Axiom/FriCAS Clifford Algebra - Associative Tests

Here are some tests to make sure that the code is Associative:

Results

Check for associativity:

(1) -> Pf := PrimeField(17)                                                  

   (1)  PrimeField(17)
                                                                 Type: Domain
(2) -> bL := matrix([[1,1],[0,1]])$SquareMatrix(2,PrimeField(17))            

        +1  1+
   (2)  |    |
        +0  1+
                                         Type: SquareMatrix(2,PrimeField(17))
(3) -> Ca := CliffordAlgebra(2,Pf,bL)

   (3)  CliffordAlgebra(2,PrimeField(17),[[1,1],[0,1]])
                                                                 Type: Domain
(4) -> randc() == reduce('+, [random()$Pf*eFromBinaryMap(i)$Ca for i in 0..3])
                                                                   Type: Void
(5) -> a := randc()
   Compiling function randc with type () -> CliffordAlgebra(2,
      PrimeField(17),[[1,1],[0,1]])

   (5)  10 + 3e  + e  + 4e e
               1    2     1 2
                        Type: CliffordAlgebra(2,PrimeField(17),[[1,1],[0,1]])
(6) -> b := randc()

   (6)  15 + 2e  + 16e  + 2e e
               1      2     1 2
                        Type: CliffordAlgebra(2,PrimeField(17),[[1,1],[0,1]])
(7) -> c := randc()

   (7)  12 + 6e  + 9e e
               1     1 2
                        Type: CliffordAlgebra(2,PrimeField(17),[[1,1],[0,1]])
(8) -> (a*b)*c - a*(b*c)

   (8)  0
                        Type: CliffordAlgebra(2,PrimeField(17),[[1,1],[0,1]])

Results 2

compare different methods for orthogonal case:

(3) -> Pf := PrimeField(17)

   (3)  PrimeField(17)
                                                                 Type: Domain
(4) -> bL := matrix([[1, 1, 0], [1, 1, 1], [0, 1, 1]])$SquareMatrix(3, PrimeField(17))

        +1  1  0+
        |       |
   (4)  |1  1  1|
        |       |
        +0  1  1+
                                         Type: SquareMatrix(3,PrimeField(17))
(5) -> Ca := CliffordAlgebra(3, Pf, bL)                                      

   (5)  CliffordAlgebra(3,PrimeField(17),[[1,1,0],[1,1,1],[0,1,1]])
                                                                 Type: Domain
(6) -> randc() == reduce('+, [random()$Pf*eFromBinaryMap(i)$Ca for i in 0..7])
                                                                   Type: Void 
(7) -> a := randc()                                                           
   Compiling function randc with type () -> CliffordAlgebra(3,                
      PrimeField(17),[[1,1,0],[1,1,1],[0,1,1]])                               

   (7)  10 + 3e  + e  + 4e e  + 15e  + 2e e  + 16e e  + 2e e e
               1    2     1 2      3     1 3      2 3     1 2 3
            Type: CliffordAlgebra(3,PrimeField(17),[[1,1,0],[1,1,1],[0,1,1]])
(8) -> b := randc()                                                          

   (8)  12 + 6e  + 9e e  + 9e  + 10e e  + 4e e
               1     1 2     3      1 3     2 3
            Type: CliffordAlgebra(3,PrimeField(17),[[1,1,0],[1,1,1],[0,1,1]])
(9) -> c := randc()                                                          

   (9)  4 + 3e  + 10e  + 2e e  + 2e  + 11e e  + 7e e  + 16e e e
              1      2     1 2     3      1 3     2 3      1 2 3
            Type: CliffordAlgebra(3,PrimeField(17),[[1,1,0],[1,1,1],[0,1,1]])
(10) -> (a*b)*c - a*(b*c)                                                    

   (10)  0
            Type: CliffordAlgebra(3,PrimeField(17),[[1,1,0],[1,1,1],[0,1,1]])
(11) -> a := randc()

   (11)  14 + 11e  + 13e  + 12e e  + 16e  + 8e e  + 5e e  + 8e e e
                 1      2      1 2      3     1 3     2 3     1 2 3
            Type: CliffordAlgebra(3,PrimeField(17),[[1,1,0],[1,1,1],[0,1,1]])
(12) -> b := randc()                                                         

   (12)  10 + 4e  + 10e  + 11e e  + 8e  + 15e e  + 14e e  + 6e e e
                1      2      1 2     3      1 3      2 3     1 2 3
            Type: CliffordAlgebra(3,PrimeField(17),[[1,1,0],[1,1,1],[0,1,1]])
(13) -> c := randc()                                                         

   (13)  15 + 2e  + 5e  + 4e e  + 15e  + 13e e  + 5e e  + 16e e e
                1     2     1 2      3      1 3     2 3      1 2 3
            Type: CliffordAlgebra(3,PrimeField(17),[[1,1,0],[1,1,1],[0,1,1]])
(14) -> (a*b)*c - a*(b*c)                                                    

   (14)  0
            Type: CliffordAlgebra(3,PrimeField(17),[[1,1,0],[1,1,1],[0,1,1]])
(15) -> B1 := CliffordAlgebra(3,Fraction(Integer),[[1,0,0],[0,1,0],[0,0,1]]) 

   (15)  CliffordAlgebra(3,Fraction(Integer),[[1,0,0],[0,1,0],[0,0,1]])
                                                                 Type: Domain
(16) -> m1 := toTable(*)$B1                                                  

         +  1       e        e       e e      e       e e      e e    e e e +
         |           1        2       1 2      3       1 3      2 3    1 2 3|
         |                                                                  |
         |  e       1       e e       e      e e       e      e e e    e e  |
         |   1               1 2       2      1 3       3      1 2 3    2 3 |
         |                                                                  |
         |  e     - e e      1       - e     e e    - e e e     e     - e e |
         |   2       1 2                1     2 3      1 2 3     3       1 3|
         |                                                                  |
         | e e     - e       e       - 1    e e e    - e e     e e     - e  |
         |  1 2       2       1              1 2 3      2 3     1 3       3 |
   (16)  |                                                                  |
         |  e     - e e    - e e    e e e     1       - e      - e     e e  |
         |   3       1 3      2 3    1 2 3               1        2     1 2 |
         |                                                                  |
         | e e     - e    - e e e    e e      e       - 1     - e e     e   |
         |  1 3       3      1 2 3    2 3      1                 1 2     2  |
         |                                                                  |
         | e e    e e e     - e     - e e     e       e e      - 1     - e  |
         |  2 3    1 2 3       3       1 3     2       1 2                1 |
         |                                                                  |
         |e e e    e e     - e e     - e     e e       e       - e     - 1  |
         + 1 2 3    2 3       1 3       3     1 2       2         1         +
 Type: Matrix(CliffordAlgebra(3,Fraction(Integer),[[1,0,0],[0,1,0],[0,0,1]]))
(17) -> setMode("orthogonal",false)$B1                                       

   (17)  true
                                                                Type: Boolean
(18) -> m1 - toTable(*)$B1                                                   

         +0  0  0  0  0  0  0  0+
         |                      |
         |0  0  0  0  0  0  0  0|
         |                      |
         |0  0  0  0  0  0  0  0|
         |                      |
         |0  0  0  0  0  0  0  0|
   (18)  |                      |
         |0  0  0  0  0  0  0  0|
         |                      |
         |0  0  0  0  0  0  0  0|
         |                      |
         |0  0  0  0  0  0  0  0|
         |                      |
         +0  0  0  0  0  0  0  0+
 Type: Matrix(CliffordAlgebra(3,Fraction(Integer),[[1,0,0],[0,1,0],[0,0,1]]))
(19) -> setMode("debug",true)$B1

   (19)  true
                                                                Type: Boolean
(20) -> toTable(*)$B1
cliffordProdTerm: e12*e12=e1*-e1-0*e12=-11
cliffordProdTerm: e12*e13=e1*-e123-0*e13=-e23
cliffordProdTerm: e12*e23=e1*e3-0*e23=e13
cliffordProdTerm: e12*e123=e1*-e13-0*e123=-e3
cliffordProdTerm: e13*e12=e1*e123-0*e12=e23
cliffordProdTerm: e13*e13=e1*-e1-0*e13=-11
cliffordProdTerm: e13*e23=e1*-e2-0*e23=-e12
cliffordProdTerm: e13*e123=e1*e12-0*e123=e2
cliffordProdTerm: e23*e12=e2*e123-0*e12=-e13
cliffordProdTerm: e23*e13=e2*-e1-0*e13=e12
cliffordProdTerm: e23*e23=e2*-e2-0*e23=-11
cliffordProdTerm: e23*e123=e2*e12-0*e123=-e1
rcProdTerm: e12Le1= e1L-e1/\e2+0=-e2
rcProdTerm: e12Le2= e1L-e2/\e2+e1=e1
rcProdTerm: e12Le3= e1L-e3/\e2+0=0
cliffordProdTerm: e12*e123=e1*-e13-0*e123=-e3
cliffordProdTerm: e123*e12=e12*e123-0*e12=-e3
rcProdTerm: e12Le3= e1L-e3/\e2+0=0
rcProdTerm: e12Le3= e1L-e3/\e2+0=0
cliffordProdTerm: e123*e13=e12*-e1-0*e13=e2
rcProdTerm: e12Le3= e1L-e3/\e2+0=0
cliffordProdTerm: e123*e23=e12*-e2-0*e23=-e1
rcProdTerm: e12Le3= e1L-e3/\e2+0=0
cliffordProdTerm: e12*e12=e1*-e1-0*e12=-11
cliffordProdTerm: e123*e123=e12*e12-0*e123=-11

         +  1       e        e       e e      e       e e      e e    e e e +
         |           1        2       1 2      3       1 3      2 3    1 2 3|
         |                                                                  |
         |  e       1       e e       e      e e       e      e e e    e e  |
         |   1               1 2       2      1 3       3      1 2 3    2 3 |
         |                                                                  |
         |  e     - e e      1       - e     e e    - e e e     e     - e e |
         |   2       1 2                1     2 3      1 2 3     3       1 3|
         |                                                                  |
         | e e     - e       e       - 1    e e e    - e e     e e     - e  |
         |  1 2       2       1              1 2 3      2 3     1 3       3 |
   (20)  |                                                                  |
         |  e     - e e    - e e    e e e     1       - e      - e     e e  |
         |   3       1 3      2 3    1 2 3               1        2     1 2 |
         |                                                                  |
         | e e     - e    - e e e    e e      e       - 1     - e e     e   |
         |  1 3       3      1 2 3    2 3      1                 1 2     2  |
         |                                                                  |
         | e e    e e e     - e     - e e     e       e e      - 1     - e  |
         |  2 3    1 2 3       3       1 3     2       1 2                1 |
         |                                                                  |
         |e e e    e e     - e e     - e     e e       e       - e     - 1  |
         + 1 2 3    2 3       1 3       3     1 2       2         1         +
 Type: Matrix(CliffordAlgebra(3,Fraction(Integer),[[1,0,0],[0,1,0],[0,0,1]]))
(21) ->

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