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# Determinant and permanent

(Redirected from Matrix arithmetic)
Determinant and permanent
You are encouraged to solve this task according to the task description, using any language you may know.

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

The determinant is given by

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

while the permanent is given by

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

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

## 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         13/05/2016MATARI   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 permanentEXITALL  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         BR     R14                return to callerSAVEAREA DS     18F                main saveareaTT       DC     F'3'               matrix size          DC     F'2',F'9',F'4',F'7',F'5',F'3',F'6',F'1',F'8' <==inputPG1      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         USING  STACK,R10          make storage addressable         LR     R10,R1             establish stack addressability         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               elseSIF1     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=1LOOPK    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=1LOOPIQ   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=1LOOPJT   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+1IF2      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 jtELOOPJT  LA     R7,1(R7)                 iq=iq+1         B      LOOPIQ                 next iqELOOPIQ  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 kELOOPK   EQU    *                    end doEIF1     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         BR     R14                return to callerIT       DS     F                  static area (out of stack)KT       DS     F                  "JQ       DS     F                  "KTI      DS     F                  "P        DS     F                  "         DROP   R12                base no longer neededSTACK    DSECT                     dynamic area (stack)SAVEAREB DS     18F                function saveareaN        DS     F                  nNM1      DS     F                  n-1R        DS     F                  determinant accuS        DS     F                  permanent accuSGN      DS     F                  signSTACKLEN EQU    *-STACKQ        DS     F                  sub matrix q((n-1)*(n-1)+1)         YREGS           END    MATARI
Output:
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 18int 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;}

## C++

Translation of: Java
#include <iostream>#include <vector> template <typename T>std::ostream &operator<<(std::ostream &os, const std::vector<T> &v) {    auto it = v.cbegin();    auto end = v.cend();     os << '[';    if (it != end) {        os << *it;        it = std::next(it);    }    while (it != end) {        os << ", " << *it;        it = std::next(it);    }    return os << ']';} using Matrix = std::vector<std::vector<double>>; Matrix squareMatrix(size_t n) {    Matrix m;    for (size_t i = 0; i < n; i++) {        std::vector<double> inner;        for (size_t j = 0; j < n; j++) {            inner.push_back(nan(""));        }        m.push_back(inner);    }    return m;} Matrix minor(const Matrix &a, int x, int y) {    auto length = a.size() - 1;    auto result = squareMatrix(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 {                result[i][j] = a[i + 1][j + 1];            }        }    }    return result;} double det(const Matrix &a) {    if (a.size() == 1) {        return a[0][0];    }     int sign = 1;    double sum = 0;    for (size_t i = 0; i < a.size(); i++) {        sum += sign * a[0][i] * det(minor(a, 0, i));        sign *= -1;    }    return sum;} double perm(const Matrix &a) {    if (a.size() == 1) {        return a[0][0];    }     double sum = 0;    for (size_t i = 0; i < a.size(); i++) {        sum += a[0][i] * perm(minor(a, 0, i));    }    return sum;} void test(const Matrix &m) {    auto p = perm(m);    auto d = det(m);     std::cout << m << '\n';    std::cout << "Permanent: " << p << ", determinant: " << d << "\n\n";} int main() {    test({ {1, 2}, {3, 4} });    test({ {1, 2, 3, 4}, {4, 5, 6, 7}, {7, 8, 9, 10}, {10, 11, 12, 13} });    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} });     return 0;}
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: 6.7788e+06, determinant: 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)) )
Output:
((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 and Permutations by swapping tasks.

Translation of: 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 @safein {    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;    return tot;} T determinant(T)(in T[][] a) nothrowin {    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;    }    return tot;} 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);    }}
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

## EchoLisp

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

 (lib 'list)(lib 'matrix) ;; adapted from Racket(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

Works with: gforth version 0.7.9_20170427

Requiring a permute.fs file from the Permutations by swapping task.

S" fsl-util.fs" REQUIREDS" fsl/dynmem.seq" REQUIRED[UNDEFINED] defines [IF] SYNONYM defines IS [THEN]S" fsl/structs.seq" REQUIREDS" fsl/lufact.seq" REQUIREDS" fsl/dets.seq" REQUIREDS" 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- }} [email protected] F*  LOOP  DROP  \ Dummy element because we're using 1-based indexing  F+ ;: permanent ( len mat -- ) ( F: -- perm )  the-mat !  0E  ['] add-perm perms ; 3 SET-PRECISION2 2 float matrix m2{{1e 2e  3e 4e  2 2 m2{{ }}fputlumatrix lmat3 3 float matrix m3{{2e 9e 4e  7e 5e 3e  6e 1e 8e  3 3 m3{{ }}fput lmat 2 lu-mallocm2{{ lmat lufactlmat det F. 2 m2{{ permanent F. CRlmat lu-free lmat 3 lu-mallocm3{{ lmat lufactlmat det F. 3 m3{{ permanent F. CRlmat 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 [email protected]$5! 2 _1 1!_1 _2 1!_1 _1 _1!! (-/ .*)_2+3 [email protected]$5  NB. determinant!7!   (+/ .*)_2+3 [email protected]$5 NB. permanent!5 !maxima example!a: matrix([2, 9, 4], [7, 5, 3], [6, 1, 8])$!determinant(a);!-360! !permanent(a);!900  ! compute permanent or determinantprogram 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) )
Output:
((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 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))}
Output:
-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}
Output:
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())}
Output:
-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})))}
Output:
-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 -> aelemPos ms i j = (ms !! i) !! j prod :: Num a => ([[a]] -> Int -> Int -> a) -> [[a]] -> [Int] -> aprod f ms = product . zipWith (f ms) [0 ..] sDeterminant :: Num a => ([[a]] -> Int -> Int -> a) -> [[a]] -> [([Int], Int)] -> asDeterminant f ms = sum . fmap (\(is, s) -> fromIntegral s * prod f ms is) determinant :: Num a => [[a]] -> adeterminant ms = sDeterminant elemPos ms . sPermutations$ [0 .. pred . length $ms] permanent :: Num a => [[a]] -> apermanent ms = sum . fmap (prod elemPos ms . fst) . sPermutations$ [0 .. pred . length $ms] -- TEST -----------------------------------------------------------------------result :: (Num a, Show a) => [[a]] -> Stringresult ms = unlines [ "Matrix:" , unlines (show <$> 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]]    ]
Output:
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 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 <$> x2) : outer f t1 x2 dot [] [] = 0dot (h1:t1) (h2:t2) = (h1*h2) + (dot t1 t2) transpose [] = []transpose ([] : xss) = transpose xsstranspose ((x:xs) : xss) = (x : [h | (h:_) <- xss]) : transpose (xs : [ t | (_:t) <- xss]) mul :: Num a => [[a]] -> [[a]] -> [[a]]mul a b = outer dot a (transpose b) delRow :: Int -> [a] -> [a]delRow i v = (first ++ rest) where (first, _:rest) = splitAt i v delCol :: Int -> [[a]] -> [[a]]delCol j m = (delRow j) <$> m -- Determinant:adj :: Num a => [[a]] -> [[a]]adj [] = []adj m =  [    [(-1)^(i+j) * det (delRow i $delCol j m) | i <- [0.. -1+length m] ] | j <- [0.. -1+length m] ]det :: Num a => [[a]] -> adet [] = 1det m = (mul m (adj m)) !! 0 !! 0 -- Permanent:padj :: Num a => [[a]] -> [[a]]padj [] = []padj m = [ [perm (delRow i$ delCol j m)    | i <- [0.. -1+length m]    ]  | j <- [0.. -1+length m]  ]perm :: Num a => [[a]] -> aperm [] = 1perm m  = (mul m (padj m)) !! 0 !! 0

## J

J has a 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  910 11 12 13 1415 16 17 18 1920 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 5x0

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 56778800

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 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:

21 23 4Determinant: -2.0Permanent: 10.0  50 1 2 3 45 6 7 8 910 11 12 13 1415 16 17 18 1920 21 22 23 24Determinant: 0.0Permanent: 6778800.0

## JavaScript

const determinant = arr =>    arr.length === 1 ? (        arr[0][0]    ) : arr[0].reduce(        (sum, v, i) => sum + v * (-1) ** i * determinant(            arr.slice(1)            .map(x => x.filter((_, j) => i !== j))        ), 0    ); const permanent = arr =>    arr.length === 1 ? (        arr[0][0]    ) : arr[0].reduce(        (sum, v, i) => sum + v * permanent(            arr.slice(1)            .map(x => x.filter((_, j) => i !== j))        ), 0    ); const 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]];console.log(determinant(M));console.log(permanent(M));
Output:
0
6778800

## jq

Works with: jq version 1.4

#### Recursive definitions

# Eliminate row i and row jdef 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) Output: $ jq -n -r -f Matrix_arithmetic.jqDeterminants: -21800Permanents:   1010295566778800

#### Determinant via LU Decomposition

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

# Requires lup/0def 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  using LinearAlgebra The determinant of a matrix A can be computed by the built-in function det(A) Translation of: 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: julia> 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")    } }
Output:
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 nend 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 0end 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 trueend   -- matrix class _MTX={}function MTX(matrix)  setmetatable(matrix,{__index=_MTX})  matrix.rows=#matrix  matrix.cols=#matrix[1]  return matrixend function _MTX:dump()  for _,r in ipairs(self) do    print(unpack(r))  endend function _MTX:perm() return self:det(1) endfunction _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 detend -- 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())
Output:
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

## 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 ## 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. ## Nim Translation of: Python Using the permutationsswap module from 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.detecho "perm: ", b.perm, " det: ", b.detecho "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 version 2.10.0+ As of version 2.10, the matrix permanent is built in: matpermanent(M) ## Perl Translation of: C #!/usr/bin/perluse 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($M) . ".\n"; determinant and det are already defined in PDL, see[1]. permanent has to be defined manually. Output: M = [ [2 9 4] [7 5 3] [6 1 8] ] det(M) = -360. det(M) = -360. perm(M) = 900.  ## Phix Translation of: Java function minor(sequence a, integer x, integer y)integer l = length(a)-1sequence result = repeat(repeat(0,l),l) for i=1 to l do for j=1 to l do result[i][j] = a[i+(i>=x)][j+(j>=y)] end for end for return resultend function function det(sequence a) if length(a)=1 then return a[1][1] end if integer sgn = 1 integer res = 0 for i=1 to length(a) do res += sgn*a[1][i]*det(minor(a,1,i)) sgn *= -1 end for return resend function function perm(sequence a) if length(a)=1 then return a[1][1] end if integer res = 0 for i=1 to length(a) do res += a[1][i]*perm(minor(a,1,i)) end for return resend function constant tests = {{{1, 2}, {3, 4}},--Determinant: -2, permanent: 10{{2, 9, 4}, {7, 5, 3}, {6, 1, 8}},--Determinant: -360, permanent: 900{{ 1, 2, 3, 4}, { 4, 5, 6, 7}, { 7, 8, 9, 10}, {10, 11, 12, 13}},--Determinant: 0, permanent: 29556{{ 0, 1, 2, 3, 4}, { 5, 6, 7, 8, 9}, {10, 11, 12, 13, 14}, {15, 16, 17, 18, 19}, {20, 21, 22, 23, 24}},--Determinant: 0, permanent: 6778800{{5}},--Determinant: 5, permanent: 5 {{1,0,0}, {0,1,0}, {0,0,1}},--Determinant: 1, permanent: 1{{0,0,1}, {0,1,0}, {1,0,0}},--Determinant: -1, Permanent: 1{{4,3}, {2,5}},--Determinant: 14, Permanent: 26{{2,5}, {4,3}},--Determinant: -14, Permanent: 26{{4,4}, {2,2}},--Determinant: 0, Permanent: 16{{7, 2, -2, 4}, {4, 4, 1, 7}, {11, -8, 9, 10}, {10, 5, 12, 13}},--det: -4319 permanent: 10723 {{-2, 2, -3}, {-1, 1, 3}, {2 , 0, -1}}--det: 18 permanent: 10}for i=1 to length(tests) do sequence ti = tests[i] ?{det(ti),perm(ti)}end for Output: {-2,10} {-360,900} {0,29556} {0,6778800} {5,5} {1,1} {-1,1} {14,26} {-14,26} {0,16} {-4319,10723} {18,10}  ## PowerShell  function det-perm ($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 5det-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. The algorithm for the determinant is a more literal translation of the expression in the task description and the Wikipedia reference.

from itertools import permutationsfrom operator import mulfrom math import fsumfrom 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))))

## Raku

(formerly Perl 6)

Works with: Rakudo version 2015.12

Uses the permutations generator from the 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;    sum σ_permutations([^@a]).race.map: {        my $permutation = .key; my$term = $op eq 'perm' ?? 1 !! .value; for$permutation.kv -> $i,$j { $term *= @a[$i][$j] };$term    }} ######### helper subs #########sub hilbert-matrix (\h) {[(1..h).map(-> \n {[(n..^n+h).map: {(1/$_).FatRat}]})]} sub rat-or-int ($num) {    return $num unless$num ~~ Rat|FatRat;    return $num.narrow if$num.narrow.WHAT ~~ Int;    $num.nude.join: '/';} sub say-it ($message, @array) {    my $max; @array.map: {$max max= max $_».&rat-or-int.comb(/\S+/)».chars}; say "\n$message";    $_».&rat-or-int.fmt(" %{$max}s").put for @array;} ########### Testing ###########my @tests = (    [        [ 1, 2 ],        [ 3, 4 ]    ],    [        [  1,  2,  3,  4 ],        [  4,  5,  6,  7 ],        [  7,  8,  9, 10 ],        [ 10, 11, 12, 13 ]    ],    hilbert-matrix 7); for @tests -> @matrix {    say-it 'Matrix:', @matrix;    say "Determinant:\t", rat-or-int @matrix.&m_arith: <det>;    say "Permanent:  \t", rat-or-int @matrix.&m_arith: <perm>;    say '-' x 40;}

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:
1   1/2   1/3   1/4   1/5   1/6   1/7
1/2   1/3   1/4   1/5   1/6   1/7   1/8
1/3   1/4   1/5   1/6   1/7   1/8   1/9
1/4   1/5   1/6   1/7   1/8   1/9  1/10
1/5   1/6   1/7   1/8   1/9  1/10  1/11
1/6   1/7   1/8   1/9  1/10  1/11  1/12
1/7   1/8   1/9  1/10  1/11  1/12  1/13
Determinant:	1/2067909047925770649600000
Permanent:  	29453515169174062608487/2067909047925770649600000
----------------------------------------

## 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,nasc=asDo 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  EndSay '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=0rowsumprod=0rowsum=0chi.=0c=2**nDo 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  EndReturn 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 asn=sqrt(words(as))a.=0Do ai=0 By 1 While as<>''  Parse Var as a.ai as  EndReturn 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  endend 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.

Translation of: 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"}
Output:
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


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

## 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 = p*x	}	return(s)}

Example:

: 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:

Library: Tcllib (Package: math::linearalgebra)
Library: Tcllib (Package: struct::list)
package require math::linearalgebrapackage 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]
Output:
1.1315223609263888e-29
29556


## Visual Basic .NET

Translation of: Java
Module Module1     Function Minor(a As Double(,), x As Integer, y As Integer) As Double(,)        Dim length = a.GetLength(0) - 1        Dim result(length - 1, length - 1) As Double        For i = 1 To length            For j = 1 To length                If i < x AndAlso j < y Then                    result(i - 1, j - 1) = a(i - 1, j - 1)                ElseIf i >= x AndAlso j < y Then                    result(i - 1, j - 1) = a(i, j - 1)                ElseIf i < x AndAlso j >= y Then                    result(i - 1, j - 1) = a(i - 1, j)                Else                    result(i - 1, j - 1) = a(i, j)                End If            Next        Next        Return result    End Function     Function Det(a As Double(,)) As Double        If a.GetLength(0) = 1 Then            Return a(0, 0)        Else            Dim sign = 1            Dim sum = 0.0            For i = 1 To a.GetLength(0)                sum += sign * a(0, i - 1) * Det(Minor(a, 0, i))                sign *= -1            Next            Return sum        End If    End Function     Function Perm(a As Double(,)) As Double        If a.GetLength(0) = 1 Then            Return a(0, 0)        Else            Dim sum = 0.0            For i = 1 To a.GetLength(0)                sum += a(0, i - 1) * Perm(Minor(a, 0, i))            Next            Return sum        End If    End Function     Sub WriteLine(a As Double(,))        For i = 1 To a.GetLength(0)            Console.Write("[")            For j = 1 To a.GetLength(1)                If j > 1 Then                    Console.Write(", ")                End If                Console.Write(a(i - 1, j - 1))            Next            Console.WriteLine("]")        Next    End Sub     Sub Test(a As Double(,))        If a.GetLength(0) <> a.GetLength(1) Then            Throw New ArgumentException("The dimensions must be equal")        End If         WriteLine(a)        Console.WriteLine("Permanant  : {0}", Perm(a))        Console.WriteLine("Determinant: {0}", Det(a))        Console.WriteLine()    End Sub     Sub Main()        Test({{1, 2}, {3, 4}})        Test({{1, 2, 3, 4}, {4, 5, 6, 7}, {7, 8, 9, 10}, {10, 11, 12, 13}})        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}})    End Sub End Module
Output:
[1, 2]
[3, 4]
Permanant  : 10
Determinant: -2

[1, 2, 3, 4]
[4, 5, 6, 7]
[7, 8, 9, 10]
[10, 11, 12, 13]
Permanant  : 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]
Permanant  : 6778800
Determinant: 0

Press any key to continue . . .

## VBA

Translation of: Phix

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

Option Base 1Private 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 = resultEnd 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 Functionerr:    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 Functionerr:    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
Output:
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);
Output:
      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