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
while the permanent is given by
In both cases the sum is over the permutations of the permutations of 1, 2, ..., n. (A permutation's sign is 1 if there are an even number of inversions and -1 otherwise; see parity of a permutation.)
More efficient algorithms for the determinant are known: LU decomposition, see for example wp:LU decomposition#Computing the determinant. Efficient methods for calculating the permanent are not known.
- Related task
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. <lang 360asm>* 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
BR R14 return to caller
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
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 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
BR R14 return to caller
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</lang>
- 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. <lang C>#include <stdio.h>
- include <stdlib.h>
- include <string.h>
double det_in(double **in, int n, int perm) { if (n == 1) return in[0][0];
double sum = 0, *m[--n]; for (int i = 0; i < n; i++) m[i] = in[i + 1] + 1;
for (int i = 0, sgn = 1; i <= n; i++) { sum += sgn * (in[i][0] * det_in(m, n, perm)); if (i == n) break;
m[i] = in[i] + 1; if (!perm) sgn = -sgn; } return sum; }
/* wrapper function */ double det(double *in, int n, int perm) { double *m[n]; for (int i = 0; i < n; i++) m[i] = in + (n * i);
return det_in(m, n, perm); }
int main(void) { double x[] = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 };
printf("det: %14.12g\n", det(x, 5, 0)); printf("perm: %14.12g\n", det(x, 5, 1));
return 0; }</lang> A method to calculate determinant that might actually be usable: <lang c>#include <stdio.h>
- include <stdlib.h>
- include <tgmath.h>
void showmat(const char *s, double **m, int n) { printf("%s:\n", s); for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) printf("%12.4f", m[i][j]); putchar('\n'); } }
int trianglize(double **m, int n) { int sign = 1; for (int i = 0; i < n; i++) { int max = 0;
for (int row = i; row < n; row++) if (fabs(m[row][i]) > fabs(m[max][i])) max = row;
if (max) { sign = -sign; double *tmp = m[i]; m[i] = m[max], m[max] = tmp; }
if (!m[i][i]) return 0;
for (int row = i + 1; row < n; row++) { double r = m[row][i] / m[i][i]; if (!r) continue;
for (int col = i; col < n; col ++) m[row][col] -= m[i][col] * r; } } return sign; }
double det(double *in, int n) { double *m[n]; m[0] = in;
for (int i = 1; i < n; i++) m[i] = m[i - 1] + n;
showmat("Matrix", m, n);
int sign = trianglize(m, n); if (!sign) return 0;
showmat("Upper triangle", m, n);
double p = 1; for (int i = 0; i < n; i++) p *= m[i][i]; return p * sign; }
- define N 18
int main(void) { double x[N * N]; srand(0); for (int i = 0; i < N * N; i++) x[i] = rand() % N;
printf("det: %19f\n", det(x, N)); return 0; }</lang>
CommonLisp
A recursive version, no libraries required, it doesn't use much consing, only for the list of columns to skip
<lang lisp> (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)) )
</lang>
- 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.
<lang d>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;
return tot;
}
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;
}
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);
}
}</lang>
- Output:
[[ 1, 2], [ 3, 4]] Permanent: 10, determinant: -2 [[ 1, 2, 3, 4], [ 4, 5, 6, 7], [ 7, 8, 9, 10], [10, 11, 12, 13]] Permanent: 29556, determinant: 0 [[ 0, 1, 2, 3, 4], [ 5, 6, 7, 8, 9], [10, 11, 12, 13, 14], [15, 16, 17, 18, 19], [20, 21, 22, 23, 24]] Permanent: 6778800, determinant: 0
EchoLisp
This requires the 'list' library for (in-permutations n) and the 'matrix' library for the built-in (determinant M). <lang lisp> (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
</lang>
Forth
Requiring a permute.fs file from the Permutations by swapping task. <lang forth>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 ['] add-perm perms ;
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</lang>
Fortran
Please find the compilation and example run at the start of the comments in the f90 source. Thank you.
<lang FORTRAN> !-*- mode: compilation; default-directory: "/tmp/" -*- !Compilation started at Sat May 18 23:25:42 ! !a=./F && make $a && $a < unixdict.txt !f95 -Wall -ffree-form F.F -o F ! j example, determinant: 7.00000000 ! j example, permanent: 5.00000000 ! maxima, determinant: -360.000000 ! maxima, permanent: 900.000000 ! !Compilation finished at Sat May 18 23:25:43
! NB. example computed by J ! NB. fixed seed random matrix ! _2+3 3?.@$5 ! 2 _1 1 !_1 _2 1 !_1 _1 _1 ! ! (-/ .*)_2+3 3?.@$5 NB. determinant !7 ! (+/ .*)_2+3 3?.@$5 NB. permanent !5
!maxima example !a: matrix([2, 9, 4], [7, 5, 3], [6, 1, 8])$ !determinant(a); !-360 ! !permanent(a); !900
! compute permanent or determinant
program f
implicit none real, dimension(3,3) :: j, m data j/ 2,-1, 1,-1,-2, 1,-1,-1,-1/ data m/2, 9, 4, 7, 5, 3, 6, 1, 8/ write(6,*) 'j example, determinant: ',det(j,3,-1) write(6,*) 'j example, permanent: ',det(j,3,1) write(6,*) 'maxima, determinant: ',det(m,3,-1) write(6,*) 'maxima, permanent: ',det(m,3,1)
contains
recursive function det(a,n,permanent) result(accumulation)
! setting permanent to 1 computes the permanent.
! setting permanent to -1 computes the determinant.
real, dimension(n,n), intent(in) :: a
integer, intent(in) :: n, permanent
real, dimension(n-1, n-1) :: b
real :: accumulation
integer :: i, sgn
if (n .eq. 1) then
accumulation = a(1,1)
else
accumulation = 0
sgn = 1
do i=1, n
b(:, :(i-1)) = a(2:, :i-1)
b(:, i:) = a(2:, i+1:)
accumulation = accumulation + sgn * a(1, i) * det(b, n-1, permanent)
sgn = sgn * permanent
enddo
endif
end function det
end program f </lang>
FunL
From the task description: <lang 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() )</lang>
Laplace expansion (recursive): <lang funl>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() )</lang>
Test using the first set of definitions (from task description): <lang funl>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) )</lang>
- 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
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. <lang go>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))
}</lang>
- Output:
-2 10 -360 900
Ryser permanent
<lang go>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
}</lang>
- Output:
10 900
Library determinant
go.matrix: <lang go>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())
}</lang>
- Output:
-2 -360
gonum/mat: <lang go>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})))
}</lang>
- Output:
-2 -360.00000000000006
Haskell
<lang Haskell>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 <$> ms) , "Determinant:" , show (determinant ms) , "Permanent:" , show (permanent ms) ]
main :: IO () main =
mapM_ (putStrLn . result) [ 5 , [[1, 0, 0], [0, 1, 0], [0, 0, 1]] , [[0, 0, 1], [0, 1, 0], [1, 0, 0]] , [[4, 3], [2, 5]] , [[2, 5], [4, 3]] , [[4, 4], [2, 2]] ]</lang>
- Output:
Matrix: [5] Determinant: 5 Permanent: 5 Matrix: [1,0,0] [0,1,0] [0,0,1] Determinant: 1 Permanent: 1 Matrix: [0,0,1] [0,1,0] [1,0,0] Determinant: -1 Permanent: 1 Matrix: [4,3] [2,5] Determinant: 14 Permanent: 26 Matrix: [2,5] [4,3] Determinant: -14 Permanent: 26 Matrix: [4,4] [2,2] Determinant: 0 Permanent: 16
Via Cramer's rule
Here is code for computing the determinant and permanent very inefficiently, via Cramer's rule (for the determinant, as well as its analog for the permanent):
<lang Haskell> outer :: (a->b->c) -> [a] -> [b] -> c outer f [] _ = [] outer f _ [] = [] outer f (h1:t1) x2 = (f h1 <$> 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 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 -> 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
</lang>
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:
<lang J> i. 5 5
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24</lang>
Its determinant is 0. When we use IEEE floating point, we only get an approximation of this result:
<lang J> -/ .* i. 5 5 _1.30277e_44</lang>
If we use exact (rational) arithmetic, we get a precise result:
<lang J> -/ .* i. 5 5x 0</lang>
Meanwhile, the permanent does not have this problem in this example (the matrix contains no negative values and permanent does not use subtraction):
<lang J> +/ .* i. 5 5 6778800</lang>
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
<lang 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)); } }</lang>
Note that the first input is the size of the matrix.
For example:
<lang>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
</lang>
jq
Recursive definitions
<lang 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 ;</lang>
Examples <lang jq>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)</lang>
- Output:
<lang sh>$ jq -n -r -f Matrix_arithmetic.jq Determinants: -2 18 0 0 Permanents: 10 10 29556 6778800</lang>
Determinant via LU Decomposition
The following uses the jq infrastructure at LU decomposition to achieve an efficient implementation of det/0: <lang jq># 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 ;
</lang> Examples
Using matrices/0 as defined above: <lang jq>matrices | det</lang>
- Output:
$ /usr/local/bin/jq -M -n -f LU.rc 2 -18 0 0
Julia
The determinant of a matrix A can be computed by the built-in function
<lang julia>det(A)</lang>
The following function computes the permanent of a matrix A from the definition: <lang julia>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</lang>
Example output: <lang julia>julia> A = [2 9 4; 7 5 3; 6 1 8] julia> det(A), perm(A) (-360.0,900)</lang>
Kotlin
<lang scala>// 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")
}
}</lang>
- 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
<lang 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()) </lang>
- 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
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.
Maple
<lang Maple>M:=<<2|9|4>,<7|5|3>,<6|1|8>>:
with(LinearAlgebra):
Determinant(M); Permanent(M);</lang> Output:
-360
900
Mathematica
Determinant is a built in function Det <lang Mathematica>Permanent[m_List] :=
With[{v = Array[x, Length[m]]},
Coefficient[Times @@ (m.v), Times @@ v]
]</lang>
Maxima
<lang maxima>a: matrix([2, 9, 4], [7, 5, 3], [6, 1, 8])$
determinant(a); -360
permanent(a); 900</lang>
Nim
Using the permutationsswap module from Permutations by swapping: <lang nim>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</lang> Output:
perm: 10.0 det: -2.0 perm: 29556.0 det: 0.0 perm: 6778800.0 det: 0.0
Ol
<lang scheme>
- 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
</lang>
PARI/GP
The determinant is built in: <lang parigp>matdet(M)</lang> and the permanent can be defined as <lang parigp>matperm(M)=my(n=#M,t);sum(i=1,n!,t=numtoperm(n,i);prod(j=1,n,M[j,t[j]]))</lang> For better performance, here's a version using Ryser's formula: <lang parigp>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; }</lang>
As of version 2.10, the matrix permanent is built in: <lang parigp>matpermanent(M)</lang>
Perl
<lang perl>#!/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";</lang>
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.
Perl 6
Uses the permutations generator from the Permutations by swapping task. This implementation is naive and brute-force (slow) but exact.
<lang perl6>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;
}</lang>
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
<lang Phix>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 Template: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</lang>
- 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
<lang 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)) </lang> 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.
<lang python>from itertools import permutations from operator import mul from math import fsum from spermutations import spermutations
def prod(lst):
return reduce(mul, lst, 1)
def perm(a):
n = len(a) r = range(n) s = permutations(r) return fsum(prod(a[i][sigma[i]] for i in r) for sigma in s)
def det(a):
n = len(a)
r = range(n)
s = spermutations(n)
return fsum(sign * prod(a[i][sigma[i]] for i in r)
for sigma, sign in s)
if __name__ == '__main__':
from pprint import pprint as pp
for a in (
[
[1, 2],
[3, 4]],
[
[1, 2, 3, 4],
[4, 5, 6, 7],
[7, 8, 9, 10],
[10, 11, 12, 13]],
[
[ 0, 1, 2, 3, 4],
[ 5, 6, 7, 8, 9],
[10, 11, 12, 13, 14],
[15, 16, 17, 18, 19],
[20, 21, 22, 23, 24]],
):
print()
pp(a)
print('Perm: %s Det: %s' % (perm(a), det(a)))</lang>
- Sample output
[[1, 2], [3, 4]] Perm: 10 Det: -2 [[1, 2, 3, 4], [4, 5, 6, 7], [7, 8, 9, 10], [10, 11, 12, 13]] Perm: 29556 Det: 0 [[0, 1, 2, 3, 4], [5, 6, 7, 8, 9], [10, 11, 12, 13, 14], [15, 16, 17, 18, 19], [20, 21, 22, 23, 24]] Perm: 6778800 Det: 0
The second matrix above is that used in the Tcl example. The third matrix is from the J language example. Note that the determinant seems to be 'exact' using this method of calculation without needing to resort to other than Pythons default numbers.
Racket
<lang racket>
- lang racket
(require math) (define determinant matrix-determinant)
(define (permanent M)
(define n (matrix-num-rows M))
(for/sum ([σ (in-permutations (range n))])
(for/product ([i n] [σi σ])
(matrix-ref M i σi))))
</lang>
REXX
<lang rexx>/* REXX ***************************************************************
- 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</lang>
<lang rexx>/* 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</lang>
<lang rexx>/* 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</lang>
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. <lang ruby>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</lang>
- Output:
determinant: -2 permanent: 10 determinant: 0 permanent: 29556 determinant: 0 permanent: 6778800
Scala
<lang 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})
</lang>
Sidef
The `determinant` method is provided by the Array class.
<lang 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"
}</lang>
- Output:
determinant: -2 permanent: 10 determinant: 0 permanent: 29556 determinant: 0 permanent: 6778800
Simula
<lang 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</lang> 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
<lang SPAD>(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</lang>
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.
<lang>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) }</lang>
Example:
<lang 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</lang>
Tcl
The determinant is provided by the linear algebra package in Tcllib. The permanent (being somewhat less common) requires definition, but is easily described:
<lang tcl>package require math::linearalgebra package require struct::list
proc permanent {matrix} {
for {set plist {};set i 0} {$i<[llength $matrix]} {incr i} {
lappend plist $i
}
foreach p [::struct::list permutations $plist] {
foreach i $plist j $p { lappend prod [lindex $matrix $i $j] } lappend sum [::tcl::mathop::* {*}$prod[set prod {}]]
}
return [::tcl::mathop::+ {*}$sum]
}</lang> Demonstrating with a sample matrix: <lang tcl>set mat {
{1 2 3 4}
{4 5 6 7}
{7 8 9 10}
{10 11 12 13}
} puts [::math::linearalgebra::det $mat] puts [permanent $mat]</lang>
- Output:
1.1315223609263888e-29 29556
VBA
As an extra, the results of the built in WorksheetFuction.MDeterm are shown. The latter does not work for scalars. <lang vb>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</lang>
- 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
<lang 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()));
};</lang> <lang zkl>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);</lang>
- 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