/usr/share/axiom-20170501/src/algebra/RECLOS.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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 | )abbrev domain RECLOS RealClosure
++ Author: Renaud Rioboo
++ Date Created: summer 1988
++ Date Last Updated: January 2004
++ References:
++ Riob92 Real algebraic closure of an ordered field,
++ Implementation in Axiom
++ Emir04 Comparing real algebraic numbers of small degree
++ Description:
++ This domain implements the real closure of an ordered field.
++ Note:
++ The code here is generic it does not depend of the way the operations
++ are done. The two macros PME and SEG should be passed as functorial
++ arguments to the domain. It does not help much to write a category
++ since non trivial methods cannot be placed there either.
RealClosure(TheField) : SIG == CODE where
TheField : Join(OrderedRing, Field, RealConstant)
E ==> OutputForm
Z ==> Integer
SE ==> Symbol
B ==> Boolean
SUP ==> SparseUnivariatePolynomial($)
N ==> PositiveInteger
RN ==> Fraction Z
LF ==> ListFunctions2($,N)
PME ==> SparseUnivariatePolynomial($)
SEG ==> RightOpenIntervalRootCharacterization($,PME)
SIG ==> Join(RealClosedField,
FullyRetractableTo TheField,
Algebra TheField) with
algebraicOf : (SEG,E) -> $
++ \axiom{algebraicOf(char)} is the external number
mainCharacterization : $ -> Union(SEG,"failed")
++ \axiom{mainCharacterization(x)} is the main algebraic
++ quantity of \axiom{x} (\axiom{SEG})
relativeApprox : ($,$) -> RN
++ \axiom{relativeApprox(n,p)} gives a relative
++ approximation of \axiom{n}
++ that has precision \axiom{p}
CODE ==> add
-- local functions
lessAlgebraic : $ -> $
newElementIfneeded : (SEG,E) -> $
-- Representation
Rec := Record(seg: SEG, val:PME, outForm:E, order:N)
Rep := Union(TheField,Rec)
-- global (mutable) variables
orderOfCreation : N := 1$N
-- it is internally used to sort the algebraic levels
instanceName : Symbol := new()$Symbol
-- this used to print the results, thus different instanciations
-- use different names
-- now the code
relativeApprox(nbe,prec) ==
nbe case TheField => retract(nbe)
appr := relativeApprox(nbe.val, nbe.seg, prec)
-- now appr has the good exact precision but is $
relativeApprox(appr,prec)
approximate(nbe,prec) ==
abs(nbe) < prec => 0
nbe case TheField => retract(nbe)
appr := approximate(nbe.val, nbe.seg, prec)
-- now appr has the good exact precision but is $
approximate(appr,prec)
newElementIfneeded(s,o) ==
p := definingPolynomial(s)
degree(p) = 1 =>
- coefficient(p,0) / leadingCoefficient(p)
res := [s, monomial(1,1), o, orderOfCreation ]$Rec
orderOfCreation := orderOfCreation + 1
res :: $
algebraicOf(s,o) ==
pol := definingPolynomial(s)
degree(pol) = 1 =>
-coefficient(pol,0) / leadingCoefficient(pol)
res := [s, monomial(1,1), o, orderOfCreation ]$Rec
orderOfCreation := orderOfCreation + 1
res :: $
rename!(x,o) ==
x.outForm := o
x
rename(x,o) ==
[x.seg, x.val, o, x.order]$Rec
rootOf(pol,n) ==
degree(pol) = 0 => "failed"
degree(pol) = 1 =>
if n=1
then
-coefficient(pol,0) / leadingCoefficient(pol)
else
"failed"
r := rootOf(pol,n)$SEG
r case "failed" => "failed"
o := hconcat(instanceName :: E , orderOfCreation :: E)$E
algebraicOf(r,o)
allRootsOf(pol:SUP):List($) ==
degree(pol)=0 => []
degree(pol)=1 => [-coefficient(pol,0) / leadingCoefficient(pol)]
liste := allRootsOf(pol)$SEG
res : List $ := []
for term in liste repeat
o := hconcat(instanceName :: E , orderOfCreation :: E)$E
res := cons(algebraicOf(term,o), res)
reverse! res
coerce(x:$):$ ==
x case TheField => x
[x.seg,x.val rem$PME definingPolynomial(x.seg),x.outForm,x.order]$Rec
positive?(x) ==
x case TheField => positive?(x)$TheField
positive?(x.val,x.seg)$SEG
negative?(x) ==
x case TheField => negative?(x)$TheField
negative?(x.val,x.seg)$SEG
abs(x) == sign(x)*x
sign(x) ==
x case TheField => sign(x)$TheField
sign(x.val,x.seg)$SEG
x < y == positive?(y-x)
x = y == zero?(x-y)
mainCharacterization(x) ==
x case TheField => "failed"
x.seg
mainDefiningPolynomial(x) ==
x case TheField => "failed"
definingPolynomial x.seg
mainForm(x) ==
x case TheField => "failed"
x.outForm
mainValue(x) ==
x case TheField => "failed"
x.val
coerce(x:$):E ==
x case TheField => x::TheField :: E
xx:$ := coerce(x)
outputForm(univariate(xx.val),x.outForm)$SUP
inv(x) ==
(res:= recip x) case "failed" => error "Division by 0"
res :: $
recip(x) ==
x case TheField =>
if ((r := recip(x)$TheField) case TheField)
then r::$
else "failed"
if ((r := recip(x.val,x.seg)$SEG) case "failed")
then "failed"
else lessAlgebraic([x.seg,r::PME,x.outForm,x.order]$Rec)
(n:Z * x:$):$ ==
x case TheField => n *$TheField x
zero?(n) => 0
one?(n) => x
[x.seg,map(z+->n*z, x.val),x.outForm,x.order]$Rec
(rn:TheField * x:$):$ ==
x case TheField => rn *$TheField x
zero?(rn) => 0
one?(rn) => x
[x.seg,map(z+->rn*z, x.val),x.outForm,x.order]$Rec
(x:$ * y:$):$ ==
(x case TheField) and (y case TheField) => x *$TheField y
(x case TheField) => x::TheField * y
-- x is no longer TheField
(y case TheField) => y::TheField * x
-- now both are algebraic
y.order > x.order =>
[y.seg,map(z+->x*z , y.val),y.outForm,y.order]$Rec
x.order > y.order =>
[x.seg,map(z+->z*y , x.val),x.outForm,x.order]$Rec
-- now x.exp = y.exp
-- we will multiply the polynomials and then reduce
-- however wee need to call lessAlgebraic
lessAlgebraic([x.seg,
(x.val * y.val) rem definingPolynomial(x.seg),
x.outForm,
x.order]$Rec)
nonNull(rep:Rec):$ ==
degree(rep.val)=0 => leadingCoefficient(rep.val)
numberOfMonomials(rep.val) = 1 => rep
zero?(rep.val,rep.seg)$SEG => 0
rep
zero?(x) ==
x case TheField => zero?(x)$TheField
false
x + y ==
(x case TheField) and (y case TheField) => x +$TheField y
(x case TheField) =>
if zero?(x)
then
y
else
nonNull([y.seg,x::PME+(y.val),y.outForm,y.order]$Rec)
-- x is no longer TheField
(y case TheField) =>
if zero?(y)
then
x
else
nonNull([x.seg,(x.val)+y::PME,x.outForm,x.order]$Rec)
-- now both are algebraic
y.order > x.order =>
nonNull([y.seg,x::PME+y.val,y.outForm,y.order]$Rec)
x.order > y.order =>
nonNull([x.seg,(x.val)+y::PME,x.outForm,x.order]$Rec)
-- now x.exp = y.exp
-- we simply add polynomials (since degree cannot increase)
-- however wee need to call lessAlgebraic
nonNull([x.seg,x.val + y.val,x.outForm,x.order])
-x ==
x case TheField => -$TheField (x::TheField)
[x.seg,-$PME x.val,x.outForm,x.order]$Rec
retractIfCan(x:$):Union(TheField,"failed") ==
x case TheField => x
o := x.order
res := lessAlgebraic x
res case TheField => res
o = res.order => "failed"
retractIfCan res
retract(x:$):TheField ==
x case TheField => x
o := x.order
res := lessAlgebraic x
res case TheField => res
o = res.order => error "Can't retract"
retract res
lessAlgebraic(x) ==
x case TheField => x
degree(x.val) = 0 => leadingCoefficient(x.val)
def := definingPolynomial(x.seg)
degree(def) = 1 =>
x.val.(- coefficient(def,0) / leadingCoefficient(def))
x
0 == (0$TheField) :: $
1 == (1$TheField) :: $
coerce(rn:TheField):$ == rn :: $
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