/usr/share/axiom-20170501/src/algebra/GBINTERN.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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 | )abbrev package GBINTERN GroebnerInternalPackage
++ References:
++ Normxx Notes 13: How to Compute a Groebner Basis
++ Description:
++ This package provides low level tools for Groebner basis computations
GroebnerInternalPackage(Dom, Expon, VarSet, Dpol) : SIG == CODE where
Dom : GcdDomain
Expon : OrderedAbelianMonoidSup
VarSet : OrderedSet
Dpol : PolynomialCategory(Dom, Expon, VarSet)
NNI ==> NonNegativeInteger
------ Definition of Record critPair and Prinp
critPair ==> Record( lcmfij: Expon, totdeg: NonNegativeInteger,
poli: Dpol, polj: Dpol )
sugarPol ==> Record( totdeg: NonNegativeInteger, pol : Dpol)
Prinp ==> Record( ci:Dpol,tci:Integer,cj:Dpol,tcj:Integer,c:Dpol,
tc:Integer,rc:Dpol,trc:Integer,tF:Integer,tD:Integer)
Prinpp ==> Record( ci:Dpol,tci:Integer,cj:Dpol,tcj:Integer,c:Dpol,
tc:Integer,rc:Dpol,trc:Integer,tF:Integer,tDD:Integer,
tDF:Integer)
SIG ==> with
credPol : (Dpol, List(Dpol)) -> Dpol
++ credPol \undocumented
redPol : (Dpol, List(Dpol)) -> Dpol
++ redPol \undocumented
gbasis : (List(Dpol), Integer, Integer) -> List(Dpol)
++ gbasis \undocumented
critT : critPair -> Boolean
++ critT \undocumented
critM : (Expon, Expon) -> Boolean
++ critM \undocumented
critB : (Expon, Expon, Expon, Expon) -> Boolean
++ critB \undocumented
critBonD : (Dpol, List(critPair)) -> List(critPair)
++ critBonD \undocumented
critMTonD1 : (List(critPair)) -> List(critPair)
++ critMTonD1 \undocumented
critMonD1 : (Expon, List(critPair)) -> List(critPair)
++ critMonD1 \undocumented
redPo : (Dpol, List(Dpol) ) -> Record(poly:Dpol, mult:Dom)
++ redPo \undocumented
hMonic : Dpol -> Dpol
++ hMonic \undocumented
updatF : (Dpol, NNI, List(sugarPol) ) -> List(sugarPol)
++ updatF \undocumented
sPol : critPair -> Dpol
++ sPol \undocumented
updatD : (List(critPair), List(critPair)) -> List(critPair)
++ updatD \undocumented
minGbasis : List(Dpol) -> List(Dpol)
++ minGbasis \undocumented
lepol : Dpol -> Integer
++ lepol \undocumented
prinshINFO : Dpol -> Void
++ prinshINFO \undocumented
prindINFO : (critPair, Dpol, Dpol,Integer,Integer,Integer) -> Integer
++ prindINFO \undocumented
fprindINFO : (critPair, Dpol, Dpol, Integer,Integer,Integer
,Integer) -> Integer
++ fprindINFO \undocumented
prinpolINFO : List(Dpol) -> Void
++ prinpolINFO \undocumented
prinb : Integer-> Void
++ prinb \undocumented
critpOrder : (critPair, critPair) -> Boolean
++ critpOrder \undocumented
makeCrit : (sugarPol, Dpol, NonNegativeInteger) -> critPair
++ makeCrit \undocumented
virtualDegree : Dpol -> NonNegativeInteger
++ virtualDegree \undocumented
CODE ==> add
Ex ==> OutputForm
import OutputForm
------ Definition of intermediate functions
if Dpol has totalDegree: Dpol -> NonNegativeInteger then
virtualDegree p == totalDegree p
else
virtualDegree p == 0
------ ordering of critpairs
critpOrder(cp1,cp2) ==
cp1.totdeg < cp2.totdeg => true
cp2.totdeg < cp1.totdeg => false
cp1.lcmfij < cp2.lcmfij
------ creating a critical pair
makeCrit(sp1, p2, totdeg2) ==
p1 := sp1.pol
deg := sup(degree(p1), degree(p2))
e1 := subtractIfCan(deg, degree(p1))::Expon
e2 := subtractIfCan(deg, degree(p2))::Expon
tdeg := max(sp1.totdeg + virtualDegree(monomial(1,e1)),
totdeg2 + virtualDegree(monomial(1,e2)))
[deg, tdeg, p1, p2]$critPair
------ calculate basis
gbasis(Pol: List(Dpol), xx1: Integer, xx2: Integer ) ==
D, D1: List(critPair)
--------- create D and Pol
Pol1:= sort((z1,z2) +-> degree z1 > degree z2, Pol)
basPols:= updatF(hMonic(first Pol1),virtualDegree(first Pol1),[])
Pol1:= rest(Pol1)
D:= nil
while _^ null Pol1 repeat
h:= hMonic(first(Pol1))
Pol1:= rest(Pol1)
toth := virtualDegree h
D1:= [makeCrit(x,h,toth) for x in basPols]
D:= updatD(critMTonD1(sort(critpOrder, D1)),
critBonD(h,D))
basPols:= updatF(h,toth,basPols)
D:= sort(critpOrder, D)
xx:= xx2
-------- loop
redPols := [x.pol for x in basPols]
while _^ null D repeat
D0:= first D
s:= hMonic(sPol(D0))
D:= rest(D)
h:= hMonic(redPol(s,redPols))
if xx1 = 1 then
prinshINFO(h)
h = 0 =>
if xx2 = 1 then
prindINFO(D0,s,h,# basPols, # D,xx)
xx:= 2
" go to top of while "
degree(h) = 0 =>
D:= nil
if xx2 = 1 then
prindINFO(D0,s,h,# basPols, # D,xx)
xx:= 2
basPols:= updatF(h,0,[])
leave "out of while"
D1:= [makeCrit(x,h,D0.totdeg) for x in basPols]
D:= updatD(critMTonD1(sort(critpOrder, D1)),
critBonD(h,D))
basPols:= updatF(h,D0.totdeg,basPols)
redPols := concat(redPols,h)
if xx2 = 1 then
prindINFO(D0,s,h,# basPols, # D,xx)
xx:= 2
Pol := [x.pol for x in basPols]
if xx2 = 1 then
prinpolINFO(Pol)
messagePrint(" THE GROEBNER BASIS POLYNOMIALS")
if xx1 = 1 and xx2 ^= 1 then
messagePrint(" THE GROEBNER BASIS POLYNOMIALS")
Pol
--------------------------------------
--- erase multiple of e in D2 using crit M
critMonD1(e: Expon, D2: List(critPair))==
null D2 => nil
x:= first(D2)
critM(e, x.lcmfij) => critMonD1(e, rest(D2))
cons(x, critMonD1(e, rest(D2)))
----------------------------
--- reduce D1 using crit T and crit M
critMTonD1(D1: List(critPair))==
null D1 => nil
f1:= first(D1)
s1:= #(D1)
cT1:= critT(f1)
s1= 1 and cT1 => nil
s1= 1 => D1
e1:= f1.lcmfij
r1:= rest(D1)
e1 = (first r1).lcmfij =>
cT1 => critMTonD1(cons(f1, rest(r1)))
critMTonD1(r1)
D1 := critMonD1(e1, r1)
cT1 => critMTonD1(D1)
cons(f1, critMTonD1(D1))
-----------------------------
--- erase elements in D fullfilling crit B
critBonD(h:Dpol, D: List(critPair))==
null D => nil
x:= first(D)
critB(degree(h), x.lcmfij, degree(x.poli), degree(x.polj)) =>
critBonD(h, rest(D))
cons(x, critBonD(h, rest(D)))
-----------------------------
--- concat F and h and erase multiples of h in F
updatF(h: Dpol, deg:NNI, F: List(sugarPol)) ==
null F => [[deg,h]]
f1:= first(F)
critM(degree(h), degree(f1.pol)) => updatF(h, deg, rest(F))
cons(f1, updatF(h, deg, rest(F)))
-----------------------------
--- concat ordered critical pair lists D1 and D2
updatD(D1: List(critPair), D2: List(critPair)) ==
null D1 => D2
null D2 => D1
dl1:= first(D1)
dl2:= first(D2)
critpOrder(dl1,dl2) => cons(dl1, updatD(D1.rest, D2))
cons(dl2, updatD(D1, D2.rest))
-----------------------------
--- remove gcd from pair of coefficients
gcdCo(c1:Dom, c2:Dom):Record(co1:Dom,co2:Dom) ==
d:=gcd(c1,c2)
[(c1 exquo d)::Dom, (c2 exquo d)::Dom]
--- calculate S-polynomial of a critical pair
sPol(p:critPair)==
Tij := p.lcmfij
fi := p.poli
fj := p.polj
cc := gcdCo(leadingCoefficient fi, leadingCoefficient fj)
reductum(fi)*monomial(cc.co2,subtractIfCan(Tij, degree fi)::Expon) -
reductum(fj)*monomial(cc.co1,subtractIfCan(Tij, degree fj)::Expon)
----------------------------
--- reduce critpair polynomial mod F
--- iterative version
redPo(s: Dpol, F: List(Dpol)) ==
m:Dom := 1
Fh := F
while _^ ( s = 0 or null F ) repeat
f1:= first(F)
s1:= degree(s)
e: Union(Expon, "failed")
(e:= subtractIfCan(s1, degree(f1))) case Expon =>
cc:=gcdCo(leadingCoefficient f1, leadingCoefficient s)
s:=cc.co1*reductum(s) - monomial(cc.co2,e)*reductum(f1)
m := m*cc.co1
F:= Fh
F:= rest F
[s,m]
redPol(s: Dpol, F: List(Dpol)) == credPol(redPo(s,F).poly,F)
----------------------------
--- crit T true, if e1 and e2 are disjoint
critT(p: critPair) == p.lcmfij = (degree(p.poli) + degree(p.polj))
----------------------------
--- crit M - true, if lcm#2 multiple of lcm#1
critM(e1: Expon, e2: Expon) ==
en: Union(Expon, "failed")
(en:=subtractIfCan(e2, e1)) case Expon
----------------------------
--- crit B - true, if eik is a multiple of eh and eik ^equal
--- lcm(eh,ei) and eik ^equal lcm(eh,ek)
critB(eh:Expon, eik:Expon, ei:Expon, ek:Expon) ==
critM(eh, eik) and (eik ^= sup(eh, ei)) and (eik ^= sup(eh, ek))
----------------------------
--- make polynomial monic case Domain a Field
hMonic(p: Dpol) ==
p= 0 => p
-- inv(leadingCoefficient(p))*p
primitivePart p
-----------------------------
--- reduce all terms of h mod F (iterative version )
credPol(h: Dpol, F: List(Dpol) ) ==
null F => h
h0:Dpol:= monomial(leadingCoefficient h, degree h)
while (h:=reductum h) ^= 0 repeat
hred:= redPo(h, F)
h := hred.poly
h0:=(hred.mult)*h0 + monomial(leadingCoefficient(h),degree h)
h0
-------------------------------
---- calculate minimal basis for ordered F
minGbasis(F: List(Dpol)) ==
null F => nil
newbas := minGbasis rest F
cons(hMonic credPol( first(F), newbas),newbas)
-------------------------------
---- calculate number of terms of polynomial
lepol(p1:Dpol)==
n: Integer
n:= 0
while p1 ^= 0 repeat
n:= n + 1
p1:= reductum(p1)
n
---- print blanc lines
prinb(n: Integer)==
for x in 1..n repeat
messagePrint(" ")
---- print reduced critpair polynom
prinshINFO(h: Dpol)==
prinb(2)
messagePrint(" reduced Critpair - Polynom :")
prinb(2)
print(h::Ex)
prinb(2)
-------------------------------
---- print info string
prindINFO(cp: critPair, ps: Dpol, ph: Dpol, i1:Integer,
i2:Integer, n:Integer) ==
ll: List Prinp
a: Dom
cpi:= cp.poli
cpj:= cp.polj
if n = 1 then
prinb(1)
messagePrint("you choose option -info- ")
messagePrint("abbrev. for the following information strings are")
messagePrint(" ci => Leading monomial for critpair calculation")
messagePrint(" tci => Number of terms of polynomial i")
messagePrint(" cj => Leading monomial for critpair calculation")
messagePrint(" tcj => Number of terms of polynomial j")
messagePrint(" c => Leading monomial of critpair polynomial")
messagePrint(" tc => Number of terms of critpair polynomial")
messagePrint(" rc => Leading monomial of redcritpair polynomial")
messagePrint(" trc => Number of terms of redcritpair polynomial")
messagePrint(" tF => Number of polynomials in reduction list F")
messagePrint(" tD => Number of critpairs still to do")
prinb(4)
n:= 2
prinb(1)
a:= 1
ph = 0 =>
ps = 0 =>
ll:= [[monomial(a,degree(cpi)),lepol(cpi),
monomial(a,degree(cpj)),
lepol(cpj),ps,0,ph,0,i1,i2]$Prinp]
print(ll::Ex)
prinb(1)
n
ll:= [[monomial(a,degree(cpi)),lepol(cpi),
monomial(a,degree(cpj)),lepol(cpj),monomial(a,degree(ps)),
lepol(ps), ph,0,i1,i2]$Prinp]
print(ll::Ex)
prinb(1)
n
ll:= [[monomial(a,degree(cpi)),lepol(cpi),
monomial(a,degree(cpj)),lepol(cpj),monomial(a,degree(ps)),
lepol(ps),monomial(a,degree(ph)),lepol(ph),i1,i2]$Prinp]
print(ll::Ex)
prinb(1)
n
-------------------------------
---- print the groebner basis polynomials
prinpolINFO(pl: List(Dpol))==
n:Integer
n:= # pl
prinb(1)
n = 1 =>
messagePrint(" There is 1 Groebner Basis Polynomial ")
prinb(2)
messagePrint(" There are ")
prinb(1)
print(n::Ex)
prinb(1)
messagePrint(" Groebner Basis Polynomials. ")
prinb(2)
fprindINFO(cp: critPair, ps: Dpol, ph: Dpol, i1:Integer,
i2:Integer, i3:Integer, n: Integer) ==
ll: List Prinpp
a: Dom
cpi:= cp.poli
cpj:= cp.polj
if n = 1 then
prinb(1)
messagePrint("you choose option -info- ")
messagePrint("abbrev. for the following information strings are")
messagePrint(" ci => Leading monomial for critpair calculation")
messagePrint(" tci => Number of terms of polynomial i")
messagePrint(" cj => Leading monomial for critpair calculation")
messagePrint(" tcj => Number of terms of polynomial j")
messagePrint(" c => Leading monomial of critpair polynomial")
messagePrint(" tc => Number of terms of critpair polynomial")
messagePrint(" rc => Leading monomial of redcritpair polynomial")
messagePrint(" trc => Number of terms of redcritpair polynomial")
messagePrint(" tF => Number of polynomials in reduction list F")
messagePrint(" tD => Number of critpairs still to do")
messagePrint(" tDF => Number of subproblems still to do")
prinb(4)
n:= 2
prinb(1)
a:= 1
ph = 0 =>
ps = 0 =>
ll:= [[monomial(a,degree(cpi)),lepol(cpi),
monomial(a,degree(cpj)),
lepol(cpj),ps,0,ph,0,i1,i2,i3]$Prinpp]
print(ll::Ex)
prinb(1)
n
ll:= [[monomial(a,degree(cpi)),lepol(cpi),
monomial(a,degree(cpj)),lepol(cpj),monomial(a,degree(ps)),
lepol(ps), ph,0,i1,i2,i3]$Prinpp]
print(ll::Ex)
prinb(1)
n
ll:= [[monomial(a,degree(cpi)),lepol(cpi),
monomial(a,degree(cpj)),lepol(cpj),monomial(a,degree(ps)),
lepol(ps),monomial(a,degree(ph)),lepol(ph),i1,i2,i3]$Prinpp]
print(ll::Ex)
prinb(1)
n
|