For a given matrix, return the determinant and the permanent of the matrix.

The determinant is given by

 



det
(
A
)
=

∑

σ


sgn
⁡
(
σ
)

∏

i
=
1


n



M

i
,

σ

i






{\displaystyle \det(A)=\sum _{\sigma }\operatorname {sgn}(\sigma )\prod _{i=1}^{n}M_{i,\sigma _{i}}}


while the permanent is given by

 



perm
⁡
(
A
)
=

∑

σ



∏

i
=
1


n



M

i
,

σ

i






{\displaystyle \operatorname {perm} (A)=\sum _{\sigma }\prod _{i=1}^{n}M_{i,\sigma _{i}}}


In both cases the sum is over the permutations 



σ


{\displaystyle \sigma }

 of the permutations of 1, 2, ..., n. (A permutation's sign is 1 if there are an even number of inversions and -1 otherwise; see parity of a permutation.)

More efficient algorithms for the determinant are known: LU decomposition, see for example wp:LU decomposition#Computing the determinant. Efficient methods for calculating the permanent are not known.



Related task

 Permutations by swapping


var [const] GSL=Import("zklGSL");	// libGSL (GNU Scientific Library)
fcn perm(A){  // should verify A is square
   numRows:=A.rows;
   Utils.Helpers.permute(numRows.toList()).reduce(  // permute(0,1,..numRows)
      'wrap(s,pm){ s + numRows.reduce('wrap(x,i){ x*A[i,pm[i]] },1.0) },
      0.0)
}
test:=fcn(A){
   println(A.format());
   println("Permanent: %.2f, determinant: %.2f".fmt(perm(A),A.det()));
};