Assuming that grassmann.spad has already been compiled (if not download from here) then the following tests are run by:
) read test2d.input
where test2d.input can be downloaded from here.
These results can be compared with the following test cases from Bertfried:
Other tests:
(1) -> )read test2d.input
-- you need to compile first time before running this
-- )compile grassmann.spad
)library CLIF
CliffordAlgebra is now explicitly exposed in frame frame1
CliffordAlgebra will be automatically loaded when needed from
/home/martin/CLIF.NRLIB/CLIF
-- two D Euclidean space
B1 := CliffordAlgebra(2,Fraction(Integer),[[1,0],[0,1]])
(1) CliffordAlgebra(2,Fraction(Integer),[[1,0],[0,1]])
Type: Domain
-- two D Hyperbolic space
B2 := CliffordAlgebra(2,Fraction(Integer),[[0,1],[1,0]])
(2) CliffordAlgebra(2,Fraction(Integer),[[0,1],[1,0]])
Type: Domain
-- non-orthogonal
B3 := CliffordAlgebra(2,Expression(Fraction(Integer)),[[1,q],[-q,1]])
(3) CliffordAlgebra(2,Expression(Fraction(Integer)),[[1,q],[-(q),1]])
Type: Domain
-- with anti-symmetric bilinear part
B4 := CliffordAlgebra(2,Expression(Fraction(Integer)),[[0,1/2+q],[1/2-q,0]])
(4)
CliffordAlgebra(2,Expression(Fraction(Integer)),[[0,q+(1/2)],[(-q)+(1/2),0]]
)
Type: Domain
--
-- the following are results for 2D Euclidean space
-- wedge for 2D Euclidean space
toTable(/\)$B1
+ 1 e e e e +
| 1 2 1 2|
| |
| e 0 e e 0 |
| 1 1 2 |
(5) | |
| e - e e 0 0 |
| 2 1 2 |
| |
|e e 0 0 0 |
+ 1 2 +
Type: Matrix(CliffordAlgebra(2,Fraction(Integer),[[1,0],[0,1]]))
--
-- regression for 2D Euclidean space
toTable(\/)$B1
+0 0 0 1 +
| |
|0 0 1 e |
| 1 |
(6) | |
|0 - 1 0 e |
| 2 |
| |
|1 e e e e |
+ 1 2 1 2+
Type: Matrix(CliffordAlgebra(2,Fraction(Integer),[[1,0],[0,1]]))
--
-- clifford multipication for 2D Euclidean space
toTable(*)$B1
+ 1 e e e e +
| 1 2 1 2|
| |
| e 1 e e e |
| 1 1 2 2 |
(7) | |
| e - e e 1 - e |
| 2 1 2 1|
| |
|e e - e e - 1 |
+ 1 2 2 1 +
Type: Matrix(CliffordAlgebra(2,Fraction(Integer),[[1,0],[0,1]]))
--
-- left contraction for 2D Euclidean space
toTable(lc)$B1
+1 e e e e +
| 1 2 1 2|
| |
|0 1 0 e |
(8) | 2 |
| |
|0 0 1 - e |
| 1|
| |
+0 0 0 - 1 +
Type: Matrix(CliffordAlgebra(2,Fraction(Integer),[[1,0],[0,1]]))
--
-- right contraction for 2D Euclidean space
toTable(rc)$B1
+ 1 0 0 0 +
| |
| e 1 0 0 |
| 1 |
(9) | |
| e 0 1 0 |
| 2 |
| |
|e e - e e - 1|
+ 1 2 2 1 +
Type: Matrix(CliffordAlgebra(2,Fraction(Integer),[[1,0],[0,1]]))
--
-- dual for 2D Euclidean space
toTable(~)$B1
+ 1 e e e e +
| 1 2 1 2|
(10) | |
|e e e - e - 1 |
+ 1 2 2 1 +
Type: Matrix(CliffordAlgebra(2,Fraction(Integer),[[1,0],[0,1]]))
--
-- grade Involution for 2D Euclidean space
toTable(gradeInvolution)$B1
+1 e e e e +
| 1 2 1 2|
(11) | |
|1 - e - e e e |
+ 1 2 1 2+
Type: Matrix(CliffordAlgebra(2,Fraction(Integer),[[1,0],[0,1]]))
--
-- the following are results for 2D Hyperbolic space
-- wedge for 2D Hyperbolic space
toTable(/\)$B2
+ 1 e e e e +
| 1 2 1 2|
| |
| e 0 e e 0 |
| 1 1 2 |
(12) | |
| e - e e 0 0 |
| 2 1 2 |
| |
|e e 0 0 0 |
+ 1 2 +
Type: Matrix(CliffordAlgebra(2,Fraction(Integer),[[0,1],[1,0]]))
--
-- regression for 2D Hyperbolic space
toTable(\/)$B2
+0 0 0 1 +
| |
|0 0 1 e |
| 1 |
(13) | |
|0 - 1 0 e |
| 2 |
| |
|1 e e e e |
+ 1 2 1 2+
Type: Matrix(CliffordAlgebra(2,Fraction(Integer),[[0,1],[1,0]]))
--
-- clifford multipication for 2D Hyperbolic space
toTable(*)$B2
+ 1 e e e e +
| 1 2 1 2|
| |
| e 0 1 + e e - e |
| 1 1 2 1|
(14) | |
| e 1 - e e 0 e |
| 2 1 2 2 |
| |
|e e e - e 1 |
+ 1 2 1 2 +
Type: Matrix(CliffordAlgebra(2,Fraction(Integer),[[0,1],[1,0]]))
--
-- left contraction for 2D Hyperbolic space
toTable(lc)$B2
+1 e e e e +
| 1 2 1 2|
| |
|0 0 1 - e |
(15) | 1|
| |
|0 1 0 e |
| 2 |
| |
+0 0 0 1 +
Type: Matrix(CliffordAlgebra(2,Fraction(Integer),[[0,1],[1,0]]))
--
-- right contraction for 2D Hyperbolic space
toTable(rc)$B2
+ 1 0 0 0+
| |
| e 0 1 0|
| 1 |
(16) | |
| e 1 0 0|
| 2 |
| |
|e e e - e 1|
+ 1 2 1 2 +
Type: Matrix(CliffordAlgebra(2,Fraction(Integer),[[0,1],[1,0]]))
--
-- dual for 2D Hyperbolic space
toTable(~)$B2
+ 1 e e e e +
| 1 2 1 2|
(17) | |
|e e - e e 1 |
+ 1 2 1 2 +
Type: Matrix(CliffordAlgebra(2,Fraction(Integer),[[0,1],[1,0]]))
--
-- grade Involution for 2D Euclidean space
toTable(gradeInvolution)$B2
+1 e e e e +
| 1 2 1 2|
(18) | |
|1 - e - e e e |
+ 1 2 1 2+
Type: Matrix(CliffordAlgebra(2,Fraction(Integer),[[0,1],[1,0]]))
--
-- the following are results for 2D non-orthogonal
-- wedge for 2D non-orthogonal space
toTable(/\)$B3
+ 1 e e e e +
| 1 2 1 2|
| |
| e 0 e e 0 |
| 1 1 2 |
(19) | |
| e - e e 0 0 |
| 2 1 2 |
| |
|e e 0 0 0 |
+ 1 2 +
Type: Matrix(CliffordAlgebra(2,Expression(Fraction(Integer)),[[1,q],[-(q),1]]))
--
-- regression for 2D non-orthogonal space
toTable(\/)$B3
(20)
4 2 3
[[0,0,0,q + 2q + 1 + (2q + 2q)e e ],
1 2
4 2 3 4 3
[0,0,q + 2q + 1 + (2q + 2q)e e ,(- q + 1)e + (2q + 2q)e ],
1 2 1 2
4 2 3 3 4
[0,- q - 2q - 1 + (- 2q - 2q)e e ,0,(- 2q - 2q)e + (- q + 1)e ],
1 2 1 2
4 2 3 4 3
[q + 2q + 1 + (2q + 2q)e e , (- q + 1)e + (2q + 2q)e ,
1 2 1 2
3 4 5 3 4 2
(- 2q - 2q)e + (- q + 1)e , - 4q - 8q - 4q + (- 7q - 6q + 1)e e ]
1 2 1 2
]
Type: Matrix(CliffordAlgebra(2,Expression(Fraction(Integer)),[[1,q],[-(q),1]]))
--
-- clifford multipication for 2D non-orthogonal space
toTable(*)$B3
+ 1 e e e e +
| 1 2 1 2 |
| |
| e 1 q + e e - q e + e |
| 1 1 2 1 2 |
(21) | |
| e - q - e e 1 - e - q e |
| 2 1 2 1 2 |
| |
| 2 |
|e e - q e - e e - q e - q - 1 - 2q e e |
+ 1 2 1 2 1 2 1 2+
Type: Matrix(CliffordAlgebra(2,Expression(Fraction(Integer)),[[1,q],[-(q),1]]))
--
-- left contraction for 2D non-orthogonal space
toTable(lc)$B3
+1 e e e e +
| 1 2 1 2 |
| |
|0 1 q - q e + e |
| 1 2|
(22) | |
|0 - q 1 - e - q e |
| 1 2|
| |
| 2 |
+0 0 0 - q - 1 +
Type: Matrix(CliffordAlgebra(2,Expression(Fraction(Integer)),[[1,q],[-(q),1]]))
--
-- right contraction for 2D non-orthogonal space
toTable(rc)$B3
+ 1 0 0 0 +
| |
| e 1 q 0 |
| 1 |
| |
(23) | e - q 1 0 |
| 2 |
| |
| 2 |
|e e - q e - e e - q e - q - 1|
+ 1 2 1 2 1 2 +
Type: Matrix(CliffordAlgebra(2,Expression(Fraction(Integer)),[[1,q],[-(q),1]]))
--
-- dual for 2D non-orthogonal space
toTable(~)$B3
+ 1 e e e e +
| 1 2 1 2 |
(24) | |
| 2 |
|e e - q e + e - e - q e - q - 1 - 2q e e |
+ 1 2 1 2 1 2 1 2+
Type: Matrix(CliffordAlgebra(2,Expression(Fraction(Integer)),[[1,q],[-(q),1]]))
--
-- grade Involution for 2D non-orthogonal space
toTable(gradeInvolution)$B3
+1 e e e e +
| 1 2 1 2|
(25) | |
|1 - e - e e e |
+ 1 2 1 2+
Type: Matrix(CliffordAlgebra(2,Expression(Fraction(Integer)),[[1,q],[-(q),1]]))
--
-- the following are results for 2D with anti-symmetric bilinear part
-- wedge for 2D part anti-symmetric bilinear space
toTable(/\)$B4
+ 1 e e e e +
| 1 2 1 2|
| |
| e 0 e e 0 |
| 1 1 2 |
(26) | |
| e - e e 0 0 |
| 2 1 2 |
| |
|e e 0 0 0 |
+ 1 2 +
Type: Matrix(CliffordAlgebra(2,Expression(Fraction(Integer)),[[0,q+(1/2)],[(-q)+(1/2),0]]))
--
-- regression for 2D part anti-symmetric bilinear space
toTable(\/)$B4
(27)
4 1 2 1 3 1
[[0,0,0,q - - q + -- + (2q - - q)e e ],
2 16 2 1 2
4 1 2 1 3 1 4 3 1 1
[0,0,q - - q + -- + (2q - - q)e e ,(- q - q + - q + --)e ],
2 16 2 1 2 4 16 1
4 1 2 1 3 1 4 3 1 1
[0,- q + - q - -- + (- 2q + - q)e e ,0,(- q + q - - q + --)e ],
2 16 2 1 2 4 16 2
4 1 2 1 3 1 4 3 1 1
[q - - q + -- + (2q - - q)e e , (- q - q + - q + --)e ,
2 16 2 1 2 4 16 1
4 3 1 1 5 3 1 4 3 2 1
(- q + q - - q + --)e , - 4q + 2q - - q + (- 7q + - q + --)e e ]
4 16 2 4 2 16 1 2
]
Type: Matrix(CliffordAlgebra(2,Expression(Fraction(Integer)),[[0,q+(1/2)],[(-q)+(1/2),0]]))
--
-- clifford multipication for 2D part anti-symmetric bilinear space
toTable(*)$B4
+ 1 e e e e +
| 1 2 1 2 |
| |
| 1 1 |
| e 0 q + - + e e (- q - -)e |
| 1 2 1 2 2 1 |
(28) | |
| 1 1 |
| e - q + - - e e 0 (- q + -)e |
| 2 2 1 2 2 2 |
| |
| 1 1 2 1 |
|e e (- q + -)e (- q - -)e - q + - - 2q e e |
+ 1 2 2 1 2 2 4 1 2+
Type: Matrix(CliffordAlgebra(2,Expression(Fraction(Integer)),[[0,q+(1/2)],[(-q)+(1/2),0]]))
--
-- left contraction for 2D part anti-symmetric bilinear space
toTable(lc)$B4
+1 e e e e +
| 1 2 1 2 |
| |
| 1 1 |
|0 0 q + - (- q - -)e |
| 2 2 1|
(29) | |
| 1 1 |
|0 - q + - 0 (- q + -)e |
| 2 2 2|
| |
| 2 1 |
|0 0 0 - q + - |
+ 4 +
Type: Matrix(CliffordAlgebra(2,Expression(Fraction(Integer)),[[0,q+(1/2)],[(-q)+(1/2),0]]))
--
-- right contraction for 2D part anti-symmetric bilinear space
toTable(rc)$B4
+ 1 0 0 0 +
| |
| 1 |
| e 0 q + - 0 |
| 1 2 |
| |
(30) | 1 |
| e - q + - 0 0 |
| 2 2 |
| |
| 1 1 2 1|
|e e (- q + -)e (- q - -)e - q + -|
+ 1 2 2 1 2 2 4+
Type: Matrix(CliffordAlgebra(2,Expression(Fraction(Integer)),[[0,q+(1/2)],[(-q)+(1/2),0]]))
--
-- dual for 2D part anti-symmetric bilinear space
toTable(~)$B4
+ 1 e e e e +
| 1 2 1 2 |
(31) | |
| 1 1 2 1 |
|e e (- q - -)e (- q + -)e - q + - - 2q e e |
+ 1 2 2 1 2 2 4 1 2+
Type: Matrix(CliffordAlgebra(2,Expression(Fraction(Integer)),[[0,q+(1/2)],[(-q)+(1/2),0]]))
--
-- grade Involution for 2D part anti-symmetric bilinear space
toTable(gradeInvolution)$B4
+1 e e e e +
| 1 2 1 2|
(32) | |
|1 - e - e e e |
+ 1 2 1 2+
Type: Matrix(CliffordAlgebra(2,Expression(Fraction(Integer)),[[0,q+(1/2)],[(-q)+(1/2),0]]))
(33) ->
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