leadingCoefficient mon ^= 1 => error "polynomial must be monic"
-- following copy code needed since FFPOLY can modify mon
copymon:Rep:= 0
while not zero? mon repeat
copymon := monomial(leadingCoefficient mon, degree mon)$Rep + copymon
mon := reductum mon
m := copymon
d := degree(m)$Rep
d1 := (d-1)::NonNegativeInteger
twod := 2*d1+1
power := computePowers()
if frobenius? then
degree(oldm)>1 and not((oldm exquo$Rep m) case "failed") =>
for i in 1..d1 repeat
frobeniusPower(i) := reduce lift frobeniusPower(i)
frobeniusPower := computeFrobeniusPowers()
m
modulus == m
if R has Finite then
size == d * size$R
random == UnVectorise([random()$R for i in 0..d1])
0 == 0$Rep
1 == 1$Rep
c * x == c *$Rep x
n * x == (n::R) *$Rep x
coerce(c:R):% == monomial(c,0)$Rep
coerce(x:%):OutputForm == coerce(x)$Rep
coefficient(x,e):R == coefficient(x,e)$Rep
reductum(x) == reductum(x)$Rep
leadingCoefficient x == (leadingCoefficient x)$Rep
degree x == (degree x)$Rep
lift(x) == x pretend Rep
reduce(p) == (monicDivide(p,m)$Rep).remainder
coerce(p) == reduce(p)
x = y == x =$Rep y
x + y == x +$Rep y
- x == -$Rep x
x * y ==
p := x *$Rep y
ans:=0$Rep
while (n:=degree p)>d1 repeat
ans:=ans + leadingCoefficient(p)*power.(n-d)
p := reductum p
ans+p
Vectorise(x) == [coefficient(lift(x),i) for i in 0..d1]
UnVectorise(vect) ==
reduce(+/[monomial(vect.(i+1),i) for i in 0..d1])
computePowers ==
mat : PrimitiveArray(%):= new(d,0)
mat.0:= reductum(-m)$Rep
w: % := monomial$Rep (1,1)
for i in 1..d1 repeat
mat.i := w *$Rep mat.(i-1)
if degree mat.i=d then
mat.i:= reductum mat.i + leadingCoefficient mat.i * mat.0
mat
if frobenius? then
computeFrobeniusPowers() ==
mat : PrimitiveArray(%):= new(d,1)
mat.1:= mult := monomial(1, size$R)$%
for i in 2..d1 repeat
mat.i := mult * mat.(i-1)
mat
frobenius(a:%):% ==
aq:% := 0
while a^=0 repeat
aq:= aq + leadingCoefficient(a)*frobeniusPower(degree a)
a := reductum a
aq
pow == power
monomial(c,e)==
if e<d then monomial(c,e)$Rep
else
if e<=twod then
c * power.(e-d)
else
k1:=e quo twod
k2 := (e-k1*twod)::NonNegativeInteger
reduce((power.d1 **k1)*monomial(c,k2))
if R has Field then
(x:% exquo y:%):Union(%, "failed") ==
uv := extendedEuclidean(y, modulus(), x)$Rep
uv case "failed" => "failed"
return reduce(uv.coef1)
recip(y:%):Union(%, "failed") == 1 exquo y
divide(x:%, y:%) ==
(q := (x exquo y)) case "failed" => error "not divisible"
[q, 0]
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