/usr/share/axiom-20170501/src/algebra/NSUP.spad is in axiom-source 20170501-3.
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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 | )abbrev domain NSUP NewSparseUnivariatePolynomial
++ Author: Marc Moreno Maza
++ Date Created: 23/07/98
++ Date Last Updated: 14/12/98
++ Description:
++ A post-facto extension for \axiomType{SUP} in order
++ to speed up operations related to pseudo-division and gcd for
++ both \axiomType{SUP} and, consequently, \axiomType{NSMP}.
NewSparseUnivariatePolynomial(R) : SIG == CODE where
R : Ring
NNI ==> NonNegativeInteger
SUPR ==> SparseUnivariatePolynomial R
SIG ==> Join(UnivariatePolynomialCategory(R),
CoercibleTo(SUPR),RetractableTo(SUPR)) with
fmecg : (%,NNI,R,%) -> %
++ \axiom{fmecg(p1,e,r,p2)} returns \axiom{p1 - r * x**e * p2}
++ where \axiom{x} is \axiom{monomial(1,1)}
monicModulo : ($, $) -> $
++ \axiom{monicModulo(a,b)} returns \axiom{r} such that \axiom{r} is
++ reduced w.r.t. \axiom{b} and \axiom{b} divides \axiom{a -r}
++ where \axiom{b} is monic.
lazyResidueClass : ($,$) -> Record(polnum:$, polden:R, power:NNI)
++ \axiom{lazyResidueClass(a,b)} returns \axiom{[r,c,n]} such that
++ \axiom{r} is reduced w.r.t. \axiom{b} and \axiom{b} divides
++ \axiom{c^n * a - r} where \axiom{c} is \axiom{leadingCoefficient(b)}
++ and \axiom{n} is as small as possible with the previous properties.
lazyPseudoRemainder : ($,$) -> $
++ \axiom{lazyPseudoRemainder(a,b)} returns \axiom{r} if
++ \axiom{lazyResidueClass(a,b)} returns \axiom{[r,c,n]}.
++ This lazy pseudo-remainder is computed by means of the
++ fmecg from NewSparseUnivariatePolynomial operation.
lazyPseudoDivide : ($,$) -> Record(coef:R,gap:NNI,quotient:$,remainder:$)
++ \axiom{lazyPseudoDivide(a,b)} returns \axiom{[c,g,q,r]} such that
++ \axiom{c^n * a = q*b +r} and \axiom{lazyResidueClass(a,b)} returns
++ \axiom{[r,c,n]} where
++ \axiom{n + g = max(0, degree(b) - degree(a) + 1)}.
lazyPseudoQuotient : ($,$) -> $
++ \axiom{lazyPseudoQuotient(a,b)} returns \axiom{q} if
++ \axiom{lazyPseudoDivide(a,b)} returns \axiom{[c,g,q,r]}
if R has IntegralDomain then
subResultantsChain : ($, $) -> List $
++ \axiom{subResultantsChain(a,b)} returns the list of the non-zero
++ sub-resultants of \axiom{a} and \axiom{b} sorted by increasing
++ degree.
lastSubResultant : ($, $) -> $
++ \axiom{lastSubResultant(a,b)} returns \axiom{resultant(a,b)}
++ if \axiom{a} and \axiom{b} has no non-trivial gcd
++ in \axiom{R^(-1) P}
++ otherwise the non-zero sub-resultant with smallest index.
extendedSubResultantGcd : ($, $) -> Record(gcd: $, coef1: $, coef2: $)
++ \axiom{extendedSubResultantGcd(a,b)} returns \axiom{[g,ca, cb]}
++ such that \axiom{g} is a gcd of \axiom{a} and \axiom{b} in
++ \axiom{R^(-1) P} and \axiom{g = ca * a + cb * b}
halfExtendedSubResultantGcd1 : ($, $) -> Record(gcd: $, coef1: $)
++ \axiom{halfExtendedSubResultantGcd1(a,b)} returns \axiom{[g,ca]}
++ such that \axiom{extendedSubResultantGcd(a,b)} returns
++ \axiom{[g,ca, cb]}
halfExtendedSubResultantGcd2 : ($, $) -> Record(gcd: $, coef2: $)
++ \axiom{halfExtendedSubResultantGcd2(a,b)} returns \axiom{[g,cb]}
++ such that \axiom{extendedSubResultantGcd(a,b)} returns
++ \axiom{[g,ca, cb]}
extendedResultant : ($, $) -> Record(resultant: R, coef1: $, coef2: $)
++ \axiom{extendedResultant(a,b)} returns \axiom{[r,ca,cb]} such that
++ \axiom{r} is the resultant of \axiom{a} and \axiom{b} and
++ \axiom{r = ca * a + cb * b}
halfExtendedResultant1 : ($, $) -> Record(resultant: R, coef1: $)
++ \axiom{halfExtendedResultant1(a,b)} returns \axiom{[r,ca]}
++ such that \axiom{extendedResultant(a,b)} returns
++ \axiom{[r,ca, cb]}
halfExtendedResultant2 : ($, $) -> Record(resultant: R, coef2: $)
++ \axiom{halfExtendedResultant2(a,b)} returns \axiom{[r,ca]} such
++ that \axiom{extendedResultant(a,b)} returns \axiom{[r,ca, cb]}
CODE ==> SparseUnivariatePolynomial(R) add
Term == Record(k:NonNegativeInteger,c:R)
Rep ==> List Term
rep(s:$):Rep == s pretend Rep
per(l:Rep):$ == l pretend $
coerce (p:$):SUPR ==
p pretend SUPR
coerce (p:SUPR):$ ==
p pretend $
retractIfCan (p:$) : Union(SUPR,"failed") ==
(p pretend SUPR)::Union(SUPR,"failed")
monicModulo(x,y) ==
zero? y =>
error "in monicModulo$NSUP: division by 0"
ground? y =>
error "in monicModulo$NSUP: ground? #2"
yy := rep y
not ((yy.first.c) = 1) =>
error "in monicModulo$NSUP: not monic #2"
xx := rep x; empty? xx => x
e := yy.first.k; y := per(yy.rest)
repeat
if (u:=subtractIfCan(xx.first.k,e)) case "failed" then break
xx:= rep fmecg(per rest(xx), u, xx.first.c, y)
if empty? xx then break
per xx
lazyResidueClass(x,y) ==
zero? y =>
error "in lazyResidueClass$NSUP: division by 0"
ground? y =>
error "in lazyResidueClass$NSUP: ground? #2"
yy := rep y; co := yy.first.c; xx: Rep := rep x
empty? xx => [x, co, 0]
pow: NNI := 0; e := yy.first.k; y := per(yy.rest);
repeat
if (u:=subtractIfCan(xx.first.k,e)) case "failed" then break
xx:= rep fmecg(co * per rest(xx), u, xx.first.c, y)
pow := pow + 1
if empty? xx then break
[per xx, co, pow]
lazyPseudoRemainder(x,y) ==
zero? y =>
error "in lazyPseudoRemainder$NSUP: division by 0"
ground? y =>
error "in lazyPseudoRemainder$NSUP: ground? #2"
ground? x => x
yy := rep y; co := yy.first.c
(co = 1) => monicModulo(x,y)
(co = -1) => - monicModulo(-x,-y)
xx:= rep x; e := yy.first.k; y := per(yy.rest)
repeat
if (u:=subtractIfCan(xx.first.k,e)) case "failed" then break
xx:= rep fmecg(co * per rest(xx), u, xx.first.c, y)
if empty? xx then break
per xx
lazyPseudoDivide(x,y) ==
zero? y =>
error "in lazyPseudoDivide$NSUP: division by 0"
ground? y =>
error "in lazyPseudoDivide$NSUP: ground? #2"
yy := rep y; e := yy.first.k;
xx: Rep := rep x; co := yy.first.c
(empty? xx) or (xx.first.k < e) => [co,0,0,x]
pow: NNI := subtractIfCan(xx.first.k,e)::NNI + 1
qq: Rep := []; y := per(yy.rest)
repeat
if (u:=subtractIfCan(xx.first.k,e)) case "failed" then break
qq := cons([u::NNI, xx.first.c]$Term, rep (co * per qq))
xx := rep fmecg(co * per rest(xx), u, xx.first.c, y)
pow := subtractIfCan(pow,1)::NNI
if empty? xx then break
[co, pow, per reverse qq, per xx]
lazyPseudoQuotient(x,y) ==
zero? y =>
error "in lazyPseudoQuotient$NSUP: division by 0"
ground? y =>
error "in lazyPseudoQuotient$NSUP: ground? #2"
yy := rep y; e := yy.first.k; xx: Rep := rep x
(empty? xx) or (xx.first.k < e) => 0
qq: Rep := []; co := yy.first.c; y := per(yy.rest)
repeat
if (u:=subtractIfCan(xx.first.k,e)) case "failed" then break
qq := cons([u::NNI, xx.first.c]$Term, rep (co * per qq))
xx := rep fmecg(co * per rest(xx), u, xx.first.c, y)
if empty? xx then break
per reverse qq
if R has IntegralDomain
then
pack ==> PseudoRemainderSequence(R, %)
subResultantGcd(p1,p2) == subResultantGcd(p1,p2)$pack
subResultantsChain(p1,p2) == chainSubResultants(p1,p2)$pack
lastSubResultant(p1,p2) == lastSubResultant(p1,p2)$pack
resultant(p1,p2) == resultant(p1,p2)$pack
extendedResultant(p1,p2) ==
re: Record(coef1: $, coef2: $, resultant: R) := _
resultantEuclidean(p1,p2)$pack
[re.resultant, re.coef1, re.coef2]
halfExtendedResultant1(p1:$, p2: $): Record(resultant: R, coef1: $) ==
re: Record(coef1: $, resultant: R) := _
semiResultantEuclidean1(p1, p2)$pack
[re.resultant, re.coef1]
halfExtendedResultant2(p1:$, p2: $): Record(resultant: R, coef2: $) ==
re: Record(coef2: $, resultant: R) := _
semiResultantEuclidean2(p1, p2)$pack
[re.resultant, re.coef2]
extendedSubResultantGcd(p1,p2) ==
re: Record(coef1: $, coef2: $, gcd: $) := _
subResultantGcdEuclidean(p1,p2)$pack
[re.gcd, re.coef1, re.coef2]
halfExtendedSubResultantGcd1(p1:$, p2: $): Record(gcd: $, coef1: $) ==
re: Record(coef1: $, gcd: $) := _
semiSubResultantGcdEuclidean1(p1, p2)$pack
[re.gcd, re.coef1]
halfExtendedSubResultantGcd2(p1:$, p2: $): Record(gcd: $, coef2: $) ==
re: Record(coef2: $, gcd: $) := _
semiSubResultantGcdEuclidean2(p1, p2)$pack
[re.gcd, re.coef2]
pseudoDivide(x,y) ==
zero? y =>
error "in pseudoDivide$NSUP: division by 0"
ground? y =>
error "in pseudoDivide$NSUP: ground? #2"
yy := rep y; e := yy.first.k
xx: Rep := rep x; co := yy.first.c
(empty? xx) or (xx.first.k < e) => [co,0,x]
pow: NNI := subtractIfCan(xx.first.k,e)::NNI + 1
qq: Rep := []; y := per(yy.rest)
repeat
if (u:=subtractIfCan(xx.first.k,e)) case "failed" then break
qq := cons([u::NNI, xx.first.c]$Term, rep (co * per qq))
xx := rep fmecg(co * per rest(xx), u, xx.first.c, y)
pow := subtractIfCan(pow,1)::NNI
if empty? xx then break
zero? pow => [co, per reverse qq, per xx]
default: R := co ** pow
q := default * (per reverse qq)
x := default * (per xx)
[co, q, x]
pseudoQuotient(x,y) ==
zero? y =>
error "in pseudoDivide$NSUP: division by 0"
ground? y =>
error "in pseudoDivide$NSUP: ground? #2"
yy := rep y; e := yy.first.k; xx: Rep := rep x
(empty? xx) or (xx.first.k < e) => 0
pow: NNI := subtractIfCan(xx.first.k,e)::NNI + 1
qq: Rep := []; co := yy.first.c; y := per(yy.rest)
repeat
if (u:=subtractIfCan(xx.first.k,e)) case "failed" then break
qq := cons([u::NNI, xx.first.c]$Term, rep (co * per qq))
xx := rep fmecg(co * per rest(xx), u, xx.first.c, y)
pow := subtractIfCan(pow,1)::NNI
if empty? xx then break
zero? pow => per reverse qq
(co ** pow) * (per reverse qq)
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