/usr/share/axiom-20170501/src/algebra/REAL0.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 | )abbrev package REAL0 RealZeroPackage
++ Author: Andy Neff
++ Description:
++ This package provides functions for finding the real zeros
++ of univariate polynomials over the integers to arbitrary user-specified
++ precision. The results are returned as a list of
++ isolating intervals which are expressed as records with
++ "left" and "right" rational number components.
RealZeroPackage(Pol) : SIG == CODE where
Pol : UnivariatePolynomialCategory Integer
RN ==> Fraction Integer
Interval ==> Record(left : RN, right : RN)
isoList ==> List(Interval)
SIG ==> with
-- next two functions find isolating intervals
realZeros : (Pol) -> isoList
++ realZeros(pol) returns a list of isolating intervals for
++ all the real zeros of the univariate polynomial pol.
realZeros : (Pol, Interval) -> isoList
++ realZeros(pol, range) returns a list of isolating intervals
++ for all the real zeros of the univariate polynomial pol which
++ lie in the interval expressed by the record range.
-- next two functions return intervals smaller then tolerence
realZeros : (Pol, RN) -> isoList
++ realZeros(pol, eps) returns a list of intervals of length less
++ than the rational number eps for all the real roots of the
++ polynomial pol.
realZeros : (Pol, Interval, RN) -> isoList
++ realZeros(pol, int, eps) returns a list of intervals of length
++ less than the rational number eps for all the real roots of the
++ polynomial pol which lie in the interval expressed by the
++ record int.
refine : (Pol, Interval, RN) -> Interval
++ refine(pol, int, eps) refines the interval int containing
++ exactly one root of the univariate polynomial pol to size less
++ than the rational number eps.
refine : (Pol, Interval, Interval) -> Union(Interval,"failed")
++ refine(pol, int, range) takes a univariate polynomial pol and
++ and isolating interval int containing exactly one real
++ root of pol; the operation returns an isolating interval which
++ is contained within range, or "failed" if no such isolating interval exists.
midpoint : Interval -> RN
++ midpoint(int) returns the midpoint of the interval int.
midpoints : isoList -> List RN
++ midpoints(isolist) returns the list of midpoints for the list
++ of intervals isolist.
CODE ==> add
--Local Functions
makeSqfr: Pol -> Pol
ReZeroSqfr: (Pol) -> isoList
PosZero: (Pol) -> isoList
Zero1: (Pol) -> isoList
transMult: (Integer, Pol) -> Pol
transMultInv: (Integer, Pol) -> Pol
transAdd1: (Pol) -> Pol
invert: (Pol) -> Pol
minus: (Pol) -> Pol
negate: Interval -> Interval
rootBound: (Pol) -> Integer
var: (Pol) -> Integer
negate(int : Interval):Interval == [-int.right,-int.left]
midpoint(i : Interval):RN == (1/2)*(i.left + i.right)
midpoints(li : isoList) : List RN ==
[midpoint x for x in li]
makeSqfr(F : Pol):Pol ==
sqfr := squareFree F
F := */[s.factor for s in factors(sqfr)]
realZeros(F : Pol) ==
ReZeroSqfr makeSqfr F
realZeros(F : Pol, rn : RN) ==
F := makeSqfr F
[refine(F,int,rn) for int in ReZeroSqfr(F)]
realZeros(F : Pol, bounds : Interval) ==
F := makeSqfr F
[rint::Interval for int in ReZeroSqfr(F) |
(rint:=refine(F,int,bounds)) case Interval]
realZeros(F : Pol, bounds : Interval, rn : RN) ==
F := makeSqfr F
[refine(F,int,rn) for int in realZeros(F,bounds)]
ReZeroSqfr(F : Pol) ==
F = 0 => error "ReZeroSqfr: zero polynomial"
L : isoList := []
degree(F) = 0 => L
if (r := minimumDegree(F)) > 0 then
L := [[0,0]$Interval]
tempF := F exquo monomial(1, r)
if not (tempF case "failed") then
F := tempF
J:isoList := [negate int for int in reverse(PosZero(minus(F)))]
K : isoList := PosZero(F)
append(append(J, L), K)
PosZero(F : Pol) == --F is square free, primitive
--and F(0) ^= 0; returns isoList for positive
--roots of F
b : Integer := rootBound(F)
F := transMult(b,F)
L : isoList := Zero1(F)
int : Interval
L := [[b*int.left, b*int.right]$Interval for int in L]
Zero1(F : Pol) == --returns isoList for roots of F in (0,1)
J : isoList
K : isoList
L : isoList
L := []
(v := var(transAdd1(invert(F)))) = 0 => []
v = 1 => L := [[0,1]$Interval]
G : Pol := transMultInv(2, F)
H : Pol := transAdd1(G)
if minimumDegree H > 0 then
-- H has a root at 0 => F has one at 1/2, and G at 1
L := [[1/2,1/2]$Interval]
Q : Pol := monomial(1, 1)
tempH : Union(Pol, "failed") := H exquo Q
if not (tempH case "failed") then H := tempH
Q := Q + monomial(-1, 0)
tempG : Union(Pol, "failed") := G exquo Q
if not (tempG case "failed") then G := tempG
int : Interval
J := [[(int.left+1)* (1/2),(int.right+1) * (1/2)]$Interval
for int in Zero1(H)]
K := [[int.left * (1/2), int.right * (1/2)]$Interval
for int in Zero1(G)]
append(append(J, L), K)
rootBound(F : Pol) == --returns power of 2 that is a bound
--for the positive roots of F
if leadingCoefficient(F) < 0 then F := -F
lcoef := leadingCoefficient(F)
F := reductum(F)
i : Integer := 0
while not (F = 0) repeat
if (an := leadingCoefficient(F)) < 0 then i := i - an
F := reductum(F)
b : Integer := 1
while (b * lcoef) <= i repeat
b := 2 * b
b
transMult(c : Integer, F : Pol) ==
--computes Pol G such that G(x) = F(c*x)
G : Pol := 0
while not (F = 0) repeat
n := degree(F)
G := G + monomial((c**n) * leadingCoefficient(F), n)
F := reductum(F)
G
transMultInv(c : Integer, F : Pol) ==
--computes Pol G such that G(x) = (c**n) * F(x/c)
d := degree(F)
cc : Integer := 1
G : Pol := monomial(leadingCoefficient F,d)
while (F:=reductum(F)) ^= 0 repeat
n := degree(F)
cc := cc*(c**(d-n):NonNegativeInteger)
G := G + monomial(cc * leadingCoefficient(F), n)
d := n
G
transAdd1(F : Pol) ==
--computes Pol G such that G(x) = F(x+1)
n := degree F
v := vectorise(F, n+1)
for i in 0..(n-1) repeat
for j in (n-i)..n repeat
qsetelt_!(v,j, qelt(v,j) + qelt(v,(j+1)))
ans : Pol := 0
for i in 0..n repeat
ans := ans + monomial(qelt(v,(i+1)),i)
ans
minus(F : Pol) ==
--computes Pol G such that G(x) = F(-x)
G : Pol := 0
while not (F = 0) repeat
n := degree(F)
coef := leadingCoefficient(F)
odd? n =>
G := G + monomial(-coef, n)
F := reductum(F)
G := G + monomial(coef, n)
F := reductum(F)
G
invert(F : Pol) ==
--computes Pol G such that G(x) = (x**n) * F(1/x)
G : Pol := 0
n := degree(F)
while not (F = 0) repeat
G := G + monomial(leadingCoefficient(F),
(n-degree(F))::NonNegativeInteger)
F := reductum(F)
G
var(F : Pol) == --number of sign variations in coefs of F
i : Integer := 0
LastCoef : Boolean
next : Boolean
LastCoef := leadingCoefficient(F) < 0
while not ((F := reductum(F)) = 0) repeat
next := leadingCoefficient(F) < 0
if ((not LastCoef) and next) or
((not next) and LastCoef) then i := i+1
LastCoef := next
i
refine(F : Pol, int : Interval, bounds : Interval) ==
lseg := min(int.right,bounds.right) - max(int.left,bounds.left)
lseg < 0 => "failed"
lseg = 0 =>
pt :=
int.left = bounds.right => int.left
int.right
elt(transMultInv(denom(pt),F),numer pt) = 0 => [pt,pt]
"failed"
lseg = int.right - int.left => int
refine(F, refine(F, int, lseg), bounds)
refine(F : Pol, int : Interval, eps : RN) ==
a := int.left
b := int.right
a=b => [a,b]$Interval
an : Integer := numer(a)
ad : Integer := denom(a)
bn : Integer := numer(b)
bd : Integer := denom(b)
xfl : Boolean := false
if (u:=elt(transMultInv(ad, F), an)) = 0 then
F := (F exquo (monomial(ad,1)-monomial(an,0)))::Pol
u:=elt(transMultInv(ad, F), an)
if (v:=elt(transMultInv(bd, F), bn)) = 0 then
F := (F exquo (monomial(bd,1)-monomial(bn,0)))::Pol
v:=elt(transMultInv(bd, F), bn)
u:=elt(transMultInv(ad, F), an)
if u > 0 then (F:=-F;v:=-v)
if v < 0 then
error [int, "is not a valid isolation interval for", F]
if eps <= 0 then error "precision must be positive"
while (b - a) >= eps repeat
mid : RN := (b + a) * (1/2)
midn : Integer := numer(mid)
midd : Integer := denom(mid)
(v := elt(transMultInv(midd, F), midn)) < 0 =>
a := mid
an := midn
ad := midd
v > 0 =>
b := mid
bn := midn
bd := midd
v = 0 =>
a := mid
b := mid
an := midn
ad := midd
xfl := true
[a, b]$Interval
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