/usr/share/axiom-20170501/src/algebra/CLIF.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 | )abbrev domain CLIF CliffordAlgebra
++ Author: Stephen M. Watt
++ Date Created: August 1988
++ Date Last Updated: May 17, 1991
++ Description:
++ CliffordAlgebra(n, K, Q) defines a vector space of dimension \spad{2**n}
++ over K, given a quadratic form Q on \spad{K**n}.
++
++ If \spad{e[i]}, \spad{1<=i<=n} is a basis for \spad{K**n} then
++ 1, \spad{e[i]} (\spad{1<=i<=n}), \spad{e[i1]*e[i2]}
++ (\spad{1<=i1<i2<=n}),...,\spad{e[1]*e[2]*..*e[n]}
++ is a basis for the Clifford Algebra.
++
++ The algebra is defined by the relations\br
++ \tab{5}\spad{e[i]*e[j] = -e[j]*e[i]} (\spad{i \~~= j}),\br
++ \tab{5}\spad{e[i]*e[i] = Q(e[i])}
++
++ Examples of Clifford Algebras are: gaussians, quaternions, exterior
++ algebras and spin algebras.
CliffordAlgebra(n, K, Q) : SIG == CODE where
n : PositiveInteger
K : Field
Q : QuadraticForm(n, K)
PI ==> PositiveInteger
NNI==> NonNegativeInteger
SIG ==> Join(Ring, Algebra(K), VectorSpace(K)) with
e : PI -> %
++ e(n) produces the appropriate unit element.
monomial : (K, List PI) -> %
++ monomial(c,[i1,i2,...,iN]) produces the value given by
++ \spad{c*e(i1)*e(i2)*...*e(iN)}.
coefficient : (%, List PI) -> K
++ coefficient(x,[i1,i2,...,iN]) extracts the coefficient of
++ \spad{e(i1)*e(i2)*...*e(iN)} in x.
recip : % -> Union(%, "failed")
++ recip(x) computes the multiplicative inverse of x or "failed"
++ if x is not invertible.
CODE ==> add
Qeelist := [Q unitVector(i::PositiveInteger) for i in 1..n]
dim := 2**n
Rep := PrimitiveArray K
New ==> new(dim, 0$K)$Rep
x, y, z: %
c: K
m: Integer
characteristic() == characteristic()$K
dimension() == dim::CardinalNumber
x = y ==
for i in 0..dim-1 repeat
if x.i ^= y.i then return false
true
x + y == (z := New; for i in 0..dim-1 repeat z.i := x.i + y.i; z)
x - y == (z := New; for i in 0..dim-1 repeat z.i := x.i - y.i; z)
- x == (z := New; for i in 0..dim-1 repeat z.i := - x.i; z)
m * x == (z := New; for i in 0..dim-1 repeat z.i := m*x.i; z)
c * x == (z := New; for i in 0..dim-1 repeat z.i := c*x.i; z)
0 == New
1 == (z := New; z.0 := 1; z)
coerce(m): % == (z := New; z.0 := m::K; z)
coerce(c): % == (z := New; z.0 := c; z)
e b ==
b::NNI > n => error "No such basis element"
iz := 2**((b-1)::NNI)
z := New; z.iz := 1; z
-- The ei*ej products could instead be precomputed in
-- a (2**n)**2 multiplication table.
addMonomProd(c1: K, b1: NNI, c2: K, b2: NNI, z: %): % ==
c := c1 * c2
bz := b2
for i in 0..n-1 | bit?(b1,i) repeat
-- Apply rule ei*ej = -ej*ei for i^=j
k := 0
for j in i+1..n-1 | bit?(b1, j) repeat k := k+1
for j in 0..i-1 | bit?(bz, j) repeat k := k+1
if odd? k then c := -c
-- Apply rule ei**2 = Q(ei)
if bit?(bz,i) then
c := c * Qeelist.(i+1)
bz:= (bz - 2**i)::NNI
else
bz:= bz + 2**i
z.bz := z.bz + c
z
x * y ==
z := New
for ix in 0..dim-1 repeat
if x.ix ^= 0 then for iy in 0..dim-1 repeat
if y.iy ^= 0 then addMonomProd(x.ix,ix,y.iy,iy,z)
z
canonMonom(c: K, lb: List PI): Record(coef: K, basel: NNI) ==
-- 0. Check input
for b in lb repeat b > n => error "No such basis element"
-- 1. Apply identity ei*ej = -ej*ei, i^=j.
-- The Rep assumes n is small so bubble sort is ok.
-- Using bubble sort keeps the exchange info obvious.
wasordered := false
exchanges := 0
while not wasordered repeat
wasordered := true
for i in 1..#lb-1 repeat
if lb.i > lb.(i+1) then
t := lb.i; lb.i := lb.(i+1); lb.(i+1) := t
exchanges := exchanges + 1
wasordered := false
if odd? exchanges then c := -c
-- 2. Prepare the basis element
-- Apply identity ei*ei = Q(ei).
bz := 0
for b in lb repeat
bn := (b-1)::NNI
if bit?(bz, bn) then
c := c * Qeelist bn
bz:= ( bz - 2**bn )::NNI
else
bz:= bz + 2**bn
[c, bz::NNI]
monomial(c, lb) ==
r := canonMonom(c, lb)
z := New
z r.basel := r.coef
z
coefficient(z, lb) ==
r := canonMonom(1, lb)
r.coef = 0 => error "Cannot take coef of 0"
z r.basel/r.coef
Ex ==> OutputForm
coerceMonom(c: K, b: NNI): Ex ==
b = 0 => c::Ex
ml := [sub("e"::Ex, i::Ex) for i in 1..n | bit?(b,i-1)]
be := reduce("*", ml)
c = 1 => be
c::Ex * be
coerce(x): Ex ==
tl := [coerceMonom(x.i,i) for i in 0..dim-1 | x.i^=0]
null tl => "0"::Ex
reduce("+", tl)
localPowerSets(j:NNI): List(List(PI)) ==
l: List List PI := list []
j = 0 => l
Sm := localPowerSets((j-1)::NNI)
Sn: List List PI := []
for x in Sm repeat Sn := cons(cons(j pretend PI, x),Sn)
append(Sn, Sm)
powerSets(j:NNI):List List PI == map(reverse, localPowerSets j)
Pn:List List PI := powerSets(n)
recip(x: %): Union(%, "failed") ==
one:% := 1
-- tmp:c := x*yC - 1$C
rhsEqs : List K := []
lhsEqs: List List K := []
lhsEqi: List K
for pi in Pn repeat
rhsEqs := cons(coefficient(one, pi), rhsEqs)
lhsEqi := []
for pj in Pn repeat
lhsEqi := cons(coefficient(x*monomial(1,pj),pi),lhsEqi)
lhsEqs := cons(reverse(lhsEqi),lhsEqs)
ans := particularSolution(matrix(lhsEqs),vector(rhsEqs)_
)$LinearSystemMatrixPackage(K, Vector K, Vector K, Matrix K)
ans case "failed" => "failed"
ansP := parts(ans)
ansC:% := 0
for pj in Pn repeat
cj:= first ansP
ansP := rest ansP
ansC := ansC + cj*monomial(1,pj)
ansC
|