Determinant and permanent

Determinant and permanent is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Given a matrix, return the determinant and the permanent of the matrix.

The determinant is given by

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

while the permanent is given by

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

In both cases the sum is over the permutations $\sigma$ 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.

J

Given the example matrix:

<lang J> i. 5 5

0  1  2  3  4
5  6  7  8  9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24</lang>

It's determinant is 0. When we use IEEE floating point, we only get an approximation of this result:

<lang J> -/ .* i. 5 5 _1.30277e_44</lang>

If we use exact (rational) arithmetic, we get a precise result:

<lang J> -/ .* i. 5 5x 0</lang>

The permanent does not have this problem in this example (the matrix contains no negative values and permanent does not use subtraction):

<lang J> +/ .* i. 5 5 6778800</lang>

PARI/GP

The determinant is built in: <lang parigp>matdet(M)</lang> and the permanent can be defined as <lang parigp>matperm(M)=my(n=#M,t);sum(i=1,n!,t=numtoperm(n,i);prod(j=1,n,M[j,t[j]]))</lang>