forked from fricas/fricas
-
Notifications
You must be signed in to change notification settings - Fork 0
/
oct.spad
382 lines (363 loc) · 14.6 KB
/
oct.spad
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
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
)abbrev category OC OctonionCategory
++ Author: R. Wisbauer, J. Grabmeier
++ Date Created: 05 September 1990
++ Basic Operations: _+, _*, octon, real, imagi, imagj, imagk,
++ imagE, imagI, imagJ, imagK
++ Related Constructors: QuaternionCategory
++ Also See:
++ AMS Classifications:
++ Keywords: octonion, non-associative algebra, Cayley-Dixon
++ References: e.g. I.L Kantor, A.S. Solodovnikov:
++ Hypercomplex Numbers, Springer Verlag Heidelberg, 1989,
++ ISBN 0-387-96980-2
++ Description:
++ OctonionCategory gives the categorial frame for the
++ octonions, and eight-dimensional non-associative algebra,
++ doubling the quaternions in the same way as doubling
++ the Complex numbers to get the quaternions.
-- Examples: octonion.input
OctonionCategory(R : CommutativeRing) : Category ==
Join(FramedNonAssociativeAlgebra(R), FullyRetractableTo R,
FullyEvalableOver R) with
conjugate : % -> %
++ conjugate(o) negates the imaginary parts i, j, k, E, I, J, K
++ of octonian o.
real : % -> R
++ real(o) extracts real part of octonion o.
imagi : % -> R
++ imagi(o) extracts the i part of octonion o.
imagj : % -> R
++ imagj(o) extracts the j part of octonion o.
imagk : % -> R
++ imagk(o) extracts the k part of octonion o.
imagE : % -> R
++ imagE(o) extracts the imaginary E part of octonion o.
imagI : % -> R
++ imagI(o) extracts the imaginary I part of octonion o.
imagJ : % -> R
++ imagJ(o) extracts the imaginary J part of octonion o.
imagK : % -> R
++ imagK(o) extracts the imaginary K part of octonion o.
norm : % -> R
++ norm(o) returns the norm of an octonion, equal to
++ the sum of the squares
++ of its coefficients.
octon : (R, R, R, R, R, R, R, R) -> %
++ octon(re, ri, rj, rk, rE, rI, rJ, rK) constructs an octonion
++ from scalars.
-- FIXME: Why we impose order here?
if R has OrderedSet then OrderedSet
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(o) computes the absolute value of an octonion, equal to
++ the square root of the \spadfunFrom{norm}{Octonion}.
if R has IntegerNumberSystem then
rational? : % -> Boolean
++ rational?(o) tests if o is rational, i.e. that all seven
++ imaginary parts are 0.
rational : % -> Fraction Integer
++ rational(o) returns the real part if all seven
++ imaginary parts are 0.
++ Error: if o is not rational.
rationalIfCan : % -> Union(Fraction Integer, "failed")
++ rationalIfCan(o) returns the real part if
++ all seven imaginary parts are 0, and "failed" otherwise.
if R has Field then
inv : % -> %
++ inv(o) returns the inverse of o if it exists.
add
characteristic() ==
characteristic()$R
rank() == 8
coordinates(x : %) : Vector(R) ==
vector([real x, imagi x, imagj x, imagk x,
imagE x, imagI x, imagJ x, imagK x])
represents(v : Vector(R)) ==
octon(v(1), v(2), v(3), v(4), v(5), v(6), v(7), v(8))
basis() : Vector(%) ==
res := zero(8)$Vector(%)
vv := zero(8)$Vector(R)
for i in 1..8 repeat
vv(i) := 1
res(i) := represents(vv)
vv(i) := 0
res
conjugate x ==
octon(real x, - imagi x, - imagj x, - imagk x, - imagE x, _
- imagI x, - imagJ x, - imagK x)
map(fn, x) ==
octon(fn real x, fn imagi x, fn imagj x, fn imagk x, fn imagE x, _
fn imagI x, fn imagJ x, fn imagK x)
norm x ==
real x * real x + imagi x * imagi x + _
imagj x * imagj x + imagk x * imagk x + _
imagE x * imagE x + imagI x * imagI x + _
imagJ x * imagJ x + imagK x * imagK x
x = y ==
(real x = real y) and (imagi x = imagi y) and _
(imagj x = imagj y) and (imagk x = imagk y) and _
(imagE x = imagE y) and (imagI x = imagI y) and _
(imagJ x = imagJ y) and (imagK x = imagK y)
x + y ==
octon(real x + real y, imagi x + imagi y, _
imagj x + imagj y, imagk x + imagk y, _
imagE x + imagE y, imagI x + imagI y, _
imagJ x + imagJ y, imagK x + imagK y)
- x ==
octon(- real x, - imagi x, - imagj x, - imagk x, _
- imagE x, - imagI x, - imagJ x, - imagK x)
r : R * x : % ==
octon(r * real x, r * imagi x, r * imagj x, r * imagk x, _
r * imagE x, r * imagI x, r * imagJ x, r * imagK x)
n : Integer * x : % ==
octon(n * real x, n * imagi x, n * imagj x, n * imagk x, _
n * imagE x, n * imagI x, n * imagJ x, n * imagK x)
coerce(r : R) ==
octon(r, 0$R, 0$R, 0$R, 0$R, 0$R, 0$R, 0$R)
coerce(n : Integer) ==
octon(n :: R, 0$R, 0$R, 0$R, 0$R, 0$R, 0$R, 0$R)
zero? x ==
zero? real x and zero? imagi x and _
zero? imagj x and zero? imagk x and _
zero? imagE x and zero? imagI x and _
zero? imagJ x and zero? imagK x
retract(x) : R ==
not (zero? imagi x and zero? imagj x and zero? imagk x and _
zero? imagE x and zero? imagI x and zero? imagJ x and zero? imagK x) =>
error "Cannot retract octonion."
real x
retractIfCan(x):Union(R,"failed") ==
not (zero? imagi x and zero? imagj x and zero? imagk x and _
zero? imagE x and zero? imagI x and zero? imagJ x and zero? imagK x) =>
"failed"
real x
coerce(x : %) : OutputForm ==
part, z : OutputForm
y : %
zero? x => (0$R) :: OutputForm
not zero?(real x) =>
y := octon(0$R, imagi(x), imagj(x), imagk(x), imagE(x),
imagI(x), imagJ(x), imagK(x))
zero? y => real(x) :: OutputForm
(real(x) :: OutputForm) + (y :: OutputForm)
-- we know that the real part is 0
not zero?(imagi(x)) =>
y := octon(0$R, 0$R, imagj(x), imagk(x), imagE(x),
imagI(x), imagJ(x), imagK(x))
z :=
part := 'i::OutputForm
(imagi(x) = 1) => part
(imagi(x) :: OutputForm) * part
zero? y => z
z + (y :: OutputForm)
-- we know that the real part and i part are 0
not zero?(imagj(x)) =>
y := octon(0$R, 0$R, 0$R, imagk(x), imagE(x),
imagI(x), imagJ(x), imagK(x))
z :=
part := 'j::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
not zero?(imagk(x)) =>
y := octon(0$R, 0$R, 0$R, 0$R, imagE(x),
imagI(x), imagJ(x), imagK(x))
z :=
part := 'k::OutputForm
(imagk(x) = 1) => part
(imagk(x) :: OutputForm) * part
zero? y => z
z + (y :: OutputForm)
-- we know that the real part, i, j, k parts are 0
not zero?(imagE(x)) =>
y := octon(0$R, 0$R, 0$R, 0$R, 0$R,
imagI(x), imagJ(x), imagK(x))
z :=
part := 'E::OutputForm
(imagE(x) = 1) => part
(imagE(x) :: OutputForm) * part
zero? y => z
z + (y :: OutputForm)
-- we know that the real part, i, j, k, E parts are 0
not zero?(imagI(x)) =>
y := octon(0$R, 0$R, 0$R, 0$R, 0$R, 0$R, imagJ(x), imagK(x))
z :=
part := 'I::OutputForm
(imagI(x) = 1) => part
(imagI(x) :: OutputForm) * part
zero? y => z
z + (y :: OutputForm)
-- we know that the real part, i, j, k, E, I parts are 0
not zero?(imagJ(x)) =>
y := octon(0$R, 0$R, 0$R, 0$R, 0$R, 0$R, 0$R, imagK(x))
z :=
part := 'J::OutputForm
(imagJ(x) = 1) => part
(imagJ(x) :: OutputForm) * part
zero? y => z
z + (y :: OutputForm)
-- we know that the real part, i, j, k, E, I, J parts are 0
part := 'K::OutputForm
(imagK(x) = 1) => part
(imagK(x) :: OutputForm) * part
if R has Field then
inv x ==
(norm x) = 0 => error "This octonion is not invertible."
(inv norm x) * conjugate x
if R has ConvertibleTo InputForm then
convert(x : %) : InputForm ==
l : List InputForm := [convert('octon),
convert(real x)$R, convert(imagi x)$R, convert(imagj x)$R, _
convert(imagk x)$R, convert(imagE 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 =>
imagE x = imagE y =>
imagI x = imagI y =>
imagJ x = imagJ y =>
imagK x < imagK y
imagJ x < imagJ y
imagI x < imagI y
imagE x < imagE 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) and _
(zero? imagE x) and (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"
)abbrev domain OCT Octonion
++ Author: R. Wisbauer, J. Grabmeier
++ Date Created: 05 September 1990
++ Basic Operations: _+, _*, octon, image, imagi, imagj, imagk,
++ imagE, imagI, imagJ, imagK
++ Related Constructors: Quaternion
++ Also See: AlgebraGivenByStructuralConstants
++ AMS Classifications:
++ Keywords: octonion, non-associative algebra, Cayley-Dixon
++ References: e.g. I.L Kantor, A.S. Solodovnikov:
++ Hypercomplex Numbers, Springer Verlag Heidelberg, 1989,
++ ISBN 0-387-96980-2
++ Description:
++ Octonion implements octonions (Cayley-Dixon algebra) over a
++ commutative ring, an eight-dimensional non-associative
++ algebra, doubling the quaternions in the same way as doubling
++ the complex numbers to get the quaternions
++ the main constructor function is {\em octon} which takes 8
++ arguments: the real part, the i imaginary part, the j
++ imaginary part, the k imaginary part, (as with quaternions)
++ and in addition the imaginary parts E, I, J, K.
-- Examples: octonion.input
Octonion(R : CommutativeRing) : Export == Impl where
QR ==> Quaternion R
Export ==> Join(OctonionCategory R, FullyRetractableTo QR) with
octon : (QR, QR) -> %
++ octon(qe, qE) constructs an octonion from two quaternions
++ using the relation {\em O = Q + QE}.
Impl ==> add
Rep := Record(e : QR, E : QR)
0 == [0, 0]
1 == [1, 0]
a, b, c, d, f, g, h, i : R
p, q : QR
x, y : %
real x == real (x.e)
imagi x == imagI (x.e)
imagj x == imagJ (x.e)
imagk x == imagK (x.e)
imagE x == real (x.E)
imagI x == imagI (x.E)
imagJ x == imagJ (x.E)
imagK x == imagK (x.E)
octon(a, b, c, d, f, g, h, i) == [quatern(a, b, c, d)$QR, quatern(f, g, h, i)$QR]
octon(p, q) == [p, q]
coerce(q) == [q, 0$QR]
retract(x) : QR ==
not(zero? imagE x and zero? imagI x and zero? imagJ x and zero? imagK x)=>
error "Cannot retract octonion to quaternion."
quatern(real x, imagi x, imagj x, imagk x)$QR
retractIfCan(x):Union(QR,"failed") ==
not(zero? imagE x and zero? imagI x and zero? imagJ x and zero? imagK x)=>
"failed"
quatern(real x, imagi x, imagj x, imagk x)$QR
x * y == [x.e*y.e-(conjugate y.E)*x.E, y.E*x.e + x.E*(conjugate y.e)]
)abbrev package OCTCT2 OctonionCategoryFunctions2
--% OctonionCategoryFunctions2
++ Author: Johannes Grabmeier
++ Date Created: 10 September 1990
++ Basic Operations: map
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: octonion, non-associative algebra, Cayley-Dixon
++ References:
++ Description:
++ OctonionCategoryFunctions2 implements functions between
++ two octonion domains defined over different rings.
++ The function map is used
++ to coerce between octonion types.
OctonionCategoryFunctions2(OR, R, OS, S) : Exports ==
Implementation where
R : CommutativeRing
S : CommutativeRing
OR : OctonionCategory R
OS : OctonionCategory S
Exports == with
map : (R -> S, OR) -> OS
++ map(f, u) maps f onto the component parts of the octonion
++ u.
Implementation == add
map(fn : R -> S, u : OR) : OS ==
octon(fn real u, fn imagi u, fn imagj u, fn imagk u, _
fn imagE u, fn imagI u, fn imagJ u, fn imagK u)$OS
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.