/usr/share/axiom-20170501/src/algebra/GDMP.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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 | )abbrev domain GDMP GeneralDistributedMultivariatePolynomial
++ Author: Barry Trager
++ References:
++ Coxx07 Ideals, varieties and algorithms
++ Description:
++ This type supports distributed multivariate polynomials
++ whose variables are from a user specified list of symbols.
++ The coefficient ring may be non commutative,
++ but the variables are assumed to commute.
++ The term ordering is specified by its third parameter.
++ Suggested types which define term orderings include:
++ \spadtype{DirectProduct}, \spadtype{HomogeneousDirectProduct},
++ \spadtype{SplitHomogeneousDirectProduct} and finally
++ \spadtype{OrderedDirectProduct} which accepts an arbitrary user
++ function to define a term ordering.
GeneralDistributedMultivariatePolynomial(vl,R,E) : SIG == CODE where
vl : List Symbol
R : Ring
E : DirectProductCategory(#vl,NonNegativeInteger)
OV ==> OrderedVariableList(vl)
SUP ==> SparseUnivariatePolynomial
NNI ==> NonNegativeInteger
SIG ==> PolynomialCategory(R,E,OV) with
reorder : (%,List Integer) -> %
++ reorder(p, perm) applies the permutation perm to the variables
++ in a polynomial and returns the new correctly ordered polynomial
CODE ==> PolynomialRing(R,E) add
--representations
Term := Record(k:E,c:R)
Rep := List Term
n := #vl
Vec ==> Vector(NonNegativeInteger)
zero?(p : %): Boolean == null(p : Rep)
totalDegree p ==
zero? p => 0
"max"/[reduce("+",(t.k)::(Vector NNI), 0) for t in p]
monomial(p:%, v: OV,e: NonNegativeInteger):% ==
locv := lookup v
p*monomial(1,
directProduct [if z=locv then e else 0 for z in 1..n]$Vec)
coerce(v: OV):% == monomial(1,v,1)
listCoef(p : %): List R ==
rec : Term
[rec.c for rec in (p:Rep)]
mainVariable(p: %) ==
zero?(p) => "failed"
for v in vl repeat
vv := variable(v)::OV
if degree(p,vv)>0 then return vv
"failed"
ground?(p) == mainVariable(p) case "failed"
retract(p : %): R ==
not ground? p => error "not a constant"
leadingCoefficient p
retractIfCan(p : %): Union(R,"failed") ==
ground?(p) => leadingCoefficient p
"failed"
degree(p: %,v: OV) == degree(univariate(p,v))
minimumDegree(p: %,v: OV) == minimumDegree(univariate(p,v))
differentiate(p: %,v: OV) ==
multivariate(differentiate(univariate(p,v)),v)
degree(p: %,lv: List OV) == [degree(p,v) for v in lv]
minimumDegree(p: %,lv: List OV) == [minimumDegree(p,v) for v in lv]
numberOfMonomials(p:%) ==
l : Rep := p : Rep
null(l) => 1
#l
monomial?(p : %): Boolean ==
l : Rep := p : Rep
null(l) or null rest(l)
if R has OrderedRing then
maxNorm(p : %): R ==
l : List R := nil
r,m : R
m := 0
for r in listCoef(p) repeat
if r > m then m := r
else if (-r) > m then m := -r
m
if R has Field then
(p : %) / (r : R) == inv(r) * p
variables(p: %) ==
maxdeg:Vector(NonNegativeInteger) := new(n,0)
while not zero?(p) repeat
tdeg := degree p
p := reductum p
for i in 1..n repeat
maxdeg.i := max(maxdeg.i, tdeg.i)
[index(i:PositiveInteger) for i in 1..n | maxdeg.i^=0]
reorder(p: %,perm: List Integer):% ==
#perm ^= n => error "must be a complete permutation of all vars"
q := [[directProduct [term.k.j for j in perm]$Vec,term.c]$Term
for term in p]
sort((z1,z2) +-> z1.k > z2.k,q)
univariate(p: %,v: OV):SUP(%) ==
zero?(p) => 0
exp := degree p
locv := lookup v
deg:NonNegativeInteger := 0
nexp := directProduct [if i=locv then (deg :=exp.i;0) else exp.i
for i in 1..n]$Vec
monomial(monomial(leadingCoefficient p,nexp),deg)+
univariate(reductum p,v)
eval(p: %,v: OV,val:%):% == univariate(p,v)(val)
eval(p: %,v: OV,val:R):% == eval(p,v,val::%)$%
eval(p: %,lv: List OV,lval: List R):% ==
lv = [] => p
eval(eval(p,first lv,(first lval)::%)$%, rest lv, rest lval)$%
-- assume Lvar are sorted correctly
evalSortedVarlist(p: %,Lvar: List OV,Lpval: List %):% ==
v := mainVariable p
v case "failed" => p
pv := v:: OV
Lvar=[] or Lpval=[] => p
mvar := Lvar.first
mvar > pv => evalSortedVarlist(p,Lvar.rest,Lpval.rest)
pval := Lpval.first
pts:SUP(%):= map(x+->evalSortedVarlist(x,Lvar,Lpval),univariate(p,pv))
mvar=pv => pts(pval)
multivariate(pts,pv)
eval(p:%,Lvar:List OV,Lpval:List %) ==
nlvar:List OV := sort((x,y) +-> x > y,Lvar)
nlpval :=
Lvar = nlvar => Lpval
nlpval := [Lpval.position(mvar,Lvar) for mvar in nlvar]
evalSortedVarlist(p,nlvar,nlpval)
multivariate(p1:SUP(%),v: OV):% ==
0=p1 => 0
degree p1 = 0 => leadingCoefficient p1
leadingCoefficient(p1)*(v::%)**degree(p1) +
multivariate(reductum p1,v)
univariate(p: %):SUP(R) ==
(v := mainVariable p) case "failed" =>
monomial(leadingCoefficient p,0)
q := univariate(p,v:: OV)
ans:SUP(R) := 0
while q ^= 0 repeat
ans := ans + monomial(ground leadingCoefficient q,degree q)
q := reductum q
ans
multivariate(p:SUP(R),v: OV):% ==
0=p => 0
(leadingCoefficient p)*monomial(1,v,degree p) +
multivariate(reductum p,v)
if R has GcdDomain then
content(p: %):R ==
zero?(p) => 0
"gcd"/[t.c for t in p]
if R has EuclideanDomain and not(R has FloatingPointSystem) then
gcd(p: %,q:%):% ==
gcd(p,q)$PolynomialGcdPackage(E,OV,R,%)
else
gcd(p: %,q:%):% ==
r : R
(pv := mainVariable(p)) case "failed" =>
(r := leadingCoefficient p) = 0$R => q
gcd(r,content q)::%
(qv := mainVariable(q)) case "failed" =>
(r := leadingCoefficient q) = 0$R => p
gcd(r,content p)::%
pv<qv => gcd(p,content univariate(q,qv))
qv<pv => gcd(q,content univariate(p,pv))
multivariate(gcd(univariate(p,pv),univariate(q,qv)),pv)
coerce(p: %) : OutputForm ==
zero?(p) => (0$R) :: OutputForm
l,lt : List OutputForm
lt := nil
vl1 := [v::OutputForm for v in vl]
for t in reverse p repeat
l := nil
for i in 1..#vl1 repeat
t.k.i = 0 => l
t.k.i = 1 => l := cons(vl1.i,l)
l := cons(vl1.i ** t.k.i ::OutputForm,l)
l := reverse l
if (t.c ^= 1) or (null l) then l := cons(t.c :: OutputForm,l)
1 = #l => lt := cons(first l,lt)
lt := cons(reduce("*",l),lt)
1 = #lt => first lt
reduce("+",lt)
|