/usr/share/axiom-20170501/src/algebra/QUATCAT.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 | )abbrev category QUATCAT QuaternionCategory
++ Author: Robert S. Sutor
++ Date Created: 23 May 1990
++ Change History: 10 September 1990
++ Description:
++ \spadtype{QuaternionCategory} describes the category of quaternions
++ and implements functions that are not representation specific.
QuaternionCategory(R) : Category == SIG where
R : CommutativeRing
AL ==> Algebra(R)
FRT ==> FullyRetractableTo(R)
DE ==> DifferentialExtension(R)
FEO ==> FullyEvalableOver(R)
FLERO ==> FullyLinearlyExplicitRingOver(R)
SIG ==> Join(AL,FRT,DE,FEO,FLERO) with
conjugate : $ -> $
++ conjugate(q) negates the imaginary parts of quaternion \spad{q}.
imagI : $ -> R
++ imagI(q) extracts the imaginary i part of quaternion \spad{q}.
imagJ : $ -> R
++ imagJ(q) extracts the imaginary j part of quaternion \spad{q}.
imagK : $ -> R
++ imagK(q) extracts the imaginary k part of quaternion \spad{q}.
norm : $ -> R
++ norm(q) computes the norm of \spad{q} (the sum of the
++ squares of the components).
quatern : (R,R,R,R) -> $
++ quatern(r,i,j,k) constructs a quaternion from scalars.
real : $ -> R
++ real(q) extracts the real part of quaternion \spad{q}.
if R has EntireRing then EntireRing
if R has OrderedSet then OrderedSet
if R has Field then DivisionRing
if R has ConvertibleTo InputForm then ConvertibleTo InputForm
if R has CharacteristicZero then CharacteristicZero
if R has CharacteristicNonZero then CharacteristicNonZero
if R has RealNumberSystem then
abs : $ -> R
++ abs(q) computes the absolute value of quaternion \spad{q}
++ (sqrt of norm).
if R has IntegerNumberSystem then
rational? : $ -> Boolean
++ rational?(q) returns {\it true} if all the imaginary
++ parts of \spad{q} are zero and the real part can be
++ converted into a rational number, and {\it false}
++ otherwise.
rational : $ -> Fraction Integer
++ rational(q) tries to convert \spad{q} into a
++ rational number. Error: if this is not
++ possible. If \spad{rational?(q)} is true, the
++ conversion will be done and the rational number returned.
rationalIfCan : $ -> Union(Fraction Integer, "failed")
++ rationalIfCan(q) returns \spad{q} as a rational number,
++ or "failed" if this is not possible.
++ Note that if \spad{rational?(q)} is true, the conversion
++ can be done and the rational number will be returned.
add
characteristic() ==
characteristic()$R
conjugate x ==
quatern(real x, - imagI x, - imagJ x, - imagK x)
map(fn, x) ==
quatern(fn real x, fn imagI x, fn imagJ x, fn imagK x)
norm x ==
real x * real x + imagI x * imagI x +
imagJ x * imagJ x + imagK x * imagK x
x = y ==
(real x = real y) and (imagI x = imagI y) and
(imagJ x = imagJ y) and (imagK x = imagK y)
x + y ==
quatern(real x + real y, imagI x + imagI y,
imagJ x + imagJ y, imagK x + imagK y)
x - y ==
quatern(real x - real y, imagI x - imagI y,
imagJ x - imagJ y, imagK x - imagK y)
- x ==
quatern(- real x, - imagI x, - imagJ x, - imagK x)
r:R * x:$ ==
quatern(r * real x, r * imagI x, r * imagJ x, r * imagK x)
n:Integer * x:$ ==
quatern(n * real x, n * imagI x, n * imagJ x, n * imagK x)
differentiate(x:$, d:R -> R) ==
quatern(d real x, d imagI x, d imagJ x, d imagK x)
coerce(r:R) ==
quatern(r,0$R,0$R,0$R)
coerce(n:Integer) ==
quatern(n :: R,0$R,0$R,0$R)
one? x ==
(real x) = 1 and zero? imagI x and
zero? imagJ x and zero? imagK x
zero? x ==
zero? real x and zero? imagI x and
zero? imagJ x and zero? imagK x
retract(x):R ==
not (zero? imagI x and zero? imagJ x and zero? imagK x) =>
error "Cannot retract quaternion."
real x
retractIfCan(x):Union(R,"failed") ==
not (zero? imagI x and zero? imagJ x and zero? imagK x) =>
"failed"
real x
coerce(x:$):OutputForm ==
part,z : OutputForm
y : $
zero? x => (0$R) :: OutputForm
not zero?(real x) =>
y := quatern(0$R,imagI(x),imagJ(x),imagK(x))
zero? y => real(x) :: OutputForm
(real(x) :: OutputForm) + (y :: OutputForm)
-- we know that the real part is 0
not zero?(imagI(x)) =>
y := quatern(0$R,0$R,imagJ(x),imagK(x))
z :=
part := "i"::Symbol::OutputForm
(imagI(x) = 1) => part
(imagI(x) :: OutputForm) * part
zero? y => z
z + (y :: OutputForm)
-- we know that the real part and i part are 0
not zero?(imagJ(x)) =>
y := quatern(0$R,0$R,0$R,imagK(x))
z :=
part := "j"::Symbol::OutputForm
(imagJ(x) = 1) => part
(imagJ(x) :: OutputForm) * part
zero? y => z
z + (y :: OutputForm)
-- we know that the real part and i and j parts are 0
part := "k"::Symbol::OutputForm
(imagK(x) = 1) => part
(imagK(x) :: OutputForm) * part
if R has Field then
inv x ==
norm x = 0 => error "This quaternion is not invertible."
(inv norm x) * conjugate x
if R has ConvertibleTo InputForm then
convert(x:$):InputForm ==
l : List InputForm := [convert("quatern" :: Symbol),
convert(real x)$R, convert(imagI x)$R, convert(imagJ x)$R,
convert(imagK x)$R]
convert(l)$InputForm
if R has OrderedSet then
x < y ==
real x = real y =>
imagI x = imagI y =>
imagJ x = imagJ y =>
imagK x < imagK y
imagJ x < imagJ y
imagI x < imagI y
real x < real y
if R has RealNumberSystem then
abs x == sqrt norm x
if R has IntegerNumberSystem then
rational? x ==
(zero? imagI x) and (zero? imagJ x) and (zero? imagK x)
rational x ==
rational? x => rational real x
error "Not a rational number"
rationalIfCan x ==
rational? x => rational real x
"failed"
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