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{{task}} [[Category:Matrices]] For a given matrix, return the [[wp:Determinant|determinant]] and the [[wp:Permanent|permanent]] of the matrix.

The determinant is given by :: $\det\left(A\right) = \sum_\sigma\sgn\left(\sigma\right)\prod_\left\{i=1\right\}^n M_\left\{i,\sigma_i\right\}$ while the permanent is given by :: $\operatorname\left\{perm\right\}\left(A\right)=\sum_\sigma\prod_\left\{i=1\right\}^n M_\left\{i,\sigma_i\right\}$ 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 [[wp:Parity of a permutation|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.

• [[Permutations by swapping]]

## 360 Assembly

For maximum compatibility, this program uses only the basic instruction set (S/360) and two ASSIST macros (XDECO,XPRNT) to keep it as short as possible. It works on OS/360 family (MVS,z/OS), on DOS/360 family (z/VSE) use GETVIS,FREEVIS instead of GETMAIN,FREEMAIN.

*        Matrix arithmetic         13/05/2016
MATARI   START
STM    R14,R12,12(R13)    save caller's registers
LR     R12,R15            set R12 as base register
USING  MATARI,R12         notify assembler
LA     R11,SAVEAREA       get the address of my savearea
ST     R13,4(R11)         save caller's savearea pointer
ST     R11,8(R13)         save my savearea pointer
LR     R13,R11            set R13 to point to my savearea
LA     R1,TT              @tt
BAL    R14,DETER          call deter(tt)
LR     R2,R0              R2=deter(tt)
LR     R3,R1              R3=perm(tt)
XDECO  R2,PG1+12          edit determinant
XPRNT  PG1,80             print determinant
XDECO  R3,PG2+12          edit permanent
XPRNT  PG2,80             print permanent
EXITALL  L      R13,SAVEAREA+4     restore caller's savearea address
LM     R14,R12,12(R13)    restore caller's registers
XR     R15,R15            set return code to 0
SAVEAREA DS     18F                main savearea
TT       DC     F'3'               matrix size
DC     F'2',F'9',F'4',F'7',F'5',F'3',F'6',F'1',F'8' <==input
PG1      DC     CL80'determinant='
PG2      DC     CL80'permanent='
XDEC     DS     CL12
*        recursive function        (R0,R1)=deter(t)   (python style)
DETER    CNOP   0,4                  returns determinant and permanent
STM    R14,R12,12(R13)    save all registers
LR     R9,R1              save R1
L      R2,0(R1)           n
BCTR   R2,0               n-1
LR     R11,R2             n-1
MR     R10,R2             (n-1)*(n-1)
SLA    R11,2              (n-1)*(n-1)*4
LA     R11,1(R11)         size of q array
A      R11,=A(STACKLEN)   R11 storage amount required
GETMAIN RU,LV=(R11)       allocate storage for stack
LA     R1,SAVEAREB        get the address of my savearea
ST     R13,4(R1)          save caller's savearea pointer
ST     R1,8(R13)          save my savearea pointer
LR     R13,R1             set R13 to point to my savearea
LR     R1,R9              restore R1
LR     R9,R1              @t
L      R4,0(R9)           t(0)
ST     R4,N               n=t(0)
IF1      CH     R4,=H'1'           if n=1
BNE    SIF1               then
L      R2,4(R9)             t(1)
ST     R2,R                 r=t(1)
ST     R2,S                 s=t(1)
B      EIF1               else
SIF1     L      R2,N                 n
BCTR   R2,0                 n-1
ST     R2,Q                 q(0)=n-1
ST     R2,NM1               nm1=n-1
LA     R0,1                 1
ST     R0,SGN               sgn=1
SR     R0,R0                0
ST     R0,R                 r=0
ST     R0,S                 s=0
LA     R6,1                 k=1
LOOPK    C      R6,N                 do k=1 to n
BH     ELOOPK               leave k
SR     R0,R0                  0
ST     R0,JQ                  jq=0
ST     R0,KTI                 kti=0
LA     R7,1                   iq=1
LOOPIQ   C      R7,NM1                 do iq=1 to n-1
BH     ELOOPIQ                leave iq
LR     R2,R7                    iq
LA     R2,1(R2)                 iq+1
ST     R2,IT                    it=iq+1
L      R2,KTI                   kti
A      R2,N                     kti+n
ST     R2,KTI                   kti=kti+n
ST     R2,KT                    kt=kti
LA     R8,1                     jt=1
LOOPJT   C      R8,N                     do jt=1 to n
BH     ELOOPJT                  leave jt
L      R2,KT                      kt
LA     R2,1(R2)                   kt+1
ST     R2,KT                      kt=kt+1
IF2      CR     R8,R6                      if jt<>k
BE     EIF2                       then
L      R2,JQ                        jq
LA     R2,1(R2)                     jq+1
ST     R2,JQ                        jq=jq+1
L      R1,KT                        kt
SLA    R1,2                         *4
L      R2,0(R1,R9)                  t(kt)
L      R1,JQ                        jq
SLA    R1,2                         *4
ST     R2,Q(R1)                     q(jq)=t(kt)
EIF2     EQU    *                          end if
LA     R8,1(R8)                   jt=jt+1
B      LOOPJT                   next jt
ELOOPJT  LA     R7,1(R7)                 iq=iq+1
B      LOOPIQ                 next iq
ELOOPIQ  LR     R1,R6                  k
SLA    R1,2                   *4
L      R5,0(R1,R9)            t(k)
LR     R2,R5                  R2,R5=t(k)
LA     R1,Q                   @q
BAL    R14,DETER              call deter(q)
LR     R3,R0                  R3=deter(q)
ST     R1,P                   p=perm(q)
MR     R4,R3                  R5=t(k)*deter(q)
M      R4,SGN                 R5=sgn*t(k)*deter(q)
A      R5,R                   +r
ST     R5,R                   r=r+sgn*t(k)*deter(q)
LR     R5,R2                  t(k)
M      R4,P                   R5=t(k)*perm(q)
A      R5,S                   +s
ST     R5,S                   s=s+t(k)*perm(q)
L      R2,SGN                 sgn
LCR    R2,R2                  -sgn
ST     R2,SGN                 sgn=-sgn
LA     R6,1(R6)               k=k+1
B      LOOPK                next k
ELOOPK   EQU    *                    end do
EIF1     EQU    *                  end if
EXIT     L      R13,SAVEAREB+4     restore caller's savearea address
L      R2,R               return value (determinant)
L      R3,S               return value (permanent)
XR     R15,R15            set return code to 0
FREEMAIN A=(R10),LV=(R11) free allocated storage
LR     R0,R2              first return value
LR     R1,R3              second return value
L      R14,12(R13)        restore caller's return address
LM     R2,R12,28(R13)     restore registers R2 to R12
IT       DS     F                  static area (out of stack)
KT       DS     F                  "
JQ       DS     F                  "
KTI      DS     F                  "
P        DS     F                  "
DROP   R12                base no longer needed
STACK    DSECT                     dynamic area (stack)
SAVEAREB DS     18F                function savearea
N        DS     F                  n
NM1      DS     F                  n-1
R        DS     F                  determinant accu
S        DS     F                  permanent accu
SGN      DS     F                  sign
STACKLEN EQU    *-STACK
Q        DS     F                  sub matrix q((n-1)*(n-1)+1)
YREGS
END    MATARI


{{out}}


determinant=        -360
permanent=           900



## C

C99 code. By no means efficient or reliable. If you need it for serious work, go find a serious library.

#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;
}


A method to calculate determinant that might actually be usable:

#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;
}


## Common Lisp

A recursive version, no libraries required, it doesn't use much consing, only for the list of columns to skip


(defun determinant (rows &optional (skip-cols nil))
(let* ((result 0) (sgn -1))
(dotimes (col (length (car rows)) result)
(unless (member col skip-cols)
(if (null (cdr rows))
(return-from determinant (elt (car rows) col))
(incf result (* (setq sgn (- sgn)) (elt (car rows) col) (determinant (cdr rows) (cons col skip-cols)))) )))))

(defun permanent (rows &optional (skip-cols nil))
(let* ((result 0))
(dotimes (col (length (car rows)) result)
(unless (member col skip-cols)
(if (null (cdr rows))
(return-from permanent (elt (car rows) col))
(incf result (* (elt (car rows) col) (permanent (cdr rows) (cons col skip-cols)))) )))))

Test using the first set of definitions (from task description):

(setq m2
'((1 2)
(3 4)))

(setq m3
'((-2 2 -3)
(-1 1  3)
( 2 0 -1)))

(setq m4
'(( 1  2  3  4)
( 4  5  6  7)
( 7  8  9 10)
(10 11 12 13)))

(setq m5
'(( 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)))

(dolist (m (list m2 m3 m4 m5))
(format t "~a determinant: ~a, permanent: ~a~%" m (determinant m) (permanent m)) )



{{out}}

((1 2) (3 4)) determinant: -2, permanent: 10
((-2 2 -3) (-1 1 3) (2 0 -1)) determinant: 18, permanent: 10
((1 2 3 4) (4 5 6 7) (7 8 9 10) (10 11 12 13)) determinant: 0, permanent: 29556
((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)) determinant: 0, permanent: 6778800



## D

This requires the modules from the [[Permutations#D|Permutations]] and [[Permutations_by_swapping#D|Permutations by swapping]] tasks. {{trans|Python}}

import std.algorithm, std.range, std.traits, permutations2,
permutations_by_swapping1;

auto prod(Range)(Range r) nothrow @safe @nogc {
return reduce!q{a * b}(ForeachType!Range(1), r);
}

T permanent(T)(in T[][] a) nothrow @safe
in {
assert(a.all!(row => row.length == a[0].length));
} body {
auto r = a.length.iota;
T tot = 0;
foreach (const sigma; r.array.permutations)
tot += r.map!(i => a[i][sigma[i]]).prod;
}

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

void main() {
import std.stdio;

foreach (/*immutable*/ 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",
a.permanent, a.determinant);
}
}


{{out}}

[[ 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


## EchoLisp

This requires the 'list' library for '''(in-permutations n)''' and the 'matrix' library for the built-in '''(determinant M)'''.


(lib 'list)
(lib 'matrix)

(define (permanent M)
(let (( n (matrix-row-num M)))
(for/sum ([σ (in-permutations n)])
(for/product ([i n] [σi σ])
(array-ref M i σi)))))

;; output
(define A (list->array '(1 2 3 4) 2 2))
(array-print A)
1  2
3  4
(determinant A) → -2
(permanent A) → 10

(define M (list->array (iota 25) 5 5))
(array-print M)
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
(determinant M) → 0
(permanent M) → 6778800



## Factor

USING: fry kernel math.combinatorics math.matrices sequences ;

: permanent ( matrix -- x )
dup square-matrix? [ "Matrix must be square." throw ] unless
[ dim first <iota> ] keep
'[ [ _ nth nth ] map-index product ] map-permutations sum ;


Example output:


IN: scratchpad USE: math.matrices.laplace   ! for determinant
{ { 2 9 4 } { 7 5 3 } { 6 1 8 } }
[ determinant ] [ permanent ] bi

--- Data stack:
-360
900



## Forth

{{libheader|Forth Scientific Library}} {{works with|gforth|0.7.9_20170427}} Requiring a permute.fs file from the [[Permutations_by_swapping#Forth|Permutations by swapping]] task.

S" fsl-util.fs" REQUIRED
S" fsl/dynmem.seq" REQUIRED
[UNDEFINED] defines [IF] SYNONYM defines IS [THEN]
S" fsl/structs.seq" REQUIRED
S" fsl/lufact.seq" REQUIRED
S" fsl/dets.seq" REQUIRED
S" permute.fs" REQUIRED

VARIABLE the-mat
: add-perm ( p0 p1 p2 ... pn n s -- )
DROP  \ sign
1E
1 DO
the-mat @ SWAP 1- I 1- }} F@ F*
LOOP
DROP  \ Dummy element because we're using 1-based indexing
F+ ;
: permanent ( len mat -- ) ( F: -- perm )
the-mat !
0E

3 SET-PRECISION
2 2 float matrix m2{{
1e 2e  3e 4e  2 2 m2{{ }}fput
lumatrix lmat
3 3 float matrix m3{{
2e 9e 4e  7e 5e 3e  6e 1e 8e  3 3 m3{{ }}fput

lmat 2 lu-malloc
m2{{ lmat lufact
lmat det F. 2 m2{{ permanent F. CR
lmat lu-free

lmat 3 lu-malloc
m3{{ lmat lufact
lmat det F. 3 m3{{ permanent F. CR
lmat lu-free


## Fortran

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


!-*- 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



## FunL

def sgn( p ) = product( (if s(0) < s(1) xor i(0) < i(1) then -1 else 1) | (s, i) <- p.combinations(2).zip( (0:p.length()).combinations(2) ) )

def perm( m ) = sum( product(m(i, sigma(i)) | i <- 0:m.length()) | sigma <- (0:m.length()).permutations() )

def det( m ) = sum( sgn(sigma)*product(m(i, sigma(i)) | i <- 0:m.length()) | sigma <- (0:m.length()).permutations() )


Laplace expansion (recursive):

def perm( m )
| m.length() == 1 and m(0).length() == 1 = m(0, 0)
| otherwise = sum( m(i, 0)*perm(m(0:i, 1:m.length()) + m(i+1:m.length(), 1:m.length())) | i <- 0:m.length() )

def det( m )
| m.length() == 1 and m(0).length() == 1 = m(0, 0)
| otherwise = sum( (-1)^i*m(i, 0)*det(m(0:i, 1:m.length()) + m(i+1:m.length(), 1:m.length())) | i <- 0:m.length() )


Test using the first set of definitions (from task description):

matrices = [
( (1, 2),
(3, 4)),
( (-2, 2, -3),
(-1, 1,  3),
( 2, 0, -1)),
( ( 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)) ]

for m <- matrices
println( m, 'perm: ' + perm(m), 'det: ' + det(m) )


{{out}}


((1, 2), (3, 4)), perm: 10, det: -2
((-2, 2, -3), (-1, 1, 3), (2, 0, -1)), perm: 10, det: 18
((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



## GLSL


mat4 m1 = mat3(1, 2, 3, 4,
5, 6, 7, 8
9,10,11,12,
13,14,15,16);

float d = det(m1);



## Go

### Implementation

This implements a naive algorithm for each that follows from the definitions. It imports the permute packge from the [[Permutations_by_swapping#Go|Permutations by swapping]] task.

package main

import (
"fmt"
"permute"
)

func determinant(m [][]float64) (d float64) {
p := make([]int, len(m))
for i := range p {
p[i] = i
}
it := permute.Iter(p)
for s := it(); s != 0; s = it() {
pr := 1.
for i, σ := range p {
pr *= m[i][σ]
}
d += float64(s) * pr
}
return
}

func permanent(m [][]float64) (d float64) {
p := make([]int, len(m))
for i := range p {
p[i] = i
}
it := permute.Iter(p)
for s := it(); s != 0; s = it() {
pr := 1.
for i, σ := range p {
pr *= m[i][σ]
}
d += pr
}
return
}

var m2 = [][]float64{
{1, 2},
{3, 4}}

var m3 = [][]float64{
{2, 9, 4},
{7, 5, 3},
{6, 1, 8}}

func main() {
fmt.Println(determinant(m2), permanent(m2))
fmt.Println(determinant(m3), permanent(m3))
}


{{out}}


-2 10
-360 900



### Ryser permanent

package main

import "fmt"

func main() {
fmt.Println(ryser([][]float64{
{1, 2},
{3, 4}}))
fmt.Println(ryser([][]float64{
{2, 9, 4},
{7, 5, 3},
{6, 1, 8}}))
}

func ryser(m [][]float64) (d float64) {
gray := 0
csum := make([]float64, len(m))
sgn := float64(len(m)&1<<1 - 1)
n2 := uint32(1) << uint(len(m))
for i := uint32(1); i < n2; i++ {
r := [...]byte{
0, 1, 28, 2, 29, 14, 24, 3, 30, 22, 20, 15, 25, 17, 4, 8,
31, 27, 13, 23, 21, 19, 16, 7, 26, 12, 18, 6, 11, 5, 10, 9,
}[i&-i*0x077CB531>>27]
b := 1 << r
if gray&b == 0 {
for c, e := range m[r] {
csum[c] += e
}
} else {
for c, e := range m[r] {
csum[c] -= e
}
}
gray ^= b
p := sgn
for _, e := range csum {
p *= e
}
d += p
sgn = -sgn
}
return
}


{{out}}


10
900



### Library determinant

'''go.matrix:'''

package main

import (
"fmt"

"github.com/skelterjohn/go.matrix"
)

func main() {
fmt.Println(matrix.MakeDenseMatrixStacked([][]float64{
{1, 2},
{3, 4}}).Det())
fmt.Println(matrix.MakeDenseMatrixStacked([][]float64{
{2, 9, 4},
{7, 5, 3},
{6, 1, 8}}).Det())
}


{{out}}


-2
-360



'''gonum/mat:'''

package main

import (
"fmt"

"gonum.org/v1/gonum/mat"
)

func main() {
fmt.Println(mat.Det(mat.NewDense(2, 2, []float64{
1, 2,
3, 4})))
fmt.Println(mat.Det(mat.NewDense(3, 3, []float64{
2, 9, 4,
7, 5, 3,
6, 1, 8})))
}


{{out}}


-2
-360.00000000000006



sPermutations :: [a] -> [([a], Int)]
sPermutations = flip zip (cycle [1, -1]) . foldl aux [[]]
where
aux items x = do
(f, item) <- zip (cycle [reverse, id]) items
f (insertEv x item)
insertEv x [] = [[x]]
insertEv x l@(y:ys) = (x : l) : ((y :) <$>) (insertEv x ys) elemPos :: [[a]] -> Int -> Int -> a elemPos ms i j = (ms !! i) !! j prod :: Num a => ([[a]] -> Int -> Int -> a) -> [[a]] -> [Int] -> a prod f ms = product . zipWith (f ms) [0 ..] sDeterminant :: Num a => ([[a]] -> Int -> Int -> a) -> [[a]] -> [([Int], Int)] -> a sDeterminant f ms = sum . fmap (\(is, s) -> fromIntegral s * prod f ms is) determinant :: Num a => [[a]] -> a determinant ms = sDeterminant elemPos ms . sPermutations$ [0 .. pred . length $ms] permanent :: Num a => [[a]] -> a permanent ms = sum . fmap (prod elemPos ms . fst) . sPermutations$ [0 .. pred . length $ms] -- TEST ----------------------------------------------------------------------- result :: (Num a, Show a) => [[a]] -> String result ms = unlines [ "Matrix:" , unlines (show <$> ms)
, "Determinant:"
, show (determinant ms)
, "Permanent:"
, show (permanent ms)
]

main :: IO ()
main =
mapM_
(putStrLn . result)
[ [[5]]
, [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
, [[0, 0, 1], [0, 1, 0], [1, 0, 0]]
, [[4, 3], [2, 5]]
, [[2, 5], [4, 3]]
, [[4, 4], [2, 2]]
]


{{Out}}

Matrix:
[5]

Determinant:
5
Permanent:
5

Matrix:
[1,0,0]
[0,1,0]
[0,0,1]

Determinant:
1
Permanent:
1

Matrix:
[0,0,1]
[0,1,0]
[1,0,0]

Determinant:
-1
Permanent:
1

Matrix:
[4,3]
[2,5]

Determinant:
14
Permanent:
26

Matrix:
[2,5]
[4,3]

Determinant:
-14
Permanent:
26

Matrix:
[4,4]
[2,2]

Determinant:
0
Permanent:
16


===Via Cramer's rule=== Here is code for computing the determinant and permanent very inefficiently, via [[wp:Cramer's rule|Cramer's rule]] (for the determinant, as well as its analog for the permanent):


outer :: (a->b->c) -> [a] -> [b] -> [[c]]
outer f [] _       = []
outer f _ []       = []
outer f (h1:t1) x2 = (f h1 <$> x2) : outer f t1 x2 dot [] [] = 0 dot (h1:t1) (h2:t2) = (h1*h2) + (dot t1 t2) transpose [] = [] transpose ([] : xss) = transpose xss transpose ((x:xs) : xss) = (x : [h | (h:_) <- xss]) : transpose (xs : [ t | (_:t) <- xss]) mul :: Num a => [[a]] -> [[a]] -> [[a]] mul a b = outer dot a (transpose b) delRow :: Int -> [a] -> [a] delRow i v = (first ++ rest) where (first, _:rest) = splitAt i v delCol :: Int -> [[a]] -> [[a]] delCol j m = (delRow j) <$> m

-- Determinant:
adj :: Num a => [[a]] -> [[a]]
[
[(-1)^(i+j) * det (delRow i $delCol j m) | i <- [0.. -1+length m] ] | j <- [0.. -1+length m] ] det :: Num a => [[a]] -> a det [] = 1 det m = (mul m (adj m)) !! 0 !! 0 -- Permanent: padj :: Num a => [[a]] -> [[a]] padj [] = [] padj m = [ [perm (delRow i$ delCol j m)
| i <- [0.. -1+length m]
]
| j <- [0.. -1+length m]
]
perm :: Num a => [[a]] -> a
perm [] = 1
perm m  = (mul m (padj m)) !! 0 !! 0



## J

J has a [[j:Vocabulary/dot|conjunction]] for defining verbs which can act as determinant (especially -/ .* ). This conjunction is symbolized as a space followed by a dot. And you can get the permanent by replacing - in that definition with +.

For example, given the matrix:

   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


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

   -/ .* i. 5 5
_1.30277e_44


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

   -/ .* i. 5 5x
0


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

   +/ .* i. 5 5
6778800


As an aside, note also that for specific verbs (like -/ .*) J uses an algorithm which is more efficient than the brute force approach implied by the [http://www.jsoftware.com/help/dictionary/d300.htm definition of  .]. (In general, where there are common, useful, concise definitions where special code can improve resource use by more than a factor of 2, the implementors of J try to make sure that that special code gets used for those definitions.)

## Java

import java.util.Scanner;

public class MatrixArithmetic {
public static double[][] minor(double[][] a, int x, int y){
int length = a.length-1;
double[][] result = new double[length][length];
for(int i=0;i<length;i++) for(int j=0;j<length;j++){
if(i<x && j<y){
result[i][j] = a[i][j];
}else if(i>=x && j<y){
result[i][j] = a[i+1][j];
}else if(i<x && j>=y){
result[i][j] = a[i][j+1];
}else{ //i>x && j>y
result[i][j] = a[i+1][j+1];
}
}
return result;
}
public static double det(double[][] a){
if(a.length == 1){
return a[0][0];
}else{
int sign = 1;
double sum = 0;
for(int i=0;i<a.length;i++){
sum += sign * a[0][i] * det(minor(a,0,i));
sign *= -1;
}
return sum;
}
}
public static double perm(double[][] a){
if(a.length == 1){
return a[0][0];
}else{
double sum = 0;
for(int i=0;i<a.length;i++){
sum += a[0][i] * perm(minor(a,0,i));
}
return sum;
}
}
public static void main(String args[]){
Scanner sc = new Scanner(System.in);
int size = sc.nextInt();
double[][] a = new double[size][size];
for(int i=0;i<size;i++) for(int j=0;j<size;j++){
a[i][j] = sc.nextDouble();
}
sc.close();
System.out.println("Determinant: "+det(a));
System.out.println("Permanent: "+perm(a));
}
}


Note that the first input is the size of the matrix.

For example:

2 1 2 3 4 Determinant: -2.0 Permanent: 10.0

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 Determinant: 0.0 Permanent: 6778800.0



## jq

{{Works with|jq|1.4}}

### =Recursive definitions=

jq
# Eliminate row i and row j
def except(i;j):
reduce del(.[i])[] as $row ([]; . + [$row | del(.[j]) ] );

def det:
def parity(i): if i % 2 == 0 then 1 else -1 end;
if length == 1 and (.[0] | length) == 1 then .[0][0]
else . as $m | reduce range(0; length) as$i
(0; . + parity($i) *$m[0][$i] * ($m | except(0;$i) | det) ) end ; def perm: if length == 1 and (.[0] | length) == 1 then .[0][0] else . as$m
| reduce range(0; length) as $i (0; . +$m[0][$i] * ($m | except(0;$i) | perm) ) end ;  '''Examples''' def matrices: [ [1, 2], [3, 4]], [ [-2, 2, -3], [-1, 1, 3], [ 2, 0, -1]], [ [ 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]] ; "Determinants: ", (matrices | det), "Permanents: ", (matrices | perm)  {{Out}} $ jq -n -r -f Matrix_arithmetic.jq
Determinants:
-2
18
0
0
Permanents:
10
10
29556
6778800


### =Determinant via LU Decomposition=

The following uses the jq infrastructure at [[LU decomposition]] to achieve an efficient implementation of det/0:

# Requires lup/0
def det:
def product_diagonal:
. as $m | reduce range(0;length) as$i (1; . * $m[$i][$i]); def tidy: if . == -0 then 0 else . end; lup | (.[0]|product_diagonal) as$l
| if $l == 0 then 0 else$l * (.[1]|product_diagonal) | tidy end ;



'''Examples'''

Using matrices/0 as defined above:

matrices | det


{{Output}} $/usr/local/bin/jq -M -n -f LU.rc 2 -18 0 0 ## Julia  The determinant of a matrix <code>A</code> can be computed by the built-in function julia det(A)  {{trans|Python}} The following function computes the permanent of a matrix A from the definition: function perm(A) m, n = size(A) if m != n; throw(ArgumentError("permanent is for square matrices only")); end sum(σ -> prod(i -> A[i,σ[i]], 1:n), permutations(1:n)) end  Example output:  A = [2 9 4; 7 5 3; 6 1 8] julia> det(A), perm(A) (-360.0,900)  ## Kotlin // version 1.1.2 typealias Matrix = Array<DoubleArray> fun johnsonTrotter(n: Int): Pair<List<IntArray>, List<Int>> { val p = IntArray(n) { it } // permutation val q = IntArray(n) { it } // inverse permutation val d = IntArray(n) { -1 } // direction = 1 or -1 var sign = 1 val perms = mutableListOf<IntArray>() val signs = mutableListOf<Int>() fun permute(k: Int) { if (k >= n) { perms.add(p.copyOf()) signs.add(sign) sign *= -1 return } permute(k + 1) for (i in 0 until k) { val z = p[q[k] + d[k]] p[q[k]] = z p[q[k] + d[k]] = k q[z] = q[k] q[k] += d[k] permute(k + 1) } d[k] *= -1 } permute(0) return perms to signs } fun determinant(m: Matrix): Double { val (sigmas, signs) = johnsonTrotter(m.size) var sum = 0.0 for ((i, sigma) in sigmas.withIndex()) { var prod = 1.0 for ((j, s) in sigma.withIndex()) prod *= m[j][s] sum += signs[i] * prod } return sum } fun permanent(m: Matrix) : Double { val (sigmas, _) = johnsonTrotter(m.size) var sum = 0.0 for (sigma in sigmas) { var prod = 1.0 for ((i, s) in sigma.withIndex()) prod *= m[i][s] sum += prod } return sum } fun main(args: Array<String>) { val m1 = arrayOf( doubleArrayOf(1.0) ) val m2 = arrayOf( doubleArrayOf(1.0, 2.0), doubleArrayOf(3.0, 4.0) ) val m3 = arrayOf( doubleArrayOf(2.0, 9.0, 4.0), doubleArrayOf(7.0, 5.0, 3.0), doubleArrayOf(6.0, 1.0, 8.0) ) val m4 = arrayOf( doubleArrayOf( 1.0, 2.0, 3.0, 4.0), doubleArrayOf( 4.0, 5.0, 6.0, 7.0), doubleArrayOf( 7.0, 8.0, 9.0, 10.0), doubleArrayOf(10.0, 11.0, 12.0, 13.0) ) val matrices = arrayOf(m1, m2, m3, m4) for (m in matrices) { println("m${m.size} -> ")
println("  determinant = ${determinant(m)}") println(" permanent =${permanent(m)}\n")
}
}


{{out}}


m1 ->
determinant = 1.0
permanent   = 1.0

m2 ->
determinant = -2.0
permanent   = 10.0

m3 ->
determinant = -360.0
permanent   = 900.0

m4 ->
determinant = 0.0
permanent   = 29556.0



## Lua

-- Johnson–Trotter permutations generator
_JT={}
function JT(dim)
local n={ values={}, positions={}, directions={}, sign=1 }
setmetatable(n,{__index=_JT})
for i=1,dim do
n.values[i]=i
n.positions[i]=i
n.directions[i]=-1
end
return n
end

function _JT:largestMobile()
for i=#self.values,1,-1 do
local loc=self.positions[i]+self.directions[i]
if loc >= 1 and loc <= #self.values and self.values[loc] < i then
return i
end
end
return 0
end

function _JT:next()
local r=self:largestMobile()
if r==0 then return false end
local rloc=self.positions[r]
local lloc=rloc+self.directions[r]
local l=self.values[lloc]
self.values[lloc],self.values[rloc] = self.values[rloc],self.values[lloc]
self.positions[l],self.positions[r] = self.positions[r],self.positions[l]
self.sign=-self.sign
for i=r+1,#self.directions do self.directions[i]=-self.directions[i] end
return true
end

-- matrix class

_MTX={}
function MTX(matrix)
setmetatable(matrix,{__index=_MTX})
matrix.rows=#matrix
matrix.cols=#matrix[1]
return matrix
end

function _MTX:dump()
for _,r in ipairs(self) do
print(unpack(r))
end
end

function _MTX:perm() return self:det(1) end
function _MTX:det(perm)
local det=0
local jt=JT(self.cols)
repeat
local pi=perm or jt.sign
for i,v in ipairs(jt.values) do
pi=pi*self[i][v]
end
det=det+pi
until not jt:next()
return det
end

-- test

matrix=MTX
{
{ 7,  2, -2,  4},
{ 4,  4,  1,  7},
{11, -8,  9, 10},
{10,  5, 12, 13}
}
matrix:dump();
print("det:",matrix:det(), "permanent:",matrix:perm(),"\n")

matrix2=MTX
{
{-2, 2,-3},
{-1, 1, 3},
{ 2, 0,-1}
}
matrix2:dump();
print("det:",matrix2:det(), "permanent:",matrix2:perm())



{{out}}

7       2       -2      4
4       4       1       7
11      -8      9       10
10      5       12      13
det:    -4319   permanent:      10723

-2      2       -3
-1      1       3
2       0       -1
det:    18      permanent:      10



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.

## Maple

M:=<<2|9|4>,<7|5|3>,<6|1|8>>:

with(LinearAlgebra):

Determinant(M);
Permanent(M);


Output:

                                    -360
900


## Mathematica

Determinant is a built in function Det

Permanent[m_List] :=
With[{v = Array[x, Length[m]]},
Coefficient[Times @@ (m.v), Times @@ v]
]


a: matrix([2, 9, 4], [7, 5, 3], [6, 1, 8])$determinant(a); -360 permanent(a); 900  ## Nim {{trans|Python}} Using the permutationsswap module from [[Permutations by swapping#Nim|Permutations by swapping]]: import sequtils, permutationsswap type Matrix[M,N: static[int]] = array[M, array[N, float]] proc det[M,N](a: Matrix[M,N]): float = let n = toSeq 0..a.high for sigma, sign in n.permutations: var x = sign.float for i in n: x *= a[i][sigma[i]] result += x proc perm[M,N](a: Matrix[M,N]): float = let n = toSeq 0..a.high for sigma, sign in n.permutations: var x = 1.0 for i in n: x *= a[i][sigma[i]] result += x const a = [ [1.0, 2.0] , [3.0, 4.0] ] b = [ [ 1.0, 2, 3, 4] , [ 4.0, 5, 6, 7] , [ 7.0, 8, 9, 10] , [10.0, 11, 12, 13] ] c = [ [ 0.0, 1, 2, 3, 4] , [ 5.0, 6, 7, 8, 9] , [10.0, 11, 12, 13, 14] , [15.0, 16, 17, 18, 19] , [20.0, 21, 22, 23, 24] ] echo "perm: ", a.perm, " det: ", a.det echo "perm: ", b.perm, " det: ", b.det echo "perm: ", c.perm, " det: ", c.det  Output: perm: 10.0 det: -2.0 perm: 29556.0 det: 0.0 perm: 6778800.0 det: 0.0  ## Ol  ; helper function that returns rest of matrix by col/row (define (rest matrix i j) (define (exclude1 l x) (append (take l (- x 1)) (drop l x))) (exclude1 (map exclude1 matrix (repeat i (length matrix))) j)) ; superfunction for determinant and permanent (define (super matrix math) (let loop ((n (length matrix)) (matrix matrix)) (if (eq? n 1) (caar matrix) (fold (lambda (x a j) (+ x (* a (lref math (mod j 2)) (super (rest matrix j 1) math)))) 0 (car matrix) (iota n 1))))) ; det/per calculators (define (det matrix) (super matrix '(-1 1))) (define (per matrix) (super matrix '( 1 1))) ; ---=( testing )=--------------------- (print (det '( (1 2) (3 4)))) ; ==> -2 (print (per '( (1 2) (3 4)))) ; ==> 10 (print (det '( ( 1 2 3 1) (-1 -1 -1 2) ( 1 3 1 1) (-2 -2 0 -1)))) ; ==> 26 (print (per '( ( 1 2 3 1) (-1 -1 -1 2) ( 1 3 1 1) (-2 -2 0 -1)))) ; ==> -10 (print (det '( ( 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)))) ; ==> 0 (print (per '( ( 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)))) ; ==> 6778800  ## PARI/GP The determinant is built in: matdet(M)  and the permanent can be defined as matperm(M)=my(n=#M,t);sum(i=1,n!,t=numtoperm(n,i);prod(j=1,n,M[j,t[j]]))  For better performance, here's a version using Ryser's formula: matperm(M)= { my(n=matsize(M)[1],innerSums=vectorv(n)); if(n==0, return(1)); sum(x=1,2^n-1, my(k=valuation(x,2),s=M[,k+1],gray=bitxor(x, x>>1)); if(bittest(gray,k), innerSums += s; , innerSums -= s; ); (-1)^hammingweight(gray)*factorback(innerSums) )*(-1)^n; }  {{works with|PARI/GP|2.10.0+}} As of version 2.10, the matrix permanent is built in: matpermanent(M)  ## Perl {{trans|C}} #!/usr/bin/perl use strict; use warnings; use PDL; use PDL::NiceSlice; sub permanent{ my$mat = shift;
my $n = shift //$mat->dim(0);
return undef if $mat->dim(0) !=$mat->dim(1);
return $mat(0,0) if$n == 1;
my $sum = 0; --$n;
my $m =$mat(1:,1:)->copy;
for(my $i = 0;$i <= $n; ++$i){
$sum +=$mat($i,0) * permanent($m, $n); last if$i == $n;$m($i,:) .=$mat($i,1:); } return sclr($sum);
}

my $M = pdl([[2,9,4], [7,5,3], [6,1,8]]); print "M =$M\n";
print "det(M) = " . $M->determinant . ".\n"; print "det(M) = " .$M->det . ".\n";
print "perm(M) = " . permanent($M) . ".\n";  determinant and det are already defined in PDL, see[http://pdl.perl.org/?docs=MatrixOps&title=the%20PDL::MatrixOps%20manpage#det]. permanent has to be defined manually. {{out}}  M = [ [2 9 4] [7 5 3] [6 1 8] ] det(M) = -360. det(M) = -360. perm(M) = 900.  ## Perl 6 {{works with|Rakudo|2015.12}} Uses the permutations generator from the [[Permutations by swapping#Perl_6|Permutations by swapping]] task. This implementation is naive and brute-force (slow) but exact. sub insert ($x, @xs) { ([flat @xs[0 ..^ $_],$x, @xs[$_ .. *]] for 0 .. @xs) } sub order ($sg, @xs) { $sg > 0 ?? @xs !! @xs.reverse } multi σ_permutations ([]) { [] => 1 } multi σ_permutations ([$x, *@xs]) {
σ_permutations(@xs).map({ |order($_.value, insert($x, $_.key)) }) Z=> |(1,-1) xx * } sub m_arith ( @a,$op ) {
note "Not a square matrix" and return
if [||] map { @a.elems cmp @a[$_].elems }, ^@a; [+] map { my$permutation = .key;
my $term =$op eq 'perm' ?? 1 !! .value;
for $permutation.kv ->$i, $j {$term *= @a[$i][$j] };
$term }, σ_permutations [^@a]; } ########### Testing ########### my @tests = ( [ [ 1, 2 ], [ 3, 4 ] ], [ [ 1, 2, 3, 4 ], [ 4, 5, 6, 7 ], [ 7, 8, 9, 10 ], [ 10, 11, 12, 13 ] ], [ [ 0, 1, 2, 3, 4 ], [ 5, 6, 7, 8, 9 ], [ 10, 11, 12, 13, 14 ], [ 15, 16, 17, 18, 19 ], [ 20, 21, 22, 23, 24 ] ] ); sub dump (@matrix) { say$_».fmt: "%3s" for @matrix;
say '';
}

for @tests -> @matrix {
say 'Matrix:';
@matrix.&dump;
say "Determinant:\t", @matrix.&m_arith: <det>;
say "Permanent:  \t", @matrix.&m_arith: <perm>;
say '-' x 25;
}


'''Output'''

Matrix:
[  1   2]
[  3   4]

Determinant:	-2
Permanent:  	10
-------------------------
Matrix:
[  1   2   3   4]
[  4   5   6   7]
[  7   8   9  10]
[ 10  11  12  13]

Determinant:	0
Permanent:  	29556
-------------------------
Matrix:
[  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]

Determinant:	0
Permanent:  	6778800
-------------------------


## Phix

{{trans|Java}}

function minor(sequence a, integer x, integer y)
integer l = length(a)-1
sequence result = repeat(repeat(0,l),l)
for i=1 to l do
for j=1 to l do
result[i][j] = a[i+(i>=x)][j+(j>=y)]
end for
end for
return result
end function

function det(sequence a)
if length(a)=1 then
return a[1][1]
end if
integer sgn = 1
integer res = 0
for i=1 to length(a) do
res += sgn*a[1][i]*det(minor(a,1,i))
sgn *= -1
end for
return res
end function

function perm(sequence a)
if length(a)=1 then
return a[1][1]
end if
integer res = 0
for i=1 to length(a) do
res += a[1][i]*perm(minor(a,1,i))
end for
return res
end function

constant tests = {
{{1,  2},
{3,  4}},
--Determinant: -2, permanent: 10
{{2, 9, 4},
{7, 5, 3},
{6, 1, 8}},
--Determinant: -360, permanent: 900
{{ 1,  2,  3,  4},
{ 4,  5,  6,  7},
{ 7,  8,  9, 10},
{10, 11, 12, 13}},
--Determinant: 0, permanent: 29556
{{ 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}},
--Determinant: 0, permanent: 6778800
{{5}},
--Determinant: 5, permanent: 5
{{1,0,0},
{0,1,0},
{0,0,1}},
--Determinant: 1, permanent: 1
{{0,0,1},
{0,1,0},
{1,0,0}},
--Determinant: -1, Permanent: 1
{{4,3},
{2,5}},
--Determinant: 14, Permanent: 26
{{2,5},
{4,3}},
--Determinant: -14, Permanent: 26
{{4,4},
{2,2}},
--Determinant: 0, Permanent: 16
{{7,    2,      -2,     4},
{4,    4,      1,      7},
{11,   -8,     9,      10},
{10,   5,      12,     13}},
--det:  -4319   permanent:      10723

{{-2,   2,      -3},
{-1,   1,      3},
{2 ,   0,      -1}}
--det:  18      permanent:      10
}
for i=1 to length(tests) do
sequence ti = tests[i]
?{det(ti),perm(ti)}
end for


{{out}}


{-2,10}
{-360,900}
{0,29556}
{0,6778800}
{5,5}
{1,1}
{-1,1}
{14,26}
{-14,26}
{0,16}
{-4319,10723}
{18,10}



## PowerShell


function det-perm ($array) { if($array) {
$size =$array.Count
function prod($A) {$prod = 1
if($A) {$A | foreach{$prod *=$_} }
$prod } function generate($sign, $n,$A) {
if($n -eq 1) {$i = 0
$prod = prod @($A | foreach{$array[$i++][$_]}) [pscustomobject]@{det =$sign*$prod; perm =$prod}
}
else{
for($i = 0;$i -lt ($n - 1);$i += 1) {
generate $sign ($n - 1) $A if($n % 2 -eq 0){
$i1,$i2 = $i, ($n-1)
$A[$i1], $A[$i2] = $A[$i2], $A[$i1]
}
else{
$i1,$i2 = 0, ($n-1)$A[$i1],$A[$i2] =$A[$i2],$A[$i1] }$sign *= -1
}
generate $sign ($n - 1) $A } }$det = $perm = 0 generate 1$size @(0..($size-1)) | foreach{$det += $_.det$perm += $_.perm } [pscustomobject]@{det = "$det"; perm = "$perm"} } else {Write-Error "empty array"} } det-perm 5 det-perm @(@(1,0,0),@(0,1,0),@(0,0,1)) det-perm @(@(0,0,1),@(0,1,0),@(1,0,0)) det-perm @(@(4,3),@(2,5)) det-perm @(@(2,5),@(4,3)) det-perm @(@(4,4),@(2,2))  Output:  det perm --- ---- 5 5 1 1 -1 1 14 26 -14 26 0 16  ## Python Using the module file spermutations.py from [[Permutations by swapping#Python|Permutations by swapping]]. The algorithm for the determinant is a more literal translation of the expression in the task description and the Wikipedia reference. 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)))  ;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))))  ## REXX /* REXX *************************************************************** * Test the two functions determinant and permanent * using the matrix specifications shown for other languages * 21.05.2013 Walter Pachl **********************************************************************/ Call test ' 1 2', ' 3 4',2 Call test ' 1 2 3 4', ' 4 5 6 7', ' 7 8 9 10', '10 11 12 13',4 Call test ' 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',5 Exit test: /********************************************************************** * Show the given matrix and compute and show determinant and permanent **********************************************************************/ Parse Arg as,n asc=as Do i=1 To n ol='' Do j=1 To n Parse Var asc a.i.j asc ol=ol right(a.i.j,3) End Say ol End Say 'determinant='right(determinant(as),7) Say ' permanent='right(permanent(as),7) Say copies('-',50) Return  /* REXX *************************************************************** * determinant.rex * compute the determinant of the given square matrix * Input: as: the representation of the matrix as vector (n**2 elements) * 21.05.2013 Walter Pachl **********************************************************************/ Parse Arg as n=sqrt(words(as)) Do i=1 To n Do j=1 To n Parse Var as a.i.j as End End Select When n=2 Then det=a.1.1*a.2.2-a.1.2*a.2.1 When n=3 Then det= a.1.1*a.2.2*a.3.3, +a.1.2*a.2.3*a.3.1, +a.1.3*a.2.1*a.3.2, -a.1.3*a.2.2*a.3.1, -a.1.2*a.2.1*a.3.3, -a.1.1*a.2.3*a.3.2 Otherwise Do det=0 Do k=1 To n det=det+((-1)**(k+1))*a.1.k*determinant(subm(k)) End End End Return det subm: Procedure Expose a. n /********************************************************************** * compute the submatrix resulting when row 1 and column k are removed * Input: a.*.*, k * Output: bs the representation of the submatrix as vector **********************************************************************/ Parse Arg k bs='' do i=2 To n Do j=1 To n If j=k Then Iterate bs=bs a.i.j End End Return bs sqrt: Procedure /********************************************************************** * compute and return the (integer) square root of the given argument * terminate the program if the argument is not a square **********************************************************************/ Parse Arg nn Do n=1 By 1 while n*n<nn End If n*n=nn Then Return n Else Do Say 'invalid number of elements:' nn 'is not a square.' Exit End  /* REXX *************************************************************** * permanent.rex * 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 arg(1) /* 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 End Return sum /********************************************************************** * 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 Return res_s init: Procedure Expose a. n /********************************************************************** * a.* (starting with index 0) contains all array elements * n is the dimension of the square matrix **********************************************************************/ Parse Arg as n=sqrt(words(as)) a.=0 Do ai=0 By 1 While as<>'' Parse Var as a.ai as End Return sqrt: Procedure /********************************************************************** * compute and return the (integer) square root of the given argument * terminate the program if the argument is not a square **********************************************************************/ Parse Arg nn Do n=1 By 1 while n*n<nn End If n*n=nn Then Return n Else Do Say 'invalid number of elements:' nn 'is not a square.' Exit End  Output:  1 2 3 4 determinant= -2 permanent= 10 -------------------------------------------------- 1 2 3 4 4 5 6 7 7 8 9 10 10 11 12 13 determinant= 0 permanent= 29556 -------------------------------------------------- 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 determinant= 0 permanent=6778800 --------------------------------------------------  ## Ruby Matrix in the standard library provides a method for the determinant, but not for the permanent. require 'matrix' class Matrix # Add "permanent" method to Matrix class def permanent r = (0...row_count).to_a # [0,1] (first example), [0,1,2,3] (second example) r.permutation.inject(0) do |sum, sigma| sum += sigma.zip(r).inject(1){|prod, (row, col)| prod *= self[row, col] } end end end m1 = Matrix[[1,2],[3,4]] # testcases from Python version m2 = Matrix[[1, 2, 3, 4], [4, 5, 6, 7], [7, 8, 9, 10], [10, 11, 12, 13]] m3 = Matrix[[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]] [m1, m2, m3].each do |m| puts "determinant:\t #{m.determinant}", "permanent:\t #{m.permanent}" puts end  {{Output}}  determinant: -2 permanent: 10 determinant: 0 permanent: 29556 determinant: 0 permanent: 6778800  ## Scala  def permutationsSgn[T]: List[T] => List[(Int,List[T])] = { case Nil => List((1,Nil)) case xs => { for { (x, i) <- xs.zipWithIndex (sgn,ys) <- permutationsSgn(xs.take(i) ++ xs.drop(1 + i)) } yield { val sgni = sgn * (2 * (i%2) - 1) (sgni, (x :: ys)) } } } def det(m:List[List[Int]]) = { val summands = for { (sgn,sigma) <- permutationsSgn((0 to m.length - 1).toList).toList } yield { val factors = for (i <- 0 to (m.length - 1)) yield m(i)(sigma(i)) factors.toList.foldLeft(sgn)({case (x,y) => x * y}) } summands.toList.foldLeft(0)({case (x,y) => x + y})  ## Sidef The determinant method is provided by the Array class. {{trans|Ruby}} class Array { method permanent { var r = @^self.len var sum = 0 r.permutations { |*a| var prod = 1 [a,r].zip {|row,col| prod *= self[row][col] } sum += prod } return sum } } var m1 = [[1,2],[3,4]] var m2 = [[1, 2, 3, 4], [4, 5, 6, 7], [7, 8, 9, 10], [10, 11, 12, 13]] var m3 = [[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]] [m1, m2, m3].each { |m| say "determinant:\t #{m.determinant}\npermanent:\t #{m.permanent}\n" }  {{out}} determinant: -2 permanent: 10 determinant: 0 permanent: 29556 determinant: 0 permanent: 6778800  ## Simula ! MATRIX ARITHMETIC ; BEGIN INTEGER PROCEDURE LENGTH(A); ARRAY A; LENGTH := UPPERBOUND(A, 1) - LOWERBOUND(A, 1) + 1; ! Set MAT to the first minor of A dropping row X and column Y ; PROCEDURE MINOR(A, X, Y, MAT); ARRAY A, MAT; INTEGER X, Y; BEGIN INTEGER I, J, rowA, M; M := LENGTH(A) - 1; ! not a constant; FOR I := 1 STEP 1 UNTIL M DO BEGIN rowA := IF I < X THEN I ELSE I + 1; FOR J := 1 STEP 1 UNTIL M DO MAT(I, J) := A(rowA, IF J < Y THEN J else J + 1); END END MINOR; REAL PROCEDURE DET(A); REAL ARRAY A; BEGIN INTEGER N; N := LENGTH(A); IF N = 1 THEN DET := A(1, 1) ELSE BEGIN INTEGER I, SIGN; REAL SUM; SIGN := 1; FOR I := 1 STEP 1 UNTIL N DO BEGIN REAL ARRAY MAT(1:N-1, 1:N-1); MINOR(A, 1, I, MAT); SUM := SUM + SIGN * A(1, I) * DET(MAT); SIGN := SIGN * -1 END; DET := SUM END END DET; REAL PROCEDURE PERM(A); REAL ARRAY A; BEGIN INTEGER N; N := LENGTH(A); IF N = 1 THEN PERM := A(1, 1) ELSE BEGIN REAL SUM; INTEGER I; FOR I := 1 STEP 1 UNTIL N DO BEGIN REAL ARRAY MAT(1:N-1, 1:N-1); MINOR(A, 1, I, MAT); SUM := SUM + A(1, I) * PERM(MAT) END; PERM := SUM END END PERM; INTEGER SIZE; SIZE := ININT; BEGIN REAL ARRAY A(1:SIZE, 1:SIZE); INTEGER I, J; FOR I := 1 STEP 1 UNTIL SIZE DO BEGIN ! may be need here: INIMAGE; FOR J := 1 STEP 1 UNTIL SIZE DO A(I, J) := INREAL END; OUTTEXT("DETERMINANT ... : "); OUTREAL(DET (A), 10, 20); OUTIMAGE; OUTTEXT("PERMANENT ..... : "); OUTREAL(PERM(A), 10, 20); OUTIMAGE; END COMMENT THE FIRST INPUT IS THE SIZE OF THE MATRIX, FOR EXAMPLE: ! 2 ! 1 2 ! 3 4 ! DETERMINANT: -2.0 ! PERMANENT: 10.0 ; COMMENT ! 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 ! DETERMINANT: 0.0 ! PERMANENT: 6778800.0 ; END  Input:  2 1 2 3 4  {{Output}}  DETERMINANT ... : -2.000000000&+000 PERMANENT ..... : 1.000000000&+001  Input:  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  {{Output}}  DETERMINANT ... : 0.000000000&+000 PERMANENT ..... : 6.778800000&+006  ## SPAD {{works with|FriCAS}} {{works with|OpenAxiom}} {{works with|Axiom}} (1) -> M:=matrix [[2, 9, 4], [7, 5, 3], [6, 1, 8]] +2 9 4+ | | (1) |7 5 3| | | +6 1 8+ Type: Matrix(Integer) (2) -> determinant M (2) - 360 Type: Integer (3) -> permanent M (3) 900 Type: PositiveInteger  [http://fricas.github.io/api/Matrix.html?highlight=matrix Domain:Matrix(R)] ## Stata Two auxiliary functions: '''range1(n,i)''' returns the column vector with numbers 1 to n except i is removed. And '''submat(a,i,j)''' returns matrix a with row i and column j removed. For x=-1, the main function '''sumrec(a,x)''' computes the determinant of a by developing the determinant along the first column. For x=1, one gets the permanent. real vector range1(real scalar n, real scalar i) { if (i < 1 | i > n) { return(1::n) } else if (i == 1) { return(2::n) } else if (i == n) { return(1::n-1) } else { return(1::i-1\i+1::n) } } real matrix submat(real matrix a, real scalar i, real scalar j) { return(a[range1(rows(a), i), range1(cols(a), j)]) } real scalar sumrec(real matrix a, real scalar x) { real scalar n, s, p n = rows(a) if (n==1) return(a[1,1]) s = 0 p = 1 for (i=1; i<=n; i++) { s = s+p*a[i,1]sumrec(submat(a, i, 1), x) p = px } return(s) }  Example: stata : a=1,1,1,0\1,1,0,1\1,0,1,1\0,1,1,1 : a [symmetric] 1 2 3 4 +-----------------+ 1 | 1 | 2 | 1 1 | 3 | 1 0 1 | 4 | 0 1 1 1 | +-----------------+ : det(a) -3 : sumrec(a,-1) -3 : sumrec(a,1) 9  ## Tcl The determinant is provided by the linear algebra package in Tcllib. The permanent (being somewhat less common) requires definition, but is easily described: {{tcllib|math::linearalgebra}} {{tcllib|struct::list}} 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]
}


Demonstrating with a sample matrix:

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]


{{out}}


1.1315223609263888e-29
29556



## VBA

{{trans|Phix}} As an extra, the results of the built in WorksheetFuction.MDeterm are shown. The latter does not work for scalars.

Option Base 1
Private Function minor(a As Variant, x As Integer, y As Integer) As Variant
Dim l As Integer: l = UBound(a) - 1
Dim result() As Double
If l > 0 Then ReDim result(l, l)
For i = 1 To l
For j = 1 To l
result(i, j) = a(i - (i >= x), j - (j >= y))
Next j
Next i
minor = result
End Function

Private Function det(a As Variant)
If IsArray(a) Then
If UBound(a) = 1 Then
On Error GoTo err
det = a(1, 1)
Exit Function
End If
Else
det = a
Exit Function
End If
Dim sgn_ As Integer: sgn_ = 1
Dim res As Integer: res = 0
Dim i As Integer
For i = 1 To UBound(a)
res = res + sgn_ * a(1, i) * det(minor(a, 1, i))
sgn_ = sgn_ * -1
Next i
det = res
Exit Function
err:
det = a(1)
End Function

Private Function perm(a As Variant) As Double
If IsArray(a) Then
If UBound(a) = 1 Then
On Error GoTo err
perm = a(1, 1)
Exit Function
End If
Else
perm = a
Exit Function
End If
Dim res As Double
Dim i As Integer
For i = 1 To UBound(a)
res = res + a(1, i) * perm(minor(a, 1, i))
Next i
perm = res
Exit Function
err:
perm = a(1)
End Function

Public Sub main()
Dim tests(13) As Variant
tests(1) = [{1,  2; 3,  4}]
'--Determinant: -2, permanent: 10
tests(2) = [{2, 9, 4; 7, 5, 3; 6, 1, 8}]
'--Determinant: -360, permanent: 900
tests(3) = [{ 1,  2,  3,  4; 4,  5,  6,  7; 7,  8,  9, 10; 10, 11, 12, 13}]
'--Determinant: 0, permanent: 29556
tests(4) = [{ 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}]
'--Determinant: 0, permanent: 6778800
tests(5) = [{5}]
'--Determinant: 5, permanent: 5
tests(6) = [{1,0,0; 0,1,0; 0,0,1}]
'--Determinant: 1, permanent: 1
tests(7) = [{0,0,1; 0,1,0; 1,0,0}]
'--Determinant: -1, Permanent: 1
tests(8) = [{4,3; 2,5}]
'--Determinant: 14, Permanent: 26
tests(9) = [{2,5; 4,3}]
'--Determinant: -14, Permanent: 26
tests(10) = [{4,4; 2,2}]
'--Determinant: 0, Permanent: 16
tests(11) = [{7,    2,      -2,     4; 4,    4,      1,      7; 11,   -8,     9,      10; 10,   5,      12,     13}]
'--det:  -4319   permanent:      10723
tests(12) = [{-2,   2,      -3; -1,   1,      3; 2 ,   0,      -1}]
'--det:  18      permanent:      10
tests(13) = 13
Debug.Print "Determinant", "Builtin det", "Permanent"
For i = 1 To 12
Debug.Print det(tests(i)), WorksheetFunction.MDeterm(tests(i)), perm(tests(i))
Next i
Debug.Print det(tests(13)), "error", perm(tests(13))
End Sub


{{out}}

Determinant   Builtin det   Permanent
-2            -2             10
-360          -360           900
0             0             29556
0             0             6778800
5             5             5
1             1             1
-1            -1             1
14            14            26
-14           -14            26
0             0             16
-4319         -4319          10723
18            18            10
13           error          13


## zkl

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()));
};

A:=GSL.Matrix(2,2).set(1,2, 3,4);
B:=GSL.Matrix(4,4).set(1,2,3,4, 4,5,6,7, 7,8,9,10, 10,11,12,13);
C:=GSL.Matrix(5,5).set( 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);
T(A,B,C).apply2(test);


{{out}}


1.00,      2.00
3.00,      4.00
Permanent: 10.00, determinant: -2.00
1.00,      2.00,      3.00,      4.00
4.00,      5.00,      6.00,      7.00
7.00,      8.00,      9.00,     10.00
10.00,     11.00,     12.00,     13.00
Permanent: 29556.00, determinant: 0.00
0.00,      1.00,      2.00,      3.00,      4.00
5.00,      6.00,      7.00,      8.00,      9.00
10.00,     11.00,     12.00,     13.00,     14.00
15.00,     16.00,     17.00,     18.00,     19.00
20.00,     21.00,     22.00,     23.00,     24.00
Permanent: 6778800.00, determinant: 0.00

`

[[Category:Geometry]]