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) ->
This site may have errors. Don't use for critical systems.
Geometric Algebra for Physicists - This is intended for physicists so it soon gets onto relativity, spacetime, electrodynamcs, quantum theory, etc. However the introduction to Geometric Algebra and classical mechanics is useful.