Various low-level helper functions

Various low-level helper functions used in Alt() and kform()

consolidate(S)
kill_trivial_rows(S)
include_perms(S)
kform_to_ktensor(S)

Arguments

S

Object of class spray

Details

Low-level helper functions, used in Alt() and kform().

• Function consolidate() takes a spray object, and combines any rows that are identical up to a permutation, respecting the sign of the permutation.

• Function kill_trivial_rows() takes a spray object and deletes any rows with a repeated entry (which have \(k\)-forms identically zero)

• Function include_perms() replaces each row of a spray object with all its permutations, respecting the sign of the permutation

• Function ktensor_to_kform() coerces a \(k\)-form to a \(k\)-tensor

Value

The functions documented here all return a spray object.

Author

Robin K. S. Hankin

See also

ktensor,kform,Alt

Examples

(S <- spray(matrix(c(1,1,2,2,1,3,3,1,3,5),ncol=2,byrow=TRUE),5:1))
#>          val
#>  3 5  =    1
#>  3 1  =    2
#>  1 3  =    3
#>  2 2  =    4
#>  1 1  =    5

kill_trivial_rows(S)  # (rows "1 1; 5" and  and "2 2; 4" killed - identically zero)
#>          val
#>  1 3  =    3
#>  3 1  =    2
#>  3 5  =    1
consolidate(S)        # (merges rows "1 3; 3" and "3 1; 2")
#>          val
#>  1 1  =    5
#>  2 2  =    4
#>  1 3  =    1
#>  3 5  =    1
include_perms(S)      # returns a spray object, not an alternating tensor.
#>          val
#>  5 3  =   -1
#>  3 5  =    1
#>  3 1  =   -1
#>  1 3  =    1

include_perms(spray(t(matrix(1:4))))
#>              val
#>  4 3 2 1  =    1
#>  4 3 1 2  =   -1
#>  4 1 3 2  =    1
#>  4 1 2 3  =   -1
#>  3 4 2 1  =   -1
#>  3 4 1 2  =    1
#>  3 1 4 2  =   -1
#>  4 2 1 3  =    1
#>  1 2 3 4  =    1
#>  3 1 2 4  =    1
#>  1 2 4 3  =   -1
#>  1 3 2 4  =   -1
#>  1 3 4 2  =    1
#>  2 1 3 4  =   -1
#>  4 2 3 1  =   -1
#>  1 4 2 3  =    1
#>  2 4 1 3  =   -1
#>  2 4 3 1  =    1
#>  3 2 4 1  =    1
#>  3 2 1 4  =   -1
#>  1 4 3 2  =   -1
#>  2 3 1 4  =    1
#>  2 1 4 3  =    1
#>  2 3 4 1  =   -1