Determinant and permanent: Difference between revisions

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For a given 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.

Cf.

Permutations by swapping

C

C99 code. By no means efficient or reliable. If you need it for serious work, go find a serious library. <lang C>#include <stdio.h>

include <stdlib.h>
include <string.h>

double det_in(double **in, int n, int perm) { if (n == 1) return in[0][0];

double sum = 0, *m[--n]; for (int i = 0; i < n; i++) m[i] = in[i + 1] + 1;

for (int i = 0, sgn = 1; i <= n; i++) { sum += sgn * (in[i][0] * det_in(m, n, perm)); if (i == n) break;

m[i] = in[i] + 1; if (!perm) sgn = -sgn; } return sum; }

/* wrapper function */ double det(double *in, int n, int perm) { double *m[n]; for (int i = 0; i < n; i++) m[i] = in + (n * i);

return det_in(m, n, perm); }

int main(void) { double x[] = { 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 };

printf("det: %14.12g\n", det(x, 5, 0)); printf("perm: %14.12g\n", det(x, 5, 1));

return 0; }</lang> A method to calculate determinant that might actually be usable: <lang c>#include <stdio.h>

include <stdlib.h>
include <tgmath.h>

void showmat(const char *s, double **m, int n) { printf("%s:\n", s); for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) printf("%12.4f", m[i][j]); putchar('\n'); } }

int trianglize(double **m, int n) { int sign = 1; for (int i = 0; i < n; i++) { int max = 0;

for (int row = i; row < n; row++) if (fabs(m[row][i]) > fabs(m[max][i])) max = row;

if (max) { sign = -sign; double *tmp = m[i]; m[i] = m[max], m[max] = tmp; }

if (!m[i][i]) return 0;

for (int row = i + 1; row < n; row++) { double r = m[row][i] / m[i][i]; if (!r) continue;

for (int col = i; col < n; col ++) m[row][col] -= m[i][col] * r; } } return sign; }

double det(double *in, int n) { double *m[n]; m[0] = in;

for (int i = 1; i < n; i++) m[i] = m[i - 1] + n;

showmat("Matrix", m, n);

int sign = trianglize(m, n); if (!sign) return 0;

showmat("Upper triangle", m, n);

double p = 1; for (int i = 0; i < n; i++) p *= m[i][i]; return p * sign; }

define N 18

int main(void) { double x[N * N]; srand(0); for (int i = 0; i < N * N; i++) x[i] = rand() % N;

printf("det: %19f\n", det(x, N)); return 0; }</lang>

D

This requires the modules from the Permutations and Permutations by swapping Tasks.

Translation of: Python

<lang d>import std.algorithm, std.range, std.traits, permutations2,

      permutations_by_swapping1;

auto prod(Range)(/*in*/ Range r) /*pure nothrow*/ {

   return reduce!q{a * b}(cast(ForeachType!Range)1, r);

}

T permanent(T)(in T[][] a) /*pure nothrow*/ in {

   foreach (const row; a)
       assert(row.length == a[0].length);

} body {

   auto r = iota(cast()a.length);
   T tot = 0;
   foreach (sigma; permutations(r.array()))
       tot += r.map!(i => a[i][sigma[i]])().prod();
   return tot;

}

T determinant(T)(in T[][] a) /*pure nothrow*/ in {

   foreach (const row; a)
       assert(row.length == a[0].length);

} body {

   immutable n = a.length;
   auto r = iota(n);
   T tot = 0;
   //foreach (sigma, sign; spermutations(n)) {
   foreach (sigma_sign; spermutations(n)) {
       const sigma = sigma_sign[0];
       immutable sign = sigma_sign[1];
       tot += sign * r.map!(i => a[i][sigma[i]])().prod();
   }
   return tot;

}

void main() {

   import std.stdio;

   foreach (const a; [[[1, 2],
                       [3, 4]],

                      [[1, 2, 3, 4],
                       [4, 5, 6, 7],
                       [7, 8, 9, 10],
                       [10, 11, 12, 13]],

                      [[ 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]]]) {
       writefln("[%([%(%2s, %)],\n %)]]", a);
       writefln("Permanent: %s, determinant: %s\n",
                permanent(a), determinant(a));
   }

}</lang>

Output:

[[ 1,  2],
 [ 3,  4]]
Permanent: 10, determinant: -2

[[ 1,  2,  3,  4],
 [ 4,  5,  6,  7],
 [ 7,  8,  9, 10],
 [10, 11, 12, 13]]
Permanent: 29556, determinant: 0

[[ 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]]
Permanent: 6778800, determinant: 0

Fortran

Please find the compilation and example run at the start of the comments in the f90 source. Thank you.

<lang FORTRAN> !-*- mode: compilation; default-directory: "/tmp/" -*- !Compilation started at Sat May 18 23:25:42 ! !a=./F && make $a && $a < unixdict.txt !f95 -Wall -ffree-form F.F -o F ! j example, determinant: 7.00000000 ! j example, permanent: 5.00000000 ! maxima, determinant: -360.000000 ! maxima, permanent: 900.000000 ! !Compilation finished at Sat May 18 23:25:43

! NB. example computed by J ! NB. fixed seed random matrix ! _2+3 3?.@$5 ! 2 _1 1 !_1 _2 1 !_1 _1 _1 ! ! (-/ .*)_2+3 3?.@$5 NB. determinant !7 ! (+/ .*)_2+3 3?.@$5 NB. permanent !5

!maxima example !a: matrix([2, 9, 4], [7, 5, 3], [6, 1, 8])$ !determinant(a); !-360 ! !permanent(a); !900

! compute permanent or determinant program f

 implicit none
 real, dimension(3,3) :: j, m
 data j/ 2,-1, 1,-1,-2, 1,-1,-1,-1/
 data m/2, 9, 4, 7, 5, 3, 6, 1, 8/
 write(6,*) 'j example, determinant: ',det(j,3,-1)
 write(6,*) 'j example, permanent:   ',det(j,3,1)
 write(6,*) 'maxima, determinant:    ',det(m,3,-1)
 write(6,*) 'maxima, permanent:      ',det(m,3,1)

contains

 recursive function det(a,n,permanent) result(accumulation)
   ! setting permanent to 1 computes the permanent.
   ! setting permanent to -1 computes the determinant.
   real, dimension(n,n), intent(in) :: a
   integer, intent(in) :: n, permanent
   real, dimension(n-1, n-1) :: b
   real :: accumulation
   integer :: i, sgn
   if (n .eq. 1) then
     accumulation = a(1,1)
   else
     accumulation = 0
     sgn = 1
     do i=1, n
       b(:, :(i-1)) = a(2:, :i-1)
       b(:, i:) = a(2:, i+1:)
       accumulation = accumulation + sgn * a(1, i) * det(b, n-1, permanent)
       sgn = sgn * permanent
     enddo
   endif
 end function det

end program f </lang>

J

J has a conjunction for defining verbs which can act as determinant. This conjunction is symbolized as a space followed by a dot.

For people who do not care to sort out what "recursive expansion by minors" means: -/ .* y gives the determinant of y and +/ .* y gives the permanent of y.

For example, given the 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>

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

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

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

As an aside, note that for specific verbs (like -/ .*) J uses an algorithm which is more efficient than the brute force approach.

МК-61/52

П4	ИПE	П2	КИП0	ИП0	П1	С/П	ИП4	/	КП2
L1	06	ИПE	П3	ИП0	П1	Сx	КП2	L1	17
ИП0	ИП2	+	П1	П2	ИП3	-	x#0	34	С/П
ПП	80	БП	21	КИП0	ИП4	С/П	КИП2	-	*
П4	ИП0	П3	x#0	35	Вx	С/П	КИП2	-	<->
/	КП1	L3	45	ИП1	ИП0	+	П3	ИПE	П1
П2	КИП1	/-/	ПП	80	ИП3	+	П3	ИП1	-
x=0	61	ИП0	П1	КИП3	КП2	L1	74	БП	12
ИП0	<->	^	КИП3	*	КИП1	+	КП2	->	L0
82	->	П0	В/О

This program calculates the determinant of the matrix of order <= 5. Prior to startup, РE entered 13, entered the order of the matrix Р0, and the elements are introduced with the launch of the program after one of them, the last on the screen will be determinant. Permanent is calculated in this way.

Mathematica

Determinant is a built in function Det <lang Mathematica>Permanent[m_List] :=

   With[{v = Array[x, Length[m]]},
     Coefficient[Times @@ (m.v), Times @@ v]
 ]</lang>

Maxima

<lang maxima>a: matrix([2, 9, 4], [7, 5, 3], [6, 1, 8])$

determinant(a); -360

permanent(a); 900</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>

Python

Using the module file spermutations.py from Permutations by swapping. The algorithm for the determinant is a more literal translation of the expression in the task description and the Wikipedia reference.

<lang python>from itertools import permutations from operator import mul from math import fsum from spermutations import spermutations

def prod(lst):

   return reduce(mul, lst, 1)

def perm(a):

   n = len(a)
   r = range(n)
   s = permutations(r)
   return fsum(prod(a[i][sigma[i]] for i in r) for sigma in s)

def det(a):

   n = len(a)
   r = range(n)
   s = spermutations(n)
   return fsum(sign * prod(a[i][sigma[i]] for i in r)
               for sigma, sign in s)

if __name__ == '__main__':

   from pprint import pprint as pp

   for a in ( 
           [
            [1, 2], 
            [3, 4]],

           [
            [1, 2, 3, 4],
            [4, 5, 6, 7],
            [7, 8, 9, 10],
            [10, 11, 12, 13]],

           [
            [ 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]],
       ):
       print()
       pp(a)
       print('Perm: %s Det: %s' % (perm(a), det(a)))</lang>

Sample output

[[1, 2], [3, 4]]
Perm: 10 Det: -2

[[1, 2, 3, 4], [4, 5, 6, 7], [7, 8, 9, 10], [10, 11, 12, 13]]
Perm: 29556 Det: 0

[[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]]
Perm: 6778800 Det: 0

The second matrix above is that used in the Tcl example. The third matrix is from the J language example. Note that the determinant seems to be 'exact' using this method of calculation without needing to resort to other than Pythons default numbers.

Racket

lang racket

(require math) (define determinant matrix-determinant)

(define (permanent M)

 (define n (matrix-num-rows M))
 (for/sum ([σ (in-permutations (range n))])
   (for/product ([i n] [σi σ])
     (matrix-ref M i σi))))

</lang>

REXX

<lang rexx>/* REXX ***************************************************************

compute the permanent of a matrix
I found an algorithm here:
http://www.codeproject.com/Articles/21282/Compute-Permanent-of-a-Matrix-with-Ryser-s-Algorit
see there for the original author.
translated it to REXX (hopefully correctly) to REXX
and believe that I can "publish" it here, on rosettacode
when I look at the copyright rules shown there:
http://www.codeproject.com/info/cpol10.aspx
20.05.2013 Walter Pachl
- - - - /

Call init /* initialize the matrix (n and a.* */ sum=0 rowsumprod=0 rowsum=0 chi.=0 c=2**n Do k=1 To c-1 /* loop all 2^n submatrices of A */

 rowsumprod = 1
 chis=dec2binarr(k,n)              /* characteristic vector         */
 Do ci=0 By 1 While chis<>
   Parse Var chis chi.ci chis
   End
 Do m=0 To n-1                     /* loop columns of submatrix #k  */
   rowsum = 0
   Do p=0 To n-1                   /* loop rows and compute rowsum  */
     mnp=m*n+p
     rowsum=rowsum+chi.p*A.mnp
     End
   rowsumprod=rowsumprod*rowsum  /* update product of rowsums     */
                           /* (optional -- use for sparse matrices) */
                           /* if (rowsumprod == 0) break;           */
   End
 sum=sum+((-1)**(n-chi.n))*rowsumprod
 /* Say rowsumprod sum */
 End

Say sum Exit /**********************************************************************

Notes
1.The submatrices are chosen by use of a characteristic vector chi
(only the columns are considered, where chi[p] == 1).
To retrieve the t from Ryser's formula, we need to save the number
n-t, as is done in chi[n]. Then we get t = n - chi[n].
2.The matrix parameter A is expected to be a one-dimensional integer
array -- should the matrix be encoded row-wise or column-wise?
-- It doesn't matter. The permanent is invariant under
row-switching and column-switching, and it is Screenshot
- per_inv.gif .
3.Further enhancements: If any rowsum equals zero,
the entire rowsumprod becomes zero, and thus the m-loop can be broken.
Since if-statements are relatively expensive compared to integer
operations, this might save time only for sparse matrices
(where most entries are zeros).
4.If anyone finds a polynomial algorithm for permanents,
he will get rich and famous (at least in the computer science world).
- - - - /

/**********************************************************************

At first, we need to transform a decimal to a binary array
with an additional element
(the last one) saving the number of ones in the array:
- - - - /

dec2binarr: Procedure

 Parse Arg n,dim
 ol='n='n 'dim='dim
 res.=0
 pos=dim-1
 Do While n>0
   res.pos=n//2
   res.dim=res.dim+res.pos
   n=n%2
   pos=pos-1
   End
 res_s=
 Do i=0 To dim
   res_s=res_s res.i
   End
 /* Say ol '->' res_s */
 Return res_s

init: /**********************************************************************

a.* (starting with index 0) contains all array elements
n is the dimension of the square matrix
- - - - /

n=5 as=' 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'

a.=0 Do ai=0 By 1 While as<>

 Parse Var as a.ai as
 End

Return</lang> Output:

Tcl

The determinant is provided by the linear algebra package in Tcllib. The permanent (being somewhat less common) requires definition, but is easily described:

Library: Tcllib (Package: math::linearalgebra)

Library: Tcllib (Package: struct::list)

<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

@@ Line 216: / Line 216: @@
 Permanent: 6778800, determinant: 0</pre>
-=={{header|FORTRAN}}==
+=={{header|Fortran}}==
 Please find the compilation and example run at the start of the comments in the f90 source.  Thank you.