Determinant and permanent: Difference between revisions
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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. |
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. |
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;Cf.: |
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* [[Permutations by swapping]] |
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=={{header|J}}== |
=={{header|J}}== |
Revision as of 09:28, 24 July 2012
Given a matrix, return the determinant and the permanent of the matrix.
The determinant is given by
while the permanent is given by
In both cases the sum is over the permutations 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.
- Cf.
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>
Tcl
The determinant is provided by the linear algebra package in Tcllib. The permanent (being somewhat less common) requires definition, but is easily described:
<lang tcl>package require math::linearalgebra package require struct::list
proc permanent {matrix} {
for {set plist {};set i 0} {$i<[llength $matrix]} {incr i} {
lappend plist $i
} foreach p [::struct::list permutations $plist] {
foreach i $plist j $p { lappend prod [lindex $matrix $i $j] } lappend sum [::tcl::mathop::* {*}$prod[set prod {}]]
} return [::tcl::mathop::+ {*}$sum]
}</lang> Demonstrating with a sample matrix: <lang tcl>set mat {
{1 2 3 4} {4 5 6 7} {7 8 9 10} {10 11 12 13}
} puts [::math::linearalgebra::det $mat] puts [permanent $mat]</lang>
- Output:
1.1315223609263888e-29 29556