/usr/share/axiom-20170501/src/algebra/SOLVEFOR.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 | )abbrev package SOLVEFOR PolynomialSolveByFormulas
++ Author: SMW June 86, BMT Sept 93
++ Description:
++ This package factors the formulas out of the general solve code,
++ allowing their recursive use over different domains.
++ Care is taken to introduce few radicals so that radical extension
++ domains can more easily simplify the results.
PolynomialSolveByFormulas(UP, F) : SIG == CODE where
UP : UnivariatePolynomialCategory F
F : Field with
"**": (%, Fraction Integer) -> %
L ==> List
SIG ==> with
solve : UP -> L F
++ solve(u) \undocumented
particularSolution : UP -> F
++ particularSolution(u) \undocumented
mapSolve : (UP, F -> F) -> Record(solns: L F,
maps: L Record(arg:F,res:F))
++ mapSolve(u,f) \undocumented
linear : UP -> L F
++ linear(u) \undocumented
quadratic : UP -> L F
++ quadratic(u) \undocumented
cubic : UP -> L F
++ cubic(u) \undocumented
quartic : UP -> L F
++ quartic(u) \undocumented
-- Arguments give coefs from high to low degree.
linear : (F, F) -> L F
++ linear(f,g) \undocumented
quadratic : (F, F, F) -> L F
++ quadratic(f,g,h) \undocumented
cubic : (F, F, F, F) -> L F
++ cubic(f,g,h,i) \undocumented
quartic : (F, F, F, F, F) -> L F
++ quartic(f,g,h,i,j) \undocumented
aLinear : (F, F) -> F
++ aLinear(f,g) \undocumented
aQuadratic : (F, F, F) -> F
++ aQuadratic(f,g,h) \undocumented
aCubic : (F, F, F, F) -> F
++ aCubic(f,g,h,j) \undocumented
aQuartic : (F, F, F, F, F) -> F
++ aQuartic(f,g,h,i,k) \undocumented
CODE ==> add
-----------------------------------------------------------------
-- Stuff for mapSolve
-----------------------------------------------------------------
id ==> (IDENTITY$Lisp)
maplist: List Record(arg: F, res: F) := []
mapSolving?: Boolean := false
-- map: F -> F := id #1 replaced with line below
map: Boolean := false
mapSolve(p, fn) ==
-- map := fn #1 replaced with line below
locmap: F -> F := x +-> fn x; map := id locmap
mapSolving? := true; maplist := []
slist := solve p
mapSolving? := false;
-- map := id #1 replaced with line below
locmap := x +-> id x; map := id locmap
[slist, maplist]
part(s: F): F ==
not mapSolving? => s
-- t := map s replaced with line below
t: F := SPADCALL(s, map)$Lisp
t = s => s
maplist := cons([t, s], maplist)
t
-----------------------------------------------------------------
-- Entry points and error handling
-----------------------------------------------------------------
cc ==> coefficient
-- local intsolve
intsolve(u:UP):L(F) ==
u := (factors squareFree u).1.factor
n := degree u
n=1 => linear (cc(u,1), cc(u,0))
n=2 => quadratic (cc(u,2), cc(u,1), cc(u,0))
n=3 => cubic (cc(u,3), cc(u,2), cc(u,1), cc(u,0))
n=4 => quartic (cc(u,4), cc(u,3), cc(u,2), cc(u,1), cc(u,0))
error "All sqfr factors of polynomial must be of degree < 5"
solve u ==
ls := nil$L(F)
for f in factors squareFree u repeat
lsf := intsolve f.factor
for i in 1..(f.exponent) repeat ls := [:lsf,:ls]
ls
particularSolution u ==
u := (factors squareFree u).1.factor
n := degree u
n=1 => aLinear (cc(u,1), cc(u,0))
n=2 => aQuadratic (cc(u,2), cc(u,1), cc(u,0))
n=3 => aCubic (cc(u,3), cc(u,2), cc(u,1), cc(u,0))
n=4 => aQuartic (cc(u,4), cc(u,3), cc(u,2), cc(u,1), cc(u,0))
error "All sqfr factors of polynomial must be of degree < 5"
needDegree(n: Integer, u: UP): Boolean ==
degree u = n => true
error concat("Polynomial must be of degree ", n::String)
needLcoef(cn: F): Boolean ==
cn ^= 0 => true
error "Leading coefficient must not be 0."
needChar0(): Boolean ==
characteristic()$F = 0 => true
error "Formula defined only for fields of characteristic 0."
linear u ==
needDegree(1, u)
linear (coefficient(u,1), coefficient(u,0))
quadratic u ==
needDegree(2, u)
quadratic (coefficient(u,2), coefficient(u,1),
coefficient(u,0))
cubic u ==
needDegree(3, u)
cubic (coefficient(u,3), coefficient(u,2),
coefficient(u,1), coefficient(u,0))
quartic u ==
needDegree(4, u)
quartic (coefficient(u,4),coefficient(u,3),
coefficient(u,2),coefficient(u,1),coefficient(u,0))
-----------------------------------------------------------------
-- The formulas
-----------------------------------------------------------------
-- local function for testing equality of radicals.
-- This function is necessary to detect at least some of the
-- situations like sqrt(9)-3 = 0 --> false.
equ(x:F,y:F):Boolean ==
( (recip(x-y)) case "failed" ) => true
false
linear(c1, c0) ==
needLcoef c1
[- c0/c1 ]
aLinear(c1, c0) ==
first linear(c1,c0)
quadratic(c2, c1, c0) ==
needLcoef c2; needChar0()
(c0 = 0) => [0$F,:linear(c2, c1)]
(c1 = 0) => [(-c0/c2)**(1/2),-(-c0/c2)**(1/2)]
D := part(c1**2 - 4*c2*c0)**(1/2)
[(-c1+D)/(2*c2), (-c1-D)/(2*c2)]
aQuadratic(c2, c1, c0) ==
needLcoef c2; needChar0()
(c0 = 0) => 0$F
(c1 = 0) => (-c0/c2)**(1/2)
D := part(c1**2 - 4*c2*c0)**(1/2)
(-c1+D)/(2*c2)
w3: F := (-1 + (-3::F)**(1/2)) / 2::F
cubic(c3, c2, c1, c0) ==
needLcoef c3; needChar0()
-- case one root = 0, not necessary but keeps result small
(c0 = 0) => [0$F,:quadratic(c3, c2, c1)]
a1 := c2/c3; a2 := c1/c3; a3 := c0/c3
-- case x**3-a3 = 0, not necessary but keeps result small
(a1 = 0 and a2 = 0) =>
[ u*(-a3)**(1/3) for u in [1, w3, w3**2 ] ]
-- case x**3 + a1*x**2 + a1**2*x/3 + a3 = 0, the general for-
-- mula is not valid in this case, but solution is easy.
P := part(-a1/3::F)
equ(a1**2,3*a2) =>
S := part((- a3 + (a1**3)/27::F)**(1/3))
[ P + S*u for u in [1,w3,w3**2] ]
-- general case
Q := part((3*a2 - a1**2)/9::F)
R := part((9*a1*a2 - 27*a3 - 2*a1**3)/54::F)
D := part(Q**3 + R**2)**(1/2)
S := part(R + D)**(1/3)
-- S = 0 is done in the previous case
[ P + S*u - Q/(S*u) for u in [1, w3, w3**2] ]
aCubic(c3, c2, c1, c0) ==
needLcoef c3; needChar0()
(c0 = 0) => 0$F
a1 := c2/c3; a2 := c1/c3; a3 := c0/c3
(a1 = 0 and a2 = 0) => (-a3)**(1/3)
P := part(-a1/3::F)
equ(a1**2,3*a2) =>
S := part((- a3 + (a1**3)/27::F)**(1/3))
P + S
Q := part((3*a2 - a1**2)/9::F)
R := part((9*a1*a2 - 27*a3 - 2*a1**3)/54::F)
D := part(Q**3 + R**2)**(1/2)
S := part(R + D)**(1/3)
P + S - Q/S
quartic(c4, c3, c2, c1, c0) ==
needLcoef c4; needChar0()
-- case one root = 0, not necessary but keeps result small
(c0 = 0) => [0$F,:cubic(c4, c3, c2, c1)]
-- Make monic:
a1 := c3/c4; a2 := c2/c4; a3 := c1/c4; a4 := c0/c4
-- case x**4 + a4 = 0 <=> (x**2-sqrt(-a4))*(x**2+sqrt(-a4))
-- not necessary but keeps result small.
(a1 = 0 and a2 = 0 and a3 = 0) =>
append( quadratic(1, 0, (-a4)**(1/2)),_
quadratic(1 ,0, -((-a4)**(1/2))) )
-- Translate w = x+a1/4 to eliminate a1: w**4+p*w**2+q*w+r
p := part(a2-3*a1*a1/8::F)
q := part(a3-a1*a2/2::F + a1**3/8::F)
r := part(a4-a1*a3/4::F + a1**2*a2/16::F - 3*a1**4/256::F)
-- t0 := the cubic resolvent of x**3-p*x**2-4*r*x+4*p*r-q**2
-- The roots of the translated polynomial are those of
-- two quadratics. (What about rt=0 ?)
-- rt=0 can be avoided by picking a root ^= p of the cubic
-- polynomial above. This is always possible provided that
-- the input is squarefree. In this case the two other roots
-- are +(-) 2*r**(1/2).
if equ(q,0) -- this means p is a root
then t0 := part(2*(r**(1/2)))
else t0 := aCubic(1, -p, -4*r, 4*p*r - q**2)
rt := part(t0 - p)**(1/2)
slist := append( quadratic( 1, rt, (-q/rt + t0)/2::F ),
quadratic( 1, -rt, ( q/rt + t0)/2::F ))
-- Translate back:
[s - a1/4::F for s in slist]
aQuartic(c4, c3, c2, c1, c0) ==
needLcoef c4; needChar0()
(c0 = 0) => 0$F
a1 := c3/c4; a2 := c2/c4; a3 := c1/c4; a4 := c0/c4
(a1 = 0 and a2 = 0 and a3 = 0) => (-a4)**(1/4)
p := part(a2-3*a1*a1/8::F)
q := part(a3-a1*a2/2::F + a1**2*a1/8::F)
r := part(a4-a1*a3/4::F + a1**2*a2/16::F - 3*a1**4/256::F)
if equ(q,0)
then t0 := part(2*(r**(1/2)))
else t0 := aCubic(1, -p, -4*r, 4*p*r - q**2)
rt := part(t0 - p)**(1/2)
s := aQuadratic( 1, rt, (-q/rt + t0)/2::F )
s - a1/4::F
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