# Reduced row echelon form

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Reduced row echelon form
You are encouraged to solve this task according to the task description, using any language you may know.

Show how to compute the reduced row echelon form (a.k.a. row canonical form) of a matrix.

The matrix can be stored in any datatype that is convenient (for most languages, this will probably be a two-dimensional array).

Built-in functions or this pseudocode (from Wikipedia) may be used:

function ToReducedRowEchelonForm(Matrix M) is
rowCount := the number of rows in M
columnCount := the number of columns in M
for 0 ≤ r < rowCount do
stop
end if
i = r
while M[i, lead] = 0 do
i = i + 1
if rowCount = i then
i = r
stop
end if
end if
end while
Swap rows i and r
If M[r, lead] is not 0 divide row r by M[r, lead]
for 0 ≤ i < rowCount do
if i ≠ r do
Subtract M[i, lead] multiplied by row r from row i
end if
end for
end for
end function


For testing purposes, the RREF of this matrix:

 1    2   -1   -4
2    3   -1   -11
-2    0   -3    22


is:

 1    0    0   -8
0    1    0    1
0    0    1   -2


## 11l

Translation of: Python
F ToReducedRowEchelonForm(&M)
V rowCount = M.len
V columnCount = M[0].len
L(r) 0 .< rowCount
R
V i = r
i++
I i == rowCount
i = r
R
swap(&M[i], &M[r])
M[r] = M[r].map(mrx -> mrx / Float(@lv))
L(i) 0 .< rowCount
I i != r
M[i] = zip(M[r], M[i]).map((rv, iv) -> iv - @lv * rv)

V mtx = [[ 1.0, 2.0, -1.0,  -4.0],
[ 2.0, 3.0, -1.0, -11.0],
[-2.0, 0.0, -3.0,  22.0]]

ToReducedRowEchelonForm(&mtx)

L(rw) mtx
print(rw.join(‘, ’))
Output:
1, 0, 0, -8
0, 1, 0, 1
0, 0, 1, -2


## 360 Assembly

Translation of: BBC BASIC
*        reduced row echelon form  27/08/2015
RREF     CSECT
USING  RREF,R12
LR     R12,R15
LA     R7,1
LOOPR    CH     R7,NROWS           do r=1 to nrows
BH     ELOOPR
BNL    ELOOPR
LR     R8,R7              i=r
WHILE    LR     R1,R8              do while m(i,lead)=0
BCTR   R1,0
MH     R1,NCOLS
BCTR   R6,0
AR     R1,R6
SLA    R1,2
LTR    R6,R6
LA     R8,1(R8)           i=i+1
CH     R8,NROWS           if i=nrows
BNE    EIF
LR     R8,R7              i=r
BE     ELOOPR
EIF      B      WHILE
EWHILE   LA     R9,1
LOOPJ1   CH     R9,NCOLS           do j=1 to ncols
BH     ELOOPJ1
LR     R1,R7              r
BCTR   R1,0
MH     R1,NCOLS
LR     R6,R9              j
BCTR   R6,0
AR     R1,R6
SLA    R1,2
LA     R3,M(R1)           R3=@m(r,j)
LR     R1,R8              i
BCTR   R1,0
MH     R1,NCOLS
LR     R6,R9              j
BCTR   R6,0
AR     R1,R6
SLA    R1,2
LA     R4,M(R1)           R4=@m(i,j)
L      R2,0(R3)
MVC    0(2,R3),0(R4)      swap m(i,j),m(r,j)
ST     R2,0(R4)
LA     R9,1(R9)           j=j+1
B      LOOPJ1
ELOOPJ1  LR     R1,R7              r
BCTR   R1,0
MH     R1,NCOLS
BCTR   R6,0
AR     R1,R6
SLA    R1,2
CH     R11,=H'1'          if n^=1
BE     ELOOPJ2
LA     R9,1
LOOPJ2   CH     R9,NCOLS           do j=1 to ncols
BH     ELOOPJ2
LR     R1,R7              r
BCTR   R1,0
MH     R1,NCOLS
LR     R6,R9              j
BCTR   R6,0
AR     R1,R6
SLA    R1,2
LA     R5,M(R1)           R5=@m(i,j)
L      R2,0(R5)           m(r,j)
LR     R1,R11             n
SRDA   R2,32
DR     R2,R1              m(r,j)/n
ST     R3,0(R5)           m(r,j)=m(r,j)/n
LA     R9,1(R9)           j=j+1
B      LOOPJ2
ELOOPJ2  LA     R8,1
LOOPI3   CH     R8,NROWS           do i=1 to nrows
BH     ELOOPI3
CR     R8,R7              if i^=r
BE     ELOOPJ3
LR     R1,R8              i
BCTR   R1,0
MH     R1,NCOLS
BCTR   R6,0
AR     R1,R6
SLA    R1,2
LA     R9,1
LOOPJ3   CH     R9,NCOLS           do j=1 to ncols
BH     ELOOPJ3
LR     R1,R8              i
BCTR   R1,0
MH     R1,NCOLS
LR     R6,R9              j
BCTR   R6,0
AR     R1,R6
SLA    R1,2
LA     R4,M(R1)           R4=@m(i,j)
L      R5,0(R4)           m(i,j)
LR     R1,R7              r
BCTR   R1,0
MH     R1,NCOLS
LR     R6,R9              j
BCTR   R6,0
AR     R1,R6
SLA    R1,2
L      R3,M(R1)           m(r,j)
MR     R2,R11             m(r,j)*n
SR     R5,R3              m(i,j)-m(r,j)*n
ST     R5,0(R4)           m(i,j)=m(i,j)-m(r,j)*n
LA     R9,1(R9)           j=j+1
B      LOOPJ3
ELOOPJ3  LA     R8,1(R8)           i=i+1
B      LOOPI3
LA     R7,1(R7)           r=r+1
B      LOOPR
ELOOPR   LA     R8,1
LOOPI4   CH     R8,NROWS           do i=1 to nrows
BH     ELOOPI4
SR     R10,R10            pgi=0
LA     R9,1
LOOPJ4   CH     R9,NCOLS           do j=1 to ncols
BH     ELOOPJ4
LR     R1,R8              i
BCTR   R1,0
MH     R1,NCOLS
LR     R6,R9              j
BCTR   R6,0
AR     R1,R6
SLA    R1,2
L      R6,M(R1)           m(i,j)
LA     R3,PG
AR     R3,R10
XDECO  R6,0(R3)           edit m(i,j)
LA     R10,12(10)         pgi=pgi+12
LA     R9,1(R9)           j=j+1
B      LOOPJ4
ELOOPJ4  XPRNT  PG,48              print m(i,j)
LA     R8,1(R8)           i=i+1
B      LOOPI4
ELOOPI4  XR     R15,R15
BR     R14
NROWS    DC     H'3'
NCOLS    DC     H'4'
M        DC     F'1',F'2',F'-1',F'-4'
DC     F'2',F'3',F'-1',F'-11'
DC     F'-2',F'0',F'-3',F'22'
PG       DC     CL48' '
YREGS
END    RREF
Output:
           1           0           0          -8
0           1           0           1
0           0           1          -2


## ActionScript

_m being of type Vector.<Vector.<Number>> the following function is a method of Matrix class. Therefore return this statements are returning the Matrix object itself.

public function RREF():Matrix {
var lead:uint, i:uint, j:uint, r:uint = 0;

for(r = 0; r < rows; r++) {
break;
i = r;

i++;

if(rows == i) {
i = r;

return this;
}
}
rowSwitch(i, r);

for(j = 0; j < columns; j++)
_m[r][j] /= val;

for(i = 0; i < rows; i++) {
if(i == r)
continue;

for(j = 0; j < columns; j++)
_m[i][j] -= val * _m[r][j];
}
}
return this;
}


generic
type Element_Type is private;
Zero : Element_Type;
with function "-" (Left, Right : in Element_Type) return Element_Type is <>;
with function "*" (Left, Right : in Element_Type) return Element_Type is <>;
with function "/" (Left, Right : in Element_Type) return Element_Type is <>;
package Matrices is
type Matrix is
array (Positive range <>, Positive range <>) of Element_Type;
function Reduced_Row_Echelon_form (Source : Matrix) return Matrix;
end Matrices;


package body Matrices is
procedure Swap_Rows (From : in out Matrix; First, Second : in Positive) is
Temporary : Element_Type;
begin
for Col in From'Range (2) loop
Temporary          := From (First, Col);
From (First, Col)  := From (Second, Col);
From (Second, Col) := Temporary;
end loop;
end Swap_Rows;

procedure Divide_Row
(From    : in out Matrix;
Row     : in Positive;
Divisor : in Element_Type)
is
begin
for Col in From'Range (2) loop
From (Row, Col) := From (Row, Col) / Divisor;
end loop;
end Divide_Row;

procedure Subtract_Rows
(From                : in out Matrix;
Subtrahend, Minuend : in Positive;
Factor              : in Element_Type)
is
begin
for Col in From'Range (2) loop
From (Minuend, Col) := From (Minuend, Col) -
From (Subtrahend, Col) * Factor;
end loop;
end Subtract_Rows;

function Reduced_Row_Echelon_form (Source : Matrix) return Matrix is
Result : Matrix   := Source;
Lead   : Positive := Result'First (2);
I      : Positive;
begin
Rows : for Row in Result'Range (1) loop
exit Rows when Lead > Result'Last (2);
I := Row;
while Result (I, Lead) = Zero loop
I := I + 1;
if I = Result'Last (1) then
I    := Row;
exit Rows when Lead = Result'Last (2);
end if;
end loop;
if I /= Row then
Swap_Rows (From => Result, First => I, Second => Row);
end if;
Divide_Row
(From    => Result,
Row     => Row,
for Other_Row in Result'Range (1) loop
if Other_Row /= Row then
Subtract_Rows
(From       => Result,
Subtrahend => Row,
Minuend    => Other_Row,
end if;
end loop;
end loop Rows;
return Result;
end Reduced_Row_Echelon_form;
end Matrices;


with Matrices;
procedure Main is
package Float_IO is new Ada.Text_IO.Float_IO (Float);
package Float_Matrices is new Matrices (
Element_Type => Float,
Zero => 0.0);
procedure Print_Matrix (Matrix : in Float_Matrices.Matrix) is
begin
for Row in Matrix'Range (1) loop
for Col in Matrix'Range (2) loop
Float_IO.Put (Matrix (Row, Col), 0, 0, 0);
end loop;
end loop;
end Print_Matrix;
My_Matrix : Float_Matrices.Matrix :=
((1.0, 2.0, -1.0, -4.0),
(2.0, 3.0, -1.0, -11.0),
(-2.0, 0.0, -3.0, 22.0));
Reduced   : Float_Matrices.Matrix :=
Float_Matrices.Reduced_Row_Echelon_form (My_Matrix);
begin
Print_Matrix (My_Matrix);
Print_Matrix (Reduced);
end Main;

Output:
1.0 2.0 -1.0 -4.0
2.0 3.0 -1.0 -11.0
-2.0 0.0 -3.0 22.0
reduced to:
1.0 0.0 0.0 -8.0
-0.0 1.0 0.0 1.0
-0.0 -0.0 1.0 -2.0

## Aime

rref(list l, integer rows, columns)
{
integer e, f, i, j, lead, r;
list u, v;

while (r < rows && lead < columns) {
i = r;
i += 1;
if (i == rows) {
i = r;
break;
}
}
}
break;
}

u = l[i];

l.spin(i, r);
if (e) {
for (j, f in u) {
u[j] = f / e;
}
}

for (i, v in l) {
if (i != r) {
for (j, f in v) {
v[j] = f - u[j] * e;
}
}
}

r += 1;
}
}

display_2(list l)
{
for (, list u in l) {
u.ucall(o_winteger, -1, 4);
o_byte('\n');
}
}

main(void)
{
list l;

l = list(list(1, 2, -1, -4),
list(2, 3, -1, -11),
list(-2, 0, -3, 22));
rref(l, 3, 4);
display_2(l);

0;
}
Output:
   1   0   0  -8
0   1   0   1
0   0   1  -2

## ALGOL 68

Translation of: Python
Works with: ALGOL 68 version Standard - no extensions to language used
Works with: ALGOL 68G version Any - tested with release mk15-0.8b.fc9.i386
MODE FIELD = REAL; # FIELD can be REAL, LONG REAL etc, or COMPL, FRAC etc #
MODE VEC = [0]FIELD;
MODE MAT = [0,0]FIELD;

PROC to reduced row echelon form = (REF MAT m)VOID: (
INT lead col := 2 LWB m;

FOR this row FROM LWB m TO UPB m DO
IF lead col > 2 UPB m THEN return FI;
INT other row := this row;
WHILE m[other row,lead col] = 0 DO
other row +:= 1;
IF other row > UPB m THEN
other row := this row;
IF lead col > 2 UPB m THEN return FI
FI
OD;
IF this row /= other row THEN
VEC swap = m[this row,lead col:];
FI;
FIELD scale = 1/m[this row,lead col];
IF scale /= 1 THEN
FOR col FROM lead col+1 TO 2 UPB m DO m[this row,col] *:= scale OD
FI;
FOR other row FROM LWB m TO UPB m DO
IF this row /= other row THEN
REAL scale = m[other row,lead col];
FOR col FROM lead col+1 TO 2 UPB m DO m[other row,col] -:= scale*m[this row,col] OD
FI
OD;
OD;
return: EMPTY
);

[3,4]FIELD mat := (
( 1, 2, -1, -4),
( 2, 3, -1, -11),
(-2, 0, -3, 22)
);

to reduced row echelon form( mat );

FORMAT
real repr = $g(-7,4)$,
vec repr = $"("n(2 UPB mat-1)(f(real repr)", ")f(real repr)")"$,
mat repr = $"("n(1 UPB mat-1)(f(vec repr)", "lx)f(vec repr)")"$;

printf((mat repr, mat, $l$))
Output:
(( 1.0000,  0.0000,  0.0000, -8.0000),
( 0.0000,  1.0000,  0.0000,  1.0000),
( 0.0000,  0.0000,  1.0000, -2.0000))


## ALGOL W

From the pseudo code.

begin
% replaces M with it's reduced row echelon form              %
% M should have bounds ( 0 :: rMax, 0 :: cMax )              %
procedure toReducedRowEchelonForm ( real    array M ( *, * )
; integer value rMax, cMax
) ;
begin
for r := 0 until rMax do begin
integer i;
if lead > cMax then goto done;
i := r;
while M( i, lead ) = 0 do begin
i := i + 1;
if rMax = i then begin
i    := r;
if cMax = lead then goto done
end if_rowCount_eq_i
% Swap rows i and r %
for c := 0 until cMax do begin
real t;
t         := M( i, c );
M( i, c ) := M( r, c );
M( r, c ) := t
end swap_rows_i_and_r ;
If M( r, lead ) not = 0 then begin
% divide row r by M[r, lead] %
for c := 0 until cMax do M( r, c ) := M( r, c ) / rLead
for i := 0 until rMax do begin
if i not = r then begin
% Subtract M[i, lead] multiplied by row r from row i %
for c := 0 until cMax do M( i, c ) := M( i, c ) - ( iLead * M( r, c ) )
end if_i_ne_r
end for_i ;
end for_r ;
done:
end toReducedRowEchelonForm ;
% test the toReducedRowEchelonForm procedure %
begin
real array m( 0 :: 2, 0 :: 3 );
M( 0, 0 ) :=  1; M( 0, 1 ) :=  2; M( 0, 2 ) := -1; M( 0, 3 ) :=  -4;
M( 1, 0 ) :=  2; M( 1, 1 ) :=  3; M( 1, 2 ) := -1; M( 1, 3 ) := -11;
M( 2, 0 ) := -2; M( 2, 1 ) :=  0; M( 2, 2 ) := -3; M( 2, 3 ) :=  22;
toReducedRowEchelonForm( M, 2, 3 );
r_format := "A"; s_w := 0; r_w := 6; r_d := 1; % set output formating %
for r := 0 until 2 do begin
write( M( r, 0 ) );
for c := 1 until 3 do writeon( " ", M( r, c ) );
end for_r
end
end.
Output:
   1.0    0.0    0.0   -8.0
0.0    1.0    0.0    1.0
0.0    0.0    1.0   -2.0


## ATS

This program was made by modifying Gauss-Jordan_matrix_inversion#ATS. (The latter program is equivalent to finding the RREF of a particular matrix.)

%{^
#include <math.h>
#include <float.h>
%}

macdef NAN = g0f2f ($extval (float, "NAN")) macdef Zero = g0i2f 0 macdef One = g0i2f 1 macdef Two = g0i2f 2 (* The following is often done by a single machine instruction. *) macdef multiply_and_add (x, y, z) = (,(x) * ,(y)) + ,(z) (*------------------------------------------------------------------*) (* A "little matrix library" *) typedef Matrix_Index_Map (m1 : int, n1 : int, m0 : int, n0 : int) = {i1, j1 : pos | i1 <= m1; j1 <= n1} (int i1, int j1) -<cloref0> [i0, j0 : pos | i0 <= m0; j0 <= n0] @(int i0, int j0) datatype Real_Matrix (tk : tkind, m1 : int, n1 : int, m0 : int, n0 : int) = | Real_Matrix of (matrixref (g0float tk, m0, n0), int m1, int n1, int m0, int n0, Matrix_Index_Map (m1, n1, m0, n0)) typedef Real_Matrix (tk : tkind, m1 : int, n1 : int) = [m0, n0 : pos] Real_Matrix (tk, m1, n1, m0, n0) typedef Real_Vector (tk : tkind, m1 : int, n1 : int) = [m1 == 1 || n1 == 1] Real_Matrix (tk, m1, n1) typedef Real_Row (tk : tkind, n1 : int) = Real_Vector (tk, 1, n1) typedef Real_Column (tk : tkind, m1 : int) = Real_Vector (tk, m1, 1) extern fn {tk : tkind} Real_Matrix_make_elt : {m0, n0 : pos} (int m0, int n0, g0float tk) -< !wrt > Real_Matrix (tk, m0, n0, m0, n0) extern fn {tk : tkind} Real_Matrix_copy : {m1, n1 : pos} Real_Matrix (tk, m1, n1) -< !refwrt > Real_Matrix (tk, m1, n1) extern fn {tk : tkind} Real_Matrix_copy_to : {m1, n1 : pos} (Real_Matrix (tk, m1, n1), (* destination *) Real_Matrix (tk, m1, n1)) -< !refwrt > void extern fn {tk : tkind} Real_Matrix_fill_with_elt : {m1, n1 : pos} (Real_Matrix (tk, m1, n1), g0float tk) -< !refwrt > void extern fn {} Real_Matrix_dimension : {tk : tkind} {m1, n1 : pos} Real_Matrix (tk, m1, n1) -<> @(int m1, int n1) extern fn {tk : tkind} Real_Matrix_get_at : {m1, n1 : pos} {i1, j1 : pos | i1 <= m1; j1 <= n1} (Real_Matrix (tk, m1, n1), int i1, int j1) -< !ref > g0float tk extern fn {tk : tkind} Real_Matrix_set_at : {m1, n1 : pos} {i1, j1 : pos | i1 <= m1; j1 <= n1} (Real_Matrix (tk, m1, n1), int i1, int j1, g0float tk) -< !refwrt > void extern fn {} Real_Matrix_apply_index_map : {tk : tkind} {m1, n1 : pos} {m0, n0 : pos} (Real_Matrix (tk, m0, n0), int m1, int n1, Matrix_Index_Map (m1, n1, m0, n0)) -<> Real_Matrix (tk, m1, n1) extern fn {} Real_Matrix_transpose : (* This is transposed INDEXING. It does NOT copy the data. *) {tk : tkind} {m1, n1 : pos} {m0, n0 : pos} Real_Matrix (tk, m1, n1, m0, n0) -<> Real_Matrix (tk, n1, m1, m0, n0) extern fn {} Real_Matrix_block : (* This is block (submatrix) INDEXING. It does NOT copy the data. *) {tk : tkind} {p0, p1 : pos | p0 <= p1} {q0, q1 : pos | q0 <= q1} {m1, n1 : pos | p1 <= m1; q1 <= n1} {m0, n0 : pos} (Real_Matrix (tk, m1, n1, m0, n0), int p0, int p1, int q0, int q1) -<> Real_Matrix (tk, p1 - p0 + 1, q1 - q0 + 1, m0, n0) extern fn {tk : tkind} Real_Matrix_unit_matrix : {m : pos} int m -< !refwrt > Real_Matrix (tk, m, m) extern fn {tk : tkind} Real_Matrix_unit_matrix_to : {m : pos} Real_Matrix (tk, m, m) -< !refwrt > void extern fn {tk : tkind} Real_Matrix_matrix_sum : {m, n : pos} (Real_Matrix (tk, m, n), Real_Matrix (tk, m, n)) -< !refwrt > Real_Matrix (tk, m, n) extern fn {tk : tkind} Real_Matrix_matrix_sum_to : {m, n : pos} (Real_Matrix (tk, m, n), (* destination*) Real_Matrix (tk, m, n), Real_Matrix (tk, m, n)) -< !refwrt > void extern fn {tk : tkind} Real_Matrix_matrix_difference : {m, n : pos} (Real_Matrix (tk, m, n), Real_Matrix (tk, m, n)) -< !refwrt > Real_Matrix (tk, m, n) extern fn {tk : tkind} Real_Matrix_matrix_difference_to : {m, n : pos} (Real_Matrix (tk, m, n), (* destination*) Real_Matrix (tk, m, n), Real_Matrix (tk, m, n)) -< !refwrt > void extern fn {tk : tkind} Real_Matrix_matrix_product : {m, n, p : pos} (Real_Matrix (tk, m, n), Real_Matrix (tk, n, p)) -< !refwrt > Real_Matrix (tk, m, p) extern fn {tk : tkind} Real_Matrix_matrix_product_to : {m, n, p : pos} (Real_Matrix (tk, m, p), (* destination*) Real_Matrix (tk, m, n), Real_Matrix (tk, n, p)) -< !refwrt > void extern fn {tk : tkind} Real_Matrix_scalar_product : {m, n : pos} (Real_Matrix (tk, m, n), g0float tk) -< !refwrt > Real_Matrix (tk, m, n) extern fn {tk : tkind} Real_Matrix_scalar_product_2 : {m, n : pos} (g0float tk, Real_Matrix (tk, m, n)) -< !refwrt > Real_Matrix (tk, m, n) extern fn {tk : tkind} Real_Matrix_scalar_product_to : {m, n : pos} (Real_Matrix (tk, m, n), (* destination*) Real_Matrix (tk, m, n), g0float tk) -< !refwrt > void extern fn {tk : tkind} (* Useful for debugging. *) Real_Matrix_fprint : {m, n : pos} (FILEref, Real_Matrix (tk, m, n)) -<1> void overload copy with Real_Matrix_copy overload copy_to with Real_Matrix_copy_to overload fill_with_elt with Real_Matrix_fill_with_elt overload dimension with Real_Matrix_dimension overload [] with Real_Matrix_get_at overload [] with Real_Matrix_set_at overload apply_index_map with Real_Matrix_apply_index_map overload transpose with Real_Matrix_transpose overload block with Real_Matrix_block overload unit_matrix with Real_Matrix_unit_matrix overload unit_matrix_to with Real_Matrix_unit_matrix_to overload matrix_sum with Real_Matrix_matrix_sum overload matrix_sum_to with Real_Matrix_matrix_sum_to overload matrix_difference with Real_Matrix_matrix_difference overload matrix_difference_to with Real_Matrix_matrix_difference_to overload matrix_product with Real_Matrix_matrix_product overload matrix_product_to with Real_Matrix_matrix_product_to overload scalar_product with Real_Matrix_scalar_product overload scalar_product with Real_Matrix_scalar_product_2 overload scalar_product_to with Real_Matrix_scalar_product_to overload + with matrix_sum overload - with matrix_difference overload * with matrix_product overload * with scalar_product (*------------------------------------------------------------------*) (* Implementation of the "little matrix library" *) implement {tk} Real_Matrix_make_elt (m0, n0, elt) = Real_Matrix (matrixref_make_elt<g0float tk> (i2sz m0, i2sz n0, elt), m0, n0, m0, n0, lam (i1, j1) => @(i1, j1)) implement {} Real_Matrix_dimension A = case+ A of Real_Matrix (_, m1, n1, _, _, _) => @(m1, n1) implement {tk} Real_Matrix_get_at (A, i1, j1) = let val+ Real_Matrix (storage, _, _, _, n0, index_map) = A val @(i0, j0) = index_map (i1, j1) in matrixref_get_at<g0float tk> (storage, pred i0, n0, pred j0) end implement {tk} Real_Matrix_set_at (A, i1, j1, x) = let val+ Real_Matrix (storage, _, _, _, n0, index_map) = A val @(i0, j0) = index_map (i1, j1) in matrixref_set_at<g0float tk> (storage, pred i0, n0, pred j0, x) end implement {} Real_Matrix_apply_index_map (A, m1, n1, index_map) = (* This is not the most efficient way to acquire new indexing, but it will work. It requires three closures, instead of the two needed by our implementations of "transpose" and "block". *) let val+ Real_Matrix (storage, m1a, n1a, m0, n0, index_map_1a) = A in Real_Matrix (storage, m1, n1, m0, n0, lam (i1, j1) => index_map_1a (i1a, j1a) where { val @(i1a, j1a) = index_map (i1, j1) }) end implement {} Real_Matrix_transpose A = let val+ Real_Matrix (storage, m1, n1, m0, n0, index_map) = A in Real_Matrix (storage, n1, m1, m0, n0, lam (i1, j1) => index_map (j1, i1)) end implement {} Real_Matrix_block (A, p0, p1, q0, q1) = let val+ Real_Matrix (storage, m1, n1, m0, n0, index_map) = A in Real_Matrix (storage, succ (p1 - p0), succ (q1 - q0), m0, n0, lam (i1, j1) => index_map (p0 + pred i1, q0 + pred j1)) end implement {tk} Real_Matrix_copy A = let val @(m1, n1) = dimension A val C = Real_Matrix_make_elt<tk> (m1, n1, A[1, 1]) val () = copy_to<tk> (C, A) in C end implement {tk} Real_Matrix_copy_to (Dst, Src) = let val @(m1, n1) = dimension Src prval [m1 : int] EQINT () = eqint_make_gint m1 prval [n1 : int] EQINT () = eqint_make_gint n1 var i : intGte 1 in for* {i : pos | i <= m1 + 1} .<(m1 + 1) - i>. (i : int i) => (i := 1; i <> succ m1; i := succ i) let var j : intGte 1 in for* {j : pos | j <= n1 + 1} .<(n1 + 1) - j>. (j : int j) => (j := 1; j <> succ n1; j := succ j) Dst[i, j] := Src[i, j] end end implement {tk} Real_Matrix_fill_with_elt (A, elt) = let val @(m1, n1) = dimension A prval [m1 : int] EQINT () = eqint_make_gint m1 prval [n1 : int] EQINT () = eqint_make_gint n1 var i : intGte 1 in for* {i : pos | i <= m1 + 1} .<(m1 + 1) - i>. (i : int i) => (i := 1; i <> succ m1; i := succ i) let var j : intGte 1 in for* {j : pos | j <= n1 + 1} .<(n1 + 1) - j>. (j : int j) => (j := 1; j <> succ n1; j := succ j) A[i, j] := elt end end implement {tk} Real_Matrix_unit_matrix {m} m = let val A = Real_Matrix_make_elt<tk> (m, m, Zero) var i : intGte 1 in for* {i : pos | i <= m + 1} .<(m + 1) - i>. (i : int i) => (i := 1; i <> succ m; i := succ i) A[i, i] := One; A end implement {tk} Real_Matrix_unit_matrix_to A = let val @(m, _) = dimension A prval [m : int] EQINT () = eqint_make_gint m var i : intGte 1 in for* {i : pos | i <= m + 1} .<(m + 1) - i>. (i : int i) => (i := 1; i <> succ m; i := succ i) let var j : intGte 1 in for* {j : pos | j <= m + 1} .<(m + 1) - j>. (j : int j) => (j := 1; j <> succ m; j := succ j) A[i, j] := (if i = j then One else Zero) end end implement {tk} Real_Matrix_matrix_sum (A, B) = let val @(m, n) = dimension A val C = Real_Matrix_make_elt<tk> (m, n, NAN) val () = matrix_sum_to<tk> (C, A, B) in C end implement {tk} Real_Matrix_matrix_sum_to (C, A, B) = let val @(m, n) = dimension A prval [m : int] EQINT () = eqint_make_gint m prval [n : int] EQINT () = eqint_make_gint n var i : intGte 1 in for* {i : pos | i <= m + 1} .<(m + 1) - i>. (i : int i) => (i := 1; i <> succ m; i := succ i) let var j : intGte 1 in for* {j : pos | j <= n + 1} .<(n + 1) - j>. (j : int j) => (j := 1; j <> succ n; j := succ j) C[i, j] := A[i, j] + B[i, j] end end implement {tk} Real_Matrix_matrix_difference (A, B) = let val @(m, n) = dimension A val C = Real_Matrix_make_elt<tk> (m, n, NAN) val () = matrix_difference_to<tk> (C, A, B) in C end implement {tk} Real_Matrix_matrix_difference_to (C, A, B) = let val @(m, n) = dimension A prval [m : int] EQINT () = eqint_make_gint m prval [n : int] EQINT () = eqint_make_gint n var i : intGte 1 in for* {i : pos | i <= m + 1} .<(m + 1) - i>. (i : int i) => (i := 1; i <> succ m; i := succ i) let var j : intGte 1 in for* {j : pos | j <= n + 1} .<(n + 1) - j>. (j : int j) => (j := 1; j <> succ n; j := succ j) C[i, j] := A[i, j] - B[i, j] end end implement {tk} Real_Matrix_matrix_product (A, B) = let val @(m, n) = dimension A and @(_, p) = dimension B val C = Real_Matrix_make_elt<tk> (m, p, NAN) val () = matrix_product_to<tk> (C, A, B) in C end implement {tk} Real_Matrix_matrix_product_to (C, A, B) = let val @(m, n) = dimension A and @(_, p) = dimension B prval [m : int] EQINT () = eqint_make_gint m prval [n : int] EQINT () = eqint_make_gint n prval [p : int] EQINT () = eqint_make_gint p var i : intGte 1 in for* {i : pos | i <= m + 1} .<(m + 1) - i>. (i : int i) => (i := 1; i <> succ m; i := succ i) let var k : intGte 1 in for* {k : pos | k <= p + 1} .<(p + 1) - k>. (k : int k) => (k := 1; k <> succ p; k := succ k) let var j : intGte 1 in C[i, k] := A[i, 1] * B[1, k]; for* {j : pos | j <= n + 1} .<(n + 1) - j>. (j : int j) => (j := 2; j <> succ n; j := succ j) C[i, k] := multiply_and_add (A[i, j], B[j, k], C[i, k]) end end end implement {tk} Real_Matrix_scalar_product (A, r) = let val @(m, n) = dimension A val C = Real_Matrix_make_elt<tk> (m, n, NAN) val () = scalar_product_to<tk> (C, A, r) in C end implement {tk} Real_Matrix_scalar_product_2 (r, A) = Real_Matrix_scalar_product<tk> (A, r) implement {tk} Real_Matrix_scalar_product_to (C, A, r) = let val @(m, n) = dimension A prval [m : int] EQINT () = eqint_make_gint m prval [n : int] EQINT () = eqint_make_gint n var i : intGte 1 in for* {i : pos | i <= m + 1} .<(m + 1) - i>. (i : int i) => (i := 1; i <> succ m; i := succ i) let var j : intGte 1 in for* {j : pos | j <= n + 1} .<(n + 1) - j>. (j : int j) => (j := 1; j <> succ n; j := succ j) C[i, j] := A[i, j] * r end end implement {tk} Real_Matrix_fprint {m, n} (outf, A) = let val @(m, n) = dimension A var i : intGte 1 in for* {i : pos | i <= m + 1} .<(m + 1) - i>. (i : int i) => (i := 1; i <> succ m; i := succ i) let var j : intGte 1 in for* {j : pos | j <= n + 1} .<(n + 1) - j>. (j : int j) => (j := 1; j <> succ n; j := succ j) let typedef FILEstar =$extype"FILE *"
extern castfn FILEref2star : FILEref -<> FILEstar
val _ = $extfcall (int, "fprintf", FILEref2star outf, "%16.6g", A[i, j]) in end; fprintln! (outf) end end (*------------------------------------------------------------------*) (* Reduced row echelon form, by Gauss-Jordan elimination *) extern fn {tk : tkind} Real_Matrix_reduced_row_echelon_form : {m, n : pos} Real_Matrix (tk, m, n) -< !refwrt > Real_Matrix (tk, m, n) implement {tk} Real_Matrix_reduced_row_echelon_form {m, n} A = let val @(m, n) = dimension A typedef one_to_m = intBtwe (1, m) typedef one_to_n = intBtwe (1, n) (* Partial pivoting, to improve the numerical stability. *) implement array_tabulate$fopr<one_to_m> i =
let
val i = g1ofg0 (sz2i (succ i))
val () = assertloc ((1 <= i) * (i <= m))
in
i
end
val rows_permutation =
$effmask_all arrayref_tabulate<one_to_m> (i2sz m) fn index_map : Matrix_Index_Map (m, n, m, n) = lam (i1, j1) =>$effmask_ref
(@(i0, j1) where { val i0 = rows_permutation[i1 - 1] })

val A = apply_index_map (copy<tk> A, m, n, index_map)

fn {}
exchange_rows (i1 : one_to_m,
i2 : one_to_m) :<!refwrt> void =
if i1 <> i2 then
let
val k1 = rows_permutation[pred i1]
and k2 = rows_permutation[pred i2]
in
rows_permutation[pred i1] := k2;
rows_permutation[pred i2] := k1
end

fn {}
normalize_pivot_row (i : one_to_m,
j : one_to_n) :<!refwrt> void =
let
prval [j : int] EQINT () = eqint_make_gint j
val pivot_val = A[i, j]
var k : intGte 1
in
A[i, j] := One;
for* {k : int | j + 1 <= k; k <= n + 1} .<(n + 1) - k>.
(k : int k) =>
(k := succ j; k <> succ n; k := succ k)
A[i, k] := A[i, k] / pivot_val
end

fn
subtract_normalized_pivot_row (ipiv : one_to_m,
i    : one_to_m,
j    : one_to_n) :<!refwrt> void =
let
prval [j : int] EQINT () = eqint_make_gint j
val factor = ~A[i, j]
var k : intGte 1
in
A[i, j] := Zero;
for* {k : int | j + 1 <= k; k <= n + 1} .<(n + 1) - k>.
(k : int k) =>
(k := succ j; k <> succ n; k := succ k)
A[i, k] := multiply_and_add (A[ipiv, k], factor, A[i, k])
end

fun
main_loop {i, j : pos | i <= m; i <= j; j <= n + 1}
.<(n + 1) - j>.
(i : int i, j : int j) :<!refwrt> void =
if j <> succ n then
let
fun
select_pivot {k : int | i <= k; k <= m + 1}
.<(m + 1) - k>.
(k         : int k,
max_abs   : g0float tk,
k_max_abs : intBtwe (i - 1, m))
:<!ref> intBtwe (i - 1, m) =
if k = succ m then
k_max_abs
else
let
val abs_akj = abs A[k, j]
in
if abs_akj > max_abs then
select_pivot (succ k, abs_akj, k)
else
select_pivot (succ k, max_abs, k_max_abs)
end

val i_pivot = select_pivot (i, Zero, pred i)
prval [i_pivot : int] EQINT () = eqint_make_gint i_pivot
in
if i_pivot = pred i then
(* There is no pivot in this column. *)
main_loop (i, succ j)
else
let
var k : intGte 1
in
exchange_rows (i_pivot, i);
normalize_pivot_row (i, j);
for* {k : int | 1 <= k; k <= i} .<i - k>.
(k : int k) =>
(k := 1; k <> i; k := succ k)
subtract_normalized_pivot_row (i, k, j);
for* {k : int | i + 1 <= k; k <= m + 1} .<(m + 1) - k>.
(k : int k) =>
(k := succ i; k <> succ m; k := succ k)
subtract_normalized_pivot_row (i, k, j);
if i <> m then
main_loop (succ i, succ j)
end
end
in
main_loop (1, 1);
A
end

Real_Matrix_reduced_row_echelon_form

(*------------------------------------------------------------------*)

implement
main0 () =
let
val () = println! ()
val () = println! ("Here is the requested solution:")
val () = println! ()
val A = Real_Matrix_make_elt (3, 4, NAN)
val () =
(A[1,1] := 1.0; A[1,2] := 2.0; A[1,3] := ~1.0; A[1,4] := ~4.0;
A[2,1] := 2.0; A[2,2] := 3.0; A[2,3] := ~1.0; A[2,4] := ~11.0;
A[3,1] := ~2.0; A[3,2] := 0.0; A[3,3] := ~3.0; A[3,4] := 22.0)
val B = reduced_row_echelon_form A
val () = Real_Matrix_fprint (stdout_ref, B)

val () = println! ()
val () = println! ("Here is a RREF with a more interesting shape:")
val () = println! ()
val A = Real_Matrix_make_elt (3, 5, NAN)
val () =
(A[1,1] := 0.0; A[1,2] := 0.0; A[1,3] := ~1.0; A[1,4] := 2.0; A[1,5] := 0.0;
A[2,1] := 0.0; A[2,2] := 0.0; A[2,3] := ~1.0; A[2,4] := 1.0; A[2,5] := 1.0;
A[3,1] := 2.0; A[3,2] := 8.0; A[3,3] := 1.0; A[3,4] := ~4.0; A[3,5] := 2.0)
val B = reduced_row_echelon_form A
val () = Real_Matrix_fprint (stdout_ref, B)

val () = println! ()
val () = println! ("It is the RREF of this matrix:")
val () = println! ()
val () = Real_Matrix_fprint (stdout_ref, A)

val () = println! ()
in
end

(*------------------------------------------------------------------*)
Output:
$patscc -std=gnu2x -g -O2 -DATS_MEMALLOC_GCBDW reduced_row_echelon_task.dats -lgc && ./a.out Here is the requested solution: 1 0 0 -8 0 1 0 1 0 0 1 -2 Here is a RREF with a more interesting shape: 1 4 0 0 0 0 0 1 0 -2 0 0 0 1 -1 It is the RREF of this matrix: 0 0 -1 2 0 0 0 -1 1 1 2 8 1 -4 2  ## AutoHotkey ToReducedRowEchelonForm(M){ rowCount := M.Count() ; the number of rows in M columnCount := M.1.Count() ; the number of columns in M r := lead := 1 while (r <= rowCount) { if (columnCount < lead) return M i := r while (M[i, lead] = 0) { i++ if (rowCount+1 = i) { i := r, lead++ if (columnCount+1 = lead) return M } } if (i<>r) for col, v in M[i] ; Swap rows i and r tempVal := M[i, col], M[i, col] := M[r, col], M[r, col] := tempVal num := M[r, lead] if (M[r, lead] <> 0) for col, val in M[r] M[r, col] /= num ; If M[r, lead] is not 0 divide row r by M[r, lead] i := 2 while (i <= rowCount) { num := M[i, lead] if (i <> r) for col, val in M[i] ; Subtract M[i, lead] multiplied by row r from row i M[i, col] -= num * M[r, col] i++ } lead++, r++ } return M }  Examples: M := [[1 , 2, -1, -4 ] , [2 , 3, -1, -11] , [-2, 0, -3, 22]] M := ToReducedRowEchelonForm(M) for row, obj in M { for col, v in obj output .= RegExReplace(v, "\.0+$|0+$") "t" output .= "n" } MsgBox % output return  Output: 1 0 0 -8 -0 1 0 1 -0 -0 1 -2  ## AutoIt Global$ivMatrix[3][4] = [[1, 2, -1, -4],[2, 3, -1, -11],[-2, 0, -3, 22]]
ToReducedRowEchelonForm($ivMatrix) Func ToReducedRowEchelonForm($matrix)
Local $clonematrix,$i
Local $lead = 0 Local$rowCount = UBound($matrix) - 1 Local$columnCount = UBound($matrix, 2) - 1 For$r = 0 To $rowCount If$columnCount = $lead Then ExitLoop$i = $r While$matrix[$i][$lead] = 0
$i += 1 If$rowCount = $i Then$i = $r$lead += 1
If $columnCount =$lead Then ExitLoop
EndIf
WEnd
; There´s no built in Function to swap Rows of a 2-Dimensional Array
; We need to clone our matrix to swap complete lines
$clonematrix =$matrix ; Swap Lines, no
For $s = 0 To$columnCount
$matrix[$r][$s] =$clonematrix[$i][$s]
$matrix[$i][$s] =$clonematrix[$r][$s]
Next
Local $m =$matrix[$r][$lead]
For $k = 0 To$columnCount
$matrix[$r][$k] =$matrix[$r][$k] / $m Next For$i = 0 To $rowCount If$i <> $r Then Local$m = $matrix[$i][$lead] For$k = 0 To $columnCount$matrix[$i][$k] -= $m *$matrix[$r][$k]
Next
EndIf
Next
$lead += 1 Next ; Console Output For$i = 0 To $rowCount ConsoleWrite("[") For$k = 0 To $columnCount ConsoleWrite($matrix[$i][$k])
If $k <>$columnCount Then ConsoleWrite(",")
Next
ConsoleWrite("]" & @CRLF)
Next
; End of Console Output
Return $matrix EndFunc ;==>ToReducedRowEchelonForm  Output: [1,0,0,-8] [-0,1,0,1] [-0,-0,1,-2] ## BASIC ### BASIC256 arraybase 1 global matrix dim matrix = {{1, 2, -1, -4}, {2, 3, -1, -11}, { -2, 0, -3, 22}} call RREF (matrix) for row = 1 to 3 for col = 1 to 4 if matrix[row, col] = 0 then print "0"; chr(9); else print matrix[row, col]; chr(9); end if next print next end subroutine RREF(m) nrows = matrix[?,] ncols = matrix[,?] lead = 1 for r = 1 to nrows if lead >= ncols then exit for i = r while matrix[i, lead] = 0 i += 1 if i = nrows then i = r lead += 1 if lead = ncols then exit for end if end while for j = 1 to ncols temp = matrix[i, j] matrix[i, j] = matrix[r, j] matrix[r, j] = temp next n = matrix[r, lead] if n <> 1 then for j = 0 to ncols matrix[r, j] /= n next end if for i = 1 to nrows if i <> r then n = matrix[i, lead] for j = 1 to ncols matrix[i, j] -= matrix[r, j] * n next end if next lead += 1 next end subroutine ### BBC BASIC  DIM matrix(2,3) matrix() = 1, 2, -1, -4, \ \ 2, 3, -1, -11, \ \ -2, 0, -3, 22 PROCrref(matrix()) FOR row% = 0 TO 2 FOR col% = 0 TO 3 PRINT matrix(row%,col%); NEXT PRINT NEXT row% END DEF PROCrref(m()) LOCAL lead%, nrows%, ncols%, i%, j%, r%, n nrows% = DIM(m(),1)+1 ncols% = DIM(m(),2)+1 FOR r% = 0 TO nrows%-1 IF lead% >= ncols% EXIT FOR i% = r% WHILE m(i%,lead%) = 0 i% += 1 IF i% = nrows% THEN i% = r% lead% += 1 IF lead% = ncols% EXIT FOR ENDIF ENDWHILE FOR j% = 0 TO ncols%-1 : SWAP m(i%,j%),m(r%,j%) : NEXT n = m(r%,lead%) IF n <> 0 FOR j% = 0 TO ncols%-1 : m(r%,j%) /= n : NEXT FOR i% = 0 TO nrows%-1 IF i% <> r% THEN n = m(i%,lead%) FOR j% = 0 TO ncols%-1 m(i%,j%) -= m(r%,j%) * n NEXT ENDIF NEXT lead% += 1 NEXT r% ENDPROC  Output:  1 0 0 -8 0 1 0 1 0 0 1 -2  ## C #include <stdio.h> #define TALLOC(n,typ) malloc(n*sizeof(typ)) #define EL_Type int typedef struct sMtx { int dim_x, dim_y; EL_Type *m_stor; EL_Type **mtx; } *Matrix, sMatrix; typedef struct sRvec { int dim_x; EL_Type *m_stor; } *RowVec, sRowVec; Matrix NewMatrix( int x_dim, int y_dim ) { int n; Matrix m; m = TALLOC( 1, sMatrix); n = x_dim * y_dim; m->dim_x = x_dim; m->dim_y = y_dim; m->m_stor = TALLOC(n, EL_Type); m->mtx = TALLOC(m->dim_y, EL_Type *); for(n=0; n<y_dim; n++) { m->mtx[n] = m->m_stor+n*x_dim; } return m; } void MtxSetRow(Matrix m, int irow, EL_Type *v) { int ix; EL_Type *mr; mr = m->mtx[irow]; for(ix=0; ix<m->dim_x; ix++) mr[ix] = v[ix]; } Matrix InitMatrix( int x_dim, int y_dim, EL_Type **v) { Matrix m; int iy; m = NewMatrix(x_dim, y_dim); for (iy=0; iy<y_dim; iy++) MtxSetRow(m, iy, v[iy]); return m; } void MtxDisplay( Matrix m ) { int iy, ix; const char *sc; for (iy=0; iy<m->dim_y; iy++) { printf(" "); sc = " "; for (ix=0; ix<m->dim_x; ix++) { printf("%s %3d", sc, m->mtx[iy][ix]); sc = ","; } printf("\n"); } printf("\n"); } void MtxMulAndAddRows(Matrix m, int ixrdest, int ixrsrc, EL_Type mplr) { int ix; EL_Type *drow, *srow; drow = m->mtx[ixrdest]; srow = m->mtx[ixrsrc]; for (ix=0; ix<m->dim_x; ix++) drow[ix] += mplr * srow[ix]; // printf("Mul row %d by %d and add to row %d\n", ixrsrc, mplr, ixrdest); // MtxDisplay(m); } void MtxSwapRows( Matrix m, int rix1, int rix2) { EL_Type *r1, *r2, temp; int ix; if (rix1 == rix2) return; r1 = m->mtx[rix1]; r2 = m->mtx[rix2]; for (ix=0; ix<m->dim_x; ix++) temp = r1[ix]; r1[ix]=r2[ix]; r2[ix]=temp; // printf("Swap rows %d and %d\n", rix1, rix2); // MtxDisplay(m); } void MtxNormalizeRow( Matrix m, int rix, int lead) { int ix; EL_Type *drow; EL_Type lv; drow = m->mtx[rix]; lv = drow[lead]; for (ix=0; ix<m->dim_x; ix++) drow[ix] /= lv; // printf("Normalize row %d\n", rix); // MtxDisplay(m); } #define MtxGet( m, rix, cix ) m->mtx[rix][cix] void MtxToReducedREForm(Matrix m) { int lead; int rix, iix; EL_Type lv; int rowCount = m->dim_y; lead = 0; for (rix=0; rix<rowCount; rix++) { if (lead >= m->dim_x) return; iix = rix; while (0 == MtxGet(m, iix,lead)) { iix++; if (iix == rowCount) { iix = rix; lead++; if (lead == m->dim_x) return; } } MtxSwapRows(m, iix, rix ); MtxNormalizeRow(m, rix, lead ); for (iix=0; iix<rowCount; iix++) { if ( iix != rix ) { lv = MtxGet(m, iix, lead ); MtxMulAndAddRows(m,iix, rix, -lv) ; } } lead++; } } int main() { Matrix m1; static EL_Type r1[] = {1,2,-1,-4}; static EL_Type r2[] = {2,3,-1,-11}; static EL_Type r3[] = {-2,0,-3,22}; static EL_Type *im[] = { r1, r2, r3 }; m1 = InitMatrix( 4,3, im ); printf("Initial\n"); MtxDisplay(m1); MtxToReducedREForm(m1); printf("Reduced R-E form\n"); MtxDisplay(m1); return 0; }  ## C# using System; namespace rref { class Program { static void Main(string[] args) { int[,] matrix = new int[3, 4]{ { 1, 2, -1, -4 }, { 2, 3, -1, -11 }, { -2, 0, -3, 22 } }; matrix = rref(matrix); } private static int[,] rref(int[,] matrix) { int lead = 0, rowCount = matrix.GetLength(0), columnCount = matrix.GetLength(1); for (int r = 0; r < rowCount; r++) { if (columnCount <= lead) break; int i = r; while (matrix[i, lead] == 0) { i++; if (i == rowCount) { i = r; lead++; if (columnCount == lead) { lead--; break; } } } for (int j = 0; j < columnCount; j++) { int temp = matrix[r, j]; matrix[r, j] = matrix[i, j]; matrix[i, j] = temp; } int div = matrix[r, lead]; if(div != 0) for (int j = 0; j < columnCount; j++) matrix[r, j] /= div; for (int j = 0; j < rowCount; j++) { if (j != r) { int sub = matrix[j, lead]; for (int k = 0; k < columnCount; k++) matrix[j, k] -= (sub * matrix[r, k]); } } lead++; } return matrix; } } }  ## C++ Note: This code is written in generic form. While it slightly complicates the code, it allows to use the same code for both built-in arrays and matrix classes. To use it with a matrix class, either program the matrix class to the specifications given in the matrix_traits comment, or specialize matrix_traits for the specific interface of your matrix class. The test code uses a built-in array for the matrix. Works with: g++ version 4.1.2 20061115 (prerelease) (Debian 4.1.1-21) #include <algorithm> // for std::swap #include <cstddef> #include <cassert> // Matrix traits: This describes how a matrix is accessed. By // externalizing this information into a traits class, the same code // can be used both with native arrays and matrix classes. To use the // default implementation of the traits class, a matrix type has to // provide the following definitions as members: // // * typedef ... index_type; // - The type used for indexing (e.g. size_t) // * typedef ... value_type; // - The element type of the matrix (e.g. double) // * index_type min_row() const; // - returns the minimal allowed row index // * index_type max_row() const; // - returns the maximal allowed row index // * index_type min_column() const; // - returns the minimal allowed column index // * index_type max_column() const; // - returns the maximal allowed column index // * value_type& operator()(index_type i, index_type k) // - returns a reference to the element i,k, where // min_row() <= i <= max_row() // min_column() <= k <= max_column() // * value_type operator()(index_type i, index_type k) const // - returns the value of element i,k // // Note that the functions are all inline and simple, so the compiler // should completely optimize them away. template<typename MatrixType> struct matrix_traits { typedef typename MatrixType::index_type index_type; typedef typename MatrixType::value_type value_type; static index_type min_row(MatrixType const& A) { return A.min_row(); } static index_type max_row(MatrixType const& A) { return A.max_row(); } static index_type min_column(MatrixType const& A) { return A.min_column(); } static index_type max_column(MatrixType const& A) { return A.max_column(); } static value_type& element(MatrixType& A, index_type i, index_type k) { return A(i,k); } static value_type element(MatrixType const& A, index_type i, index_type k) { return A(i,k); } }; // specialization of the matrix traits for built-in two-dimensional // arrays template<typename T, std::size_t rows, std::size_t columns> struct matrix_traits<T[rows][columns]> { typedef std::size_t index_type; typedef T value_type; static index_type min_row(T const (&)[rows][columns]) { return 0; } static index_type max_row(T const (&)[rows][columns]) { return rows-1; } static index_type min_column(T const (&)[rows][columns]) { return 0; } static index_type max_column(T const (&)[rows][columns]) { return columns-1; } static value_type& element(T (&A)[rows][columns], index_type i, index_type k) { return A[i][k]; } static value_type element(T const (&A)[rows][columns], index_type i, index_type k) { return A[i][k]; } }; // Swap rows i and k of a matrix A // Note that due to the reference, both dimensions are preserved for // built-in arrays template<typename MatrixType> void swap_rows(MatrixType& A, typename matrix_traits<MatrixType>::index_type i, typename matrix_traits<MatrixType>::index_type k) { matrix_traits<MatrixType> mt; typedef typename matrix_traits<MatrixType>::index_type index_type; // check indices assert(mt.min_row(A) <= i); assert(i <= mt.max_row(A)); assert(mt.min_row(A) <= k); assert(k <= mt.max_row(A)); for (index_type col = mt.min_column(A); col <= mt.max_column(A); ++col) std::swap(mt.element(A, i, col), mt.element(A, k, col)); } // divide row i of matrix A by v template<typename MatrixType> void divide_row(MatrixType& A, typename matrix_traits<MatrixType>::index_type i, typename matrix_traits<MatrixType>::value_type v) { matrix_traits<MatrixType> mt; typedef typename matrix_traits<MatrixType>::index_type index_type; assert(mt.min_row(A) <= i); assert(i <= mt.max_row(A)); assert(v != 0); for (index_type col = mt.min_column(A); col <= mt.max_column(A); ++col) mt.element(A, i, col) /= v; } // in matrix A, add v times row k to row i template<typename MatrixType> void add_multiple_row(MatrixType& A, typename matrix_traits<MatrixType>::index_type i, typename matrix_traits<MatrixType>::index_type k, typename matrix_traits<MatrixType>::value_type v) { matrix_traits<MatrixType> mt; typedef typename matrix_traits<MatrixType>::index_type index_type; assert(mt.min_row(A) <= i); assert(i <= mt.max_row(A)); assert(mt.min_row(A) <= k); assert(k <= mt.max_row(A)); for (index_type col = mt.min_column(A); col <= mt.max_column(A); ++col) mt.element(A, i, col) += v * mt.element(A, k, col); } // convert A to reduced row echelon form template<typename MatrixType> void to_reduced_row_echelon_form(MatrixType& A) { matrix_traits<MatrixType> mt; typedef typename matrix_traits<MatrixType>::index_type index_type; index_type lead = mt.min_row(A); for (index_type row = mt.min_row(A); row <= mt.max_row(A); ++row) { if (lead > mt.max_column(A)) return; index_type i = row; while (mt.element(A, i, lead) == 0) { ++i; if (i > mt.max_row(A)) { i = row; ++lead; if (lead > mt.max_column(A)) return; } } swap_rows(A, i, row); divide_row(A, row, mt.element(A, row, lead)); for (i = mt.min_row(A); i <= mt.max_row(A); ++i) { if (i != row) add_multiple_row(A, i, row, -mt.element(A, i, lead)); } } } // test code #include <iostream> int main() { double M[3][4] = { { 1, 2, -1, -4 }, { 2, 3, -1, -11 }, { -2, 0, -3, 22 } }; to_reduced_row_echelon_form(M); for (int i = 0; i < 3; ++i) { for (int j = 0; j < 4; ++j) std::cout << M[i][j] << '\t'; std::cout << "\n"; } return EXIT_SUCCESS; }  Output: 1 0 0 -8 -0 1 0 1 -0 -0 1 -2  ## Common Lisp Direct implementation of the pseudo-code given. (defun convert-to-row-echelon-form (matrix) (let* ((dimensions (array-dimensions matrix)) (row-count (first dimensions)) (column-count (second dimensions)) (lead 0)) (labels ((find-pivot (start lead) (let ((i start)) (loop :while (zerop (aref matrix i lead)) :do (progn (incf i) (when (= i row-count) (setf i start) (incf lead) (when (= lead column-count) (return-from convert-to-row-echelon-form matrix)))) :finally (return (values i lead))))) (swap-rows (r1 r2) (loop :for c :upfrom 0 :below column-count :do (rotatef (aref matrix r1 c) (aref matrix r2 c)))) (divide-row (r value) (loop :for c :upfrom 0 :below column-count :do (setf (aref matrix r c) (/ (aref matrix r c) value))))) (loop :for r :upfrom 0 :below row-count :when (<= column-count lead) :do (return matrix) :do (multiple-value-bind (i nlead) (find-pivot r lead) (setf lead nlead) (swap-rows i r) (divide-row r (aref matrix r lead)) (loop :for i :upfrom 0 :below row-count :when (/= i r) :do (let ((scale (aref matrix i lead))) (loop :for c :upfrom 0 :below column-count :do (decf (aref matrix i c) (* scale (aref matrix r c)))))) (incf lead)) :finally (return matrix)))))  ## D import std.stdio, std.algorithm, std.array, std.conv; void toReducedRowEchelonForm(T)(T[][] M) pure nothrow @nogc { if (M.empty) return; immutable nrows = M.length; immutable ncols = M[0].length; size_t lead; foreach (immutable r; 0 .. nrows) { if (ncols <= lead) return; { size_t i = r; while (M[i][lead] == 0) { i++; if (nrows == i) { i = r; lead++; if (ncols == lead) return; } } swap(M[i], M[r]); } M[r][] /= M[r][lead]; foreach (j, ref mj; M) if (j != r) mj[] -= M[r][] * mj[lead]; lead++; } } void main() { auto A = [[ 1, 2, -1, -4], [ 2, 3, -1, -11], [-2, 0, -3, 22]]; A.toReducedRowEchelonForm; writefln("%(%(%2d %)\n%)", A); }  Output:  1 0 0 -8 0 1 0 1 0 0 1 -2 ## EasyLang Translation of: Go proc rref . m[][] . nrow = len m[][] ncol = len m[1][] lead = 1 for r to nrow if lead > ncol return . i = r while m[i][lead] = 0 i += 1 if i > nrow i = r lead += 1 if lead > ncol return . . . swap m[i][] m[r][] m = m[r][lead] for k to ncol m[r][k] /= m . for i to nrow if i <> r m = m[i][lead] for k to ncol m[i][k] -= m * m[r][k] . . . lead += 1 . . test[][] = [ [ 1 2 -1 -4 ] [ 2 3 -1 -11 ] [ -2 0 -3 22 ] ] rref test[][] print test[][] ## Euphoria function ToReducedRowEchelonForm(sequence M) integer lead,rowCount,columnCount,i sequence temp lead = 1 rowCount = length(M) columnCount = length(M[1]) for r = 1 to rowCount do if columnCount <= lead then exit end if i = r while M[i][lead] = 0 do i += 1 if rowCount = i then i = r lead += 1 if columnCount = lead then exit end if end if end while temp = M[i] M[i] = M[r] M[r] = temp M[r] /= M[r][lead] for j = 1 to rowCount do if j != r then M[j] -= M[j][lead]*M[r] end if end for lead += 1 end for return M end function ? ToReducedRowEchelonForm( { { 1, 2, -1, -4 }, { 2, 3, -1, -11 }, { -2, 0, -3, 22 } }) Output: { {1,0,0,-8}, {0,1,0,1}, {0,0,1,-2} } ## Factor USE: math.matrices.elimination { { 1 2 -1 -4 } { 2 3 -1 -11 } { -2 0 -3 22 } } solution .  Output: { { 1 0 0 -8 } { 0 1 0 1 } { 0 0 1 -2 } }  ## Fortran module Rref implicit none contains subroutine to_rref(matrix) real, dimension(:,:), intent(inout) :: matrix integer :: pivot, norow, nocolumn integer :: r, i real, dimension(:), allocatable :: trow pivot = 1 norow = size(matrix, 1) nocolumn = size(matrix, 2) allocate(trow(nocolumn)) do r = 1, norow if ( nocolumn <= pivot ) exit i = r do while ( matrix(i, pivot) == 0 ) i = i + 1 if ( norow == i ) then i = r pivot = pivot + 1 if ( nocolumn == pivot ) return end if end do trow = matrix(i, :) matrix(i, :) = matrix(r, :) matrix(r, :) = trow matrix(r, :) = matrix(r, :) / matrix(r, pivot) do i = 1, norow if ( i /= r ) matrix(i, :) = matrix(i, :) - matrix(r, :) * matrix(i, pivot) end do pivot = pivot + 1 end do deallocate(trow) end subroutine to_rref end module Rref  program prg_test use rref implicit none real, dimension(3, 4) :: m = reshape( (/ 1, 2, -1, -4, & 2, 3, -1, -11, & -2, 0, -3, 22 /), & (/ 3, 4 /), order = (/ 2, 1 /) ) integer :: i print *, "Original matrix" do i = 1, size(m,1) print *, m(i, :) end do call to_rref(m) print *, "Reduced row echelon form" do i = 1, size(m,1) print *, m(i, :) end do end program prg_test  ## FreeBASIC Include the code from Matrix multiplication#FreeBASIC because this function uses the matrix type defined there and I don't want to reproduce it all here. #include once "matmult.bas" sub rowswap( byval M as Matrix, i as uinteger, j as uinteger ) dim as integer k for k = 0 to ubound(M.m, 2) swap M.m(j, k), M.m(i, k) next k end sub function rowech(byval M as Matrix) as Matrix dim as uinteger lead = 0, rowCount = 1+ubound(M.m, 1), colCount = 1+ubound(M.m, 2) dim as uinteger r, i, j dim as double K for r = 0 to rowCount-1 if lead >= colCount then exit for i = r while M.m(i, lead) = 0 i += 1 if i = rowCount then i = r lead += 1 if lead = colCount then exit for endif wend rowswap M, r, i K = M.m(r,lead) if K <> 0 then for j = 0 to colCount-1 M.m(r,j) /= K next j endif for i = 0 to rowCount-1 if i <> r then K = M.m(i, lead) for j = 0 to colCount-1 M.m(i,j) -= M.m(r,j) * K next j endif next i lead += 1 next r return M end function dim as Matrix M = Matrix (3, 4) dim as Matrix N M.m(0,0) = 1 : M.m(0,1) = 2 : M.m(0,2) = -1 : M.M(0,3) = -4 M.m(1,0) = 2 : M.m(1,1) = 3 : M.m(1,2) = -1 : M.m(1,3) = -11 M.m(2,0) = -2: M.m(2,1) = 0 : M.m(2,2) = -3 : M.m(2,3) = 22 dim as integer i, j N = rowech(M) for i=0 to 2 for j = 0 to 3 print N.m(i, j), next j print next i Output:  1 0 0 -8 -0 1 0 1 -0 -0 1 -2  ## Go ### 2D representation From WP pseudocode: package main import "fmt" type matrix [][]float64 func (m matrix) print() { for _, r := range m { fmt.Println(r) } fmt.Println("") } func main() { m := matrix{ { 1, 2, -1, -4}, { 2, 3, -1, -11}, {-2, 0, -3, 22}, } m.print() rref(m) m.print() } func rref(m matrix) { lead := 0 rowCount := len(m) columnCount := len(m[0]) for r := 0; r < rowCount; r++ { if lead >= columnCount { return } i := r for m[i][lead] == 0 { i++ if rowCount == i { i = r lead++ if columnCount == lead { return } } } m[i], m[r] = m[r], m[i] f := 1 / m[r][lead] for j, _ := range m[r] { m[r][j] *= f } for i = 0; i < rowCount; i++ { if i != r { f = m[i][lead] for j, e := range m[r] { m[i][j] -= e * f } } } lead++ } }  Output: (not so pretty, sorry) [1 2 -1 -4] [2 3 -1 -11] [-2 0 -3 22] [1 0 0 -8] [-0 1 0 1] [-0 -0 1 -2]  ### Flat representation package main import "fmt" type matrix struct { ele []float64 stride int } func matrixFromRows(rows [][]float64) *matrix { if len(rows) == 0 { return &matrix{nil, 0} } m := &matrix{make([]float64, len(rows)*len(rows[0])), len(rows[0])} for rx, row := range rows { copy(m.ele[rx*m.stride:(rx+1)*m.stride], row) } return m } func (m *matrix) print(heading string) { if heading > "" { fmt.Print("\n", heading, "\n") } for e := 0; e < len(m.ele); e += m.stride { fmt.Printf("%6.2f ", m.ele[e:e+m.stride]) fmt.Println() } } func (m *matrix) rref() { lead := 0 for rxc0 := 0; rxc0 < len(m.ele); rxc0 += m.stride { if lead >= m.stride { return } ixc0 := rxc0 for m.ele[ixc0+lead] == 0 { ixc0 += m.stride if ixc0 == len(m.ele) { ixc0 = rxc0 lead++ if lead == m.stride { return } } } for c, ix, rx := 0, ixc0, rxc0; c < m.stride; c++ { m.ele[ix], m.ele[rx] = m.ele[rx], m.ele[ix] ix++ rx++ } if d := m.ele[rxc0+lead]; d != 0 { d := 1 / d for c, rx := 0, rxc0; c < m.stride; c++ { m.ele[rx] *= d rx++ } } for ixc0 = 0; ixc0 < len(m.ele); ixc0 += m.stride { if ixc0 != rxc0 { f := m.ele[ixc0+lead] for c, ix, rx := 0, ixc0, rxc0; c < m.stride; c++ { m.ele[ix] -= m.ele[rx] * f ix++ rx++ } } } lead++ } } func main() { m := matrixFromRows([][]float64{ {1, 2, -1, -4}, {2, 3, -1, -11}, {-2, 0, -3, 22}, }) m.print("Input:") m.rref() m.print("Reduced:") }  Output: Input: [ 1.00 2.00 -1.00 -4.00] [ 2.00 3.00 -1.00 -11.00] [ -2.00 0.00 -3.00 22.00] Reduced: [ 1.00 0.00 0.00 -8.00] [ -0.00 1.00 0.00 1.00] [ -0.00 -0.00 1.00 -2.00]  ## Groovy This solution implements the transformation to reduced row echelon form with optional pivoting. Options are provided for both partial pivoting and scaled partial pivoting. The default option is no pivoting at all. enum Pivoting { NONE({ i, it -> 1 }), PARTIAL({ i, it -> - (it[i].abs()) }), SCALED({ i, it -> - it[i].abs()/(it.inject(0) { sum, elt -> sum + elt.abs() } ) }); public final Closure comparer private Pivoting(Closure c) { comparer = c } } def isReducibleMatrix = { matrix -> def m = matrix.size() m > 1 && matrix[0].size() > m && matrix[1..<m].every { row -> row.size() == matrix[0].size() } } def reducedRowEchelonForm = { matrix, Pivoting pivoting = Pivoting.NONE -> assert isReducibleMatrix(matrix) def m = matrix.size() def n = matrix[0].size() (0..<m).each { i -> matrix[i..<m].sort(pivoting.comparer.curry(i)) matrix[i][i..<n] = matrix[i][i..<n].collect { it/matrix[i][i] } ((0..<i) + ((i+1)..<m)).each { k -> (i..<n).reverse().each { j -> matrix[k][j] -= matrix[i][j]*matrix[k][i] } } } matrix }  This test first demonstrates the test case provided, and then demonstrates another test case designed to show the dangers of not using pivoting on an otherwise solvable matrix. Both test cases exercise all three pivoting options. def matrixCopy = { matrix -> matrix.collect { row -> row.collect { it } } } println "Tests for matrix A:" def a = [ [1, 2, -1, -4], [2, 3, -1, -11], [-2, 0, -3, 22] ] a.each { println it } println() println "pivoting == Pivoting.NONE" reducedRowEchelonForm(matrixCopy(a)).each { println it } println() println "pivoting == Pivoting.PARTIAL" reducedRowEchelonForm(matrixCopy(a), Pivoting.PARTIAL).each { println it } println() println "pivoting == Pivoting.SCALED" reducedRowEchelonForm(matrixCopy(a), Pivoting.SCALED).each { println it } println() println "Tests for matrix B (divides by 0 without pivoting):" def b = [ [1, 2, -1, -4], [2, 4, -1, -11], [-2, 0, -6, 24] ] b.each { println it } println() println "pivoting == Pivoting.NONE" try { reducedRowEchelonForm(matrixCopy(b)).each { println it } println() } catch (e) { println "KABOOM!${e.message}"
println()
}

println "pivoting == Pivoting.PARTIAL"
reducedRowEchelonForm(matrixCopy(b), Pivoting.PARTIAL).each { println it }
println()
println "pivoting == Pivoting.SCALED"
reducedRowEchelonForm(matrixCopy(b), Pivoting.SCALED).each { println it }
println()

Output:
Tests for matrix A:
[1, 2, -1, -4]
[2, 3, -1, -11]
[-2, 0, -3, 22]

pivoting == Pivoting.NONE
[1, 0, 0, -8]
[0, 1, 0, 1]
[0, 0, 1, -2]

pivoting == Pivoting.PARTIAL
[1, 0.0, 0E-11, -7.9999999997000000000150]
[0, 1, 0E-10, 0.999999999700000000010]
[0, 0.0, 1, -2.00000000030]

pivoting == Pivoting.SCALED
[1, 0, 0, -8]
[0, 1, 0, 1]
[0, 0, 1, -2]

Tests for matrix B (divides by 0 without pivoting):
[1, 2, -1, -4]
[2, 4, -1, -11]
[-2, 0, -6, 24]

pivoting == Pivoting.NONE
KABOOM! Division undefined

pivoting == Pivoting.PARTIAL
[1, 0, 0.00, -3.00]
[0, 1, 0.00, -2.00]
[0, 0, 1, -3]

pivoting == Pivoting.SCALED
[1, 0, 0, -3]
[0, 1, 0, -2]
[0, 0, 1, -3]

This program was produced by translating from the Python and gradually refactoring the result into a more functional style.

import Data.List (find)

rref :: Fractional a => [[a]] -> [[a]]
rref m = f m 0 [0 .. rows - 1]
where rows = length m
cols = length $head m f m _ [] = m f m lead (r : rs) | indices == Nothing = m | otherwise = f m' (lead' + 1) rs where indices = find p l p (col, row) = m !! row !! col /= 0 l = [(col, row) | col <- [lead .. cols - 1], row <- [r .. rows - 1]] Just (lead', i) = indices newRow = map (/ m !! i !! lead')$ m !! i

m' = zipWith g [0..] $replace r newRow$
replace i (m !! r) m
g n row
| n == r    = row
| otherwise = zipWith h newRow row
where h = subtract . (* row !! lead')

replace :: Int -> a -> [a] -> [a]
{- Replaces the element at the given index. -}
replace n e l = a ++ e : b
where (a, _ : b) = splitAt n l


## Icon and Unicon

Works in both languages:

procedure main(A)
tM := [[  1,  2, -1, -4],
[  2,  3, -1,-11],
[ -2,  0, -3, 22]]
showMat(rref(tM))
end

procedure rref(M)
rCount := *\M | stop("no Matrix?")
cCount := *(M[1]) | 0
every r := !rCount do {
i := r
while M[i,lead] = 0 do {
if (i+:=1) > rCount then {
i := r
if cCount < (lead +:= 1) then stop("can't reduce")
}
}
M[i] :=: M[r]
if 0 ~= (m0 := M[r,lead]) then every !M[r] /:= real(m0)
every r ~= (i := !rCount) do {
every !(mr := copy(M[r])) *:= M[i,lead]
every M[i,j := !cCount] -:= mr[j]
}
}
return M
end

procedure showMat(M)
every r := !M do every writes(right(!r,5)||" " | "\n")
end

Output:
->rref
1.0   0.0   0.0  -8.0
0.0   1.0   0.0   1.0
0.0   0.0   1.0  -2.0
->


## J

The reduced row echelon form of a matrix can be obtained using the gauss_jordan verb from the linear.ijs script, available as part of the math/misc addon. gauss_jordan and the verb pivot are shown below (in a mediawiki "[Expand]" region) for completeness:

Implementation:

NB.*pivot v Pivot at row, column
NB. form: (row,col) pivot M
'r c'=. x
col=. c{"1 y
y - (col - r = i.#y) */ (r{y) % r{col
)

NB.*gauss_jordan v Gauss-Jordan elimination (full pivoting)
NB. y is: matrix
NB. x is: optional minimum tolerance, default 1e_15.
NB.   If a column below the current pivot has numbers of magnitude all
NB.   less then x, it is treated as all zeros.
gauss_jordan=: verb define
1e_15 gauss_jordan y
:
mtx=. y
'r c'=. $mtx rows=. i.r i=. j=. 0 max=. i.>./ while. (i<r) *. j<c do. k=. max col=. | i}. j{"1 mtx if. 0 < x-k{col do. NB. if all col < tol, set to 0: mtx=. 0 (<(i}.rows);j) } mtx else. NB. otherwise sort and pivot: if. k do. mtx=. (<i,i+k) C. mtx end. mtx=. (i,j) pivot mtx i=. >:i end. j=. >:j end. mtx )  Usage:  require 'math/misc/linear' ]A=: 1 2 _1 _4 , 2 3 _1 _11 ,: _2 0 _3 22 1 2 _1 _4 2 3 _1 _11 _2 0 _3 22 gauss_jordan A 1 0 0 _8 0 1 0 1 0 0 1 _2  Additional examples, recommended on talk page:  gauss_jordan 2 0 _1 0 0,1 0 0 _1 0,3 0 0 _2 _1,0 1 0 0 _2,:0 1 _1 0 0 1 0 0 0 _1 0 1 0 0 _2 0 0 1 0 _2 0 0 0 1 _1 0 0 0 0 0 gauss_jordan 1 2 3 4 3 1,2 4 6 2 6 2,3 6 18 9 9 _6,4 8 12 10 12 4,:5 10 24 11 15 _4 1 2 0 0 3 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 gauss_jordan 0 1,1 2,:0 5 1 0 0 1 0 0  And: mat=: 0 ". ];._2 noun define 1 0 0 0 0 0 1 0 0 0 0 _1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 _1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 _1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 _1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 _1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 _1 0 0 0 1 0 0 0 1 0 0 0 0 0 _1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 _1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 _1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 _1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 _1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 _1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 _1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 _1 0 0 0 ) gauss_jordan mat 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.435897 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.307692 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.512821 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0.717949 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0.487179 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0.205128 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0.282051 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0.333333 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0.512821 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0.641026 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0.717949 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0.769231 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0.512821 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0.820513  ## Java This requires Apache Commons 2.2+ import java.util.*; import java.lang.Math; import org.apache.commons.math.fraction.Fraction; import org.apache.commons.math.fraction.FractionConversionException; /* Matrix class * Handles elementary Matrix operations: * Interchange * Multiply and Add * Scale * Reduced Row Echelon Form */ class Matrix { LinkedList<LinkedList<Fraction>> matrix; int numRows; int numCols; static class Coordinate { int row; int col; Coordinate(int r, int c) { row = r; col = c; } public String toString() { return "(" + row + ", " + col + ")"; } } Matrix(double [][] m) { numRows = m.length; numCols = m[0].length; matrix = new LinkedList<LinkedList<Fraction>>(); for (int i = 0; i < numRows; i++) { matrix.add(new LinkedList<Fraction>()); for (int j = 0; j < numCols; j++) { try { matrix.get(i).add(new Fraction(m[i][j])); } catch (FractionConversionException e) { System.err.println("Fraction could not be converted from double by apache commons . . ."); } } } } public void Interchange(Coordinate a, Coordinate b) { LinkedList<Fraction> temp = matrix.get(a.row); matrix.set(a.row, matrix.get(b.row)); matrix.set(b.row, temp); int t = a.row; a.row = b.row; b.row = t; } public void Scale(Coordinate x, Fraction d) { LinkedList<Fraction> row = matrix.get(x.row); for (int i = 0; i < numCols; i++) { row.set(i, row.get(i).multiply(d)); } } public void MultiplyAndAdd(Coordinate to, Coordinate from, Fraction scalar) { LinkedList<Fraction> row = matrix.get(to.row); LinkedList<Fraction> rowMultiplied = matrix.get(from.row); for (int i = 0; i < numCols; i++) { row.set(i, row.get(i).add((rowMultiplied.get(i).multiply(scalar)))); } } public void RREF() { Coordinate pivot = new Coordinate(0,0); int submatrix = 0; for (int x = 0; x < numCols; x++) { pivot = new Coordinate(pivot.row, x); //Step 1 //Begin with the leftmost nonzero column. This is a pivot column. The pivot position is at the top. for (int i = x; i < numCols; i++) { if (isColumnZeroes(pivot) == false) { break; } else { pivot.col = i; } } //Step 2 //Select a nonzero entry in the pivot column with the highest absolute value as a pivot. pivot = findPivot(pivot); if (getCoordinate(pivot).doubleValue() == 0.0) { pivot.row++; continue; } //If necessary, interchange rows to move this entry into the pivot position. //move this row to the top of the submatrix if (pivot.row != submatrix) { Interchange(new Coordinate(submatrix, pivot.col), pivot); } //Force pivot to be 1 if (getCoordinate(pivot).doubleValue() != 1) { /* System.out.println(getCoordinate(pivot)); System.out.println(pivot); System.out.println(matrix); */ Fraction scalar = getCoordinate(pivot).reciprocal(); Scale(pivot, scalar); } //Step 3 //Use row replacement operations to create zeroes in all positions below the pivot. //belowPivot = belowPivot + (Pivot * -belowPivot) for (int i = pivot.row; i < numRows; i++) { if (i == pivot.row) { continue; } Coordinate belowPivot = new Coordinate(i, pivot.col); Fraction complement = (getCoordinate(belowPivot).negate().divide(getCoordinate(pivot))); MultiplyAndAdd(belowPivot, pivot, complement); } //Step 5 //Beginning with the rightmost pivot and working upward and to the left, create zeroes above each pivot. //If a pivot is not 1, make it 1 by a scaling operation. //Use row replacement operations to create zeroes in all positions above the pivot for (int i = pivot.row; i >= 0; i--) { if (i == pivot.row) { if (getCoordinate(pivot).doubleValue() != 1.0) { Scale(pivot, getCoordinate(pivot).reciprocal()); } continue; } if (i == pivot.row) { continue; } Coordinate abovePivot = new Coordinate(i, pivot.col); Fraction complement = (getCoordinate(abovePivot).negate().divide(getCoordinate(pivot))); MultiplyAndAdd(abovePivot, pivot, complement); } //Step 4 //Ignore the row containing the pivot position and cover all rows, if any, above it. //Apply steps 1-3 to the remaining submatrix. Repeat until there are no more nonzero entries. if ((pivot.row + 1) >= numRows || isRowZeroes(new Coordinate(pivot.row+1, pivot.col))) { break; } submatrix++; pivot.row++; } } public boolean isColumnZeroes(Coordinate a) { for (int i = 0; i < numRows; i++) { if (matrix.get(i).get(a.col).doubleValue() != 0.0) { return false; } } return true; } public boolean isRowZeroes(Coordinate a) { for (int i = 0; i < numCols; i++) { if (matrix.get(a.row).get(i).doubleValue() != 0.0) { return false; } } return true; } public Coordinate findPivot(Coordinate a) { int first_row = a.row; Coordinate pivot = new Coordinate(a.row, a.col); Coordinate current = new Coordinate(a.row, a.col); for (int i = a.row; i < (numRows - first_row); i++) { current.row = i; if (getCoordinate(current).doubleValue() == 1.0) { Interchange(current, a); } } current.row = a.row; for (int i = current.row; i < (numRows - first_row); i++) { current.row = i; if (getCoordinate(current).doubleValue() != 0) { pivot.row = i; break; } } return pivot; } public Fraction getCoordinate(Coordinate a) { return matrix.get(a.row).get(a.col); } public String toString() { return matrix.toString().replace("], ", "]\n"); } public static void main (String[] args) { double[][] matrix_1 = { {1, 2, -1, -4}, {2, 3, -1, -11}, {-2, 0, -3, 22} }; Matrix x = new Matrix(matrix_1); System.out.println("before\n" + x.toString() + "\n"); x.RREF(); System.out.println("after\n" + x.toString() + "\n"); double matrix_2 [][] = { {2, 0, -1, 0, 0}, {1, 0, 0, -1, 0}, {3, 0, 0, -2, -1}, {0, 1, 0, 0, -2}, {0, 1, -1, 0, 0} }; Matrix y = new Matrix(matrix_2); System.out.println("before\n" + y.toString() + "\n"); y.RREF(); System.out.println("after\n" + y.toString() + "\n"); double matrix_3 [][] = { {1, 2, 3, 4, 3, 1}, {2, 4, 6, 2, 6, 2}, {3, 6, 18, 9, 9, -6}, {4, 8, 12, 10, 12, 4}, {5, 10, 24, 11, 15, -4} }; Matrix z = new Matrix(matrix_3); System.out.println("before\n" + z.toString() + "\n"); z.RREF(); System.out.println("after\n" + z.toString() + "\n"); double matrix_4 [][] = { {0, 1}, {1, 2}, {0,5} }; Matrix a = new Matrix(matrix_4); System.out.println("before\n" + a.toString() + "\n"); a.RREF(); System.out.println("after\n" + a.toString() + "\n"); } }  ## JavaScript Works with: SpiderMonkey for the print() function. Extends the Matrix class defined at Matrix Transpose#JavaScript // modifies the matrix in-place Matrix.prototype.toReducedRowEchelonForm = function() { var lead = 0; for (var r = 0; r < this.rows(); r++) { if (this.columns() <= lead) { return; } var i = r; while (this.mtx[i][lead] == 0) { i++; if (this.rows() == i) { i = r; lead++; if (this.columns() == lead) { return; } } } var tmp = this.mtx[i]; this.mtx[i] = this.mtx[r]; this.mtx[r] = tmp; var val = this.mtx[r][lead]; for (var j = 0; j < this.columns(); j++) { this.mtx[r][j] /= val; } for (var i = 0; i < this.rows(); i++) { if (i == r) continue; val = this.mtx[i][lead]; for (var j = 0; j < this.columns(); j++) { this.mtx[i][j] -= val * this.mtx[r][j]; } } lead++; } return this; } var m = new Matrix([ [ 1, 2, -1, -4], [ 2, 3, -1,-11], [-2, 0, -3, 22] ]); print(m.toReducedRowEchelonForm()); print(); m = new Matrix([ [ 1, 2, 3, 7], [-4, 7,-2, 7], [ 3, 3, 0, 7] ]); print(m.toReducedRowEchelonForm());  Output: 1,0,0,-8 0,1,0,1 0,0,1,-2 1,0,0,0.6666666666666663 0,1,0,1.666666666666667 0,0,1,1 ## jq Works with: jq Also works with gojq, the Go implementation of jq, and with fq. Generic Functions # swap .[$i] and .[$j] def array_swap($i; $j): if$i == $j then . elif$i < $j then array_swap($j; $i) else .[$i] as $t | .[:$j] + [$t] + .[$j:$i] + .[$i + 1:]
end ;

# element-wise subtraction: $a -$b
def array_subtract($a;$b):
$a | [range(0;length) as$i | .[$i] -$b[$i]]; def lpad($len):
tostring | ($len - length) as$l | (" " * $l)[:$l] + .;

# Ensure -0 prints as 0
def matrix_print:
([.[][] | tostring | length] | max) as $max | .[] | map(if . == 0 then 0 else . end | lpad($max))
| join("  ");

# RREF
# assume input is a rectangular numeric matrix
def toReducedRowEchelonForm:
length as $nr | (.[0]|length) as$nc
| { lead: 0, r: -1, a: .}
| until ($nc == .lead or .r ==$nr;
.r += 1
| .r as $r | .i =$r
| until ($nc == .lead or .a[.i][.lead] != 0; .i += 1 | if$nr == .i
then .i = $r | .lead += 1 else . end ) | if$nc > .lead and $nr >$r
then .i as $i | .a |= array_swap($i; $r) | .a[$r][.lead] as $div | if$div != 0
then .a[$r] |= map(. /$div)
else .
end
| reduce range(0; $nr) as$k (.;
if $k !=$r
then .a[$k][.lead] as$mult
| .a[$k] = array_subtract(.a[$k]; (.a[$r] | map(. *$mult)))
else .
end )
else .
end )
| .a;

[   [ 1,  2,  -1,  -4],
[ 2,  3,  -1, -11],
[-2,  0,  -3,  22]  ],
[   [1, 2, -1, -4],
[2, 4, -1, -11],
[-2, 0, -6, 24] ]

| "Original:", matrix_print, "",
"RREF:",  (toReducedRowEchelonForm|matrix_print), "\n"
Output:

Invocation: jq -nrc -f reduced-row-echelon-form.jq

Original:
1    2   -1   -4
2    3   -1  -11
-2    0   -3   22

RREF:
1   0   0  -8
0   1   0   1
0   0   1  -2

Original:
1    2   -1   -4
2    4   -1  -11
-2    0   -6   24

RREF:
1   0   0  -3
0   1   0  -2
0   0   1  -3


## Julia

RowEchelon.jl offers the function rref to compute the reduced-row echelon form:

julia> matrix = [1 2 -1 -4 ; 2 3 -1 -11 ; -2 0 -3 22]
3x4 Int32 Array:
1  2  -1   -4
2  3  -1  -11
-2  0  -3   22

julia> rref(matrix)
3x4 Array{Float64,2}:
1.0  0.0  0.0  -8.0
0.0  1.0  0.0   1.0
0.0  0.0  1.0  -2.0



## Kotlin

// version 1.1.51

typealias Matrix = Array<DoubleArray>

/* changes the matrix to RREF 'in place' */
fun Matrix.toReducedRowEchelonForm() {
val rowCount = this.size
val colCount = this[0].size
for (r in 0 until rowCount) {
var i = r

i++
if (rowCount == i) {
i = r
}
}

val temp = this[i]
this[i] = this[r]
this[r] = temp

for (j in 0 until colCount) this[r][j] /= div
}

for (k in 0 until rowCount) {
if (k != r) {
for (j in 0 until colCount) this[k][j] -= this[r][j] * mult
}
}

}
}

fun Matrix.printf(title: String) {
println(title)
val rowCount = this.size
val colCount = this[0].size

for (r in 0 until rowCount) {
for (c in 0 until colCount) {
if (this[r][c] == -0.0) this[r][c] = 0.0  // get rid of negative zeros
print("${"% 6.2f".format(this[r][c])} ") } println() } println() } fun main(args: Array<String>) { val matrices = listOf( arrayOf( doubleArrayOf( 1.0, 2.0, -1.0, -4.0), doubleArrayOf( 2.0, 3.0, -1.0, -11.0), doubleArrayOf(-2.0, 0.0, -3.0, 22.0) ), arrayOf( doubleArrayOf(1.0, 2.0, 3.0, 4.0, 3.0, 1.0), doubleArrayOf(2.0, 4.0, 6.0, 2.0, 6.0, 2.0), doubleArrayOf(3.0, 6.0, 18.0, 9.0, 9.0, -6.0), doubleArrayOf(4.0, 8.0, 12.0, 10.0, 12.0, 4.0), doubleArrayOf(5.0, 10.0, 24.0, 11.0, 15.0, -4.0) ) ) for (m in matrices) { m.printf("Original matrix:") m.toReducedRowEchelonForm() m.printf("Reduced row echelon form:") } }  Output: Original matrix: 1.00 2.00 -1.00 -4.00 2.00 3.00 -1.00 -11.00 -2.00 0.00 -3.00 22.00 Reduced row echelon form: 1.00 0.00 0.00 -8.00 0.00 1.00 0.00 1.00 0.00 0.00 1.00 -2.00 Original matrix: 1.00 2.00 3.00 4.00 3.00 1.00 2.00 4.00 6.00 2.00 6.00 2.00 3.00 6.00 18.00 9.00 9.00 -6.00 4.00 8.00 12.00 10.00 12.00 4.00 5.00 10.00 24.00 11.00 15.00 -4.00 Reduced row echelon form: 1.00 2.00 0.00 0.00 3.00 4.00 0.00 0.00 1.00 0.00 0.00 -1.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  ## Lua function ToReducedRowEchelonForm ( M ) local lead = 1 local n_rows, n_cols = #M, #M[1] for r = 1, n_rows do if n_cols <= lead then break end local i = r while M[i][lead] == 0 do i = i + 1 if n_rows == i then i = r lead = lead + 1 if n_cols == lead then break end end end M[i], M[r] = M[r], M[i] local m = M[r][lead] for k = 1, n_cols do M[r][k] = M[r][k] / m end for i = 1, n_rows do if i ~= r then local m = M[i][lead] for k = 1, n_cols do M[i][k] = M[i][k] - m * M[r][k] end end end lead = lead + 1 end end M = { { 1, 2, -1, -4 }, { 2, 3, -1, -11 }, { -2, 0, -3, 22 } } res = ToReducedRowEchelonForm( M ) for i = 1, #M do for j = 1, #M[1] do io.write( M[i][j], " " ) end io.write( "\n" ) end  Output: 1 0 0 -8 0 1 0 1 0 0 1 -2  ## M2000 Interpreter low bound 1 for array Module Base1 { dim base 1, A(3, 4) A(1, 1)= 1, 2, -1, -4, 2 , 3, -1, -11, -2 , 0 , -3, 22 lead=1 rowcount=3 columncount=4 gosub disp() for r=1 to rowcount { if columncount<lead then exit i=r while A(i,lead)=0 { i++ if rowcount=i then i=r : lead++ : if columncount<lead then exit } for c =1 to columncount { swap A(i, c), A(r, c) } if A(r, lead)<>0 then { div1=A(r,lead) For c =1 to columncount { A( r, c)/=div1 } } for i=1 to rowcount { if i<>r then { mult=A(i,lead) for j=1 to columncount { A(i,j)-=A(r,j)*mult } } } lead=lead+1 } disp() sub disp() local i, j for i=1 to rowcount for j=1 to columncount Print A(i, j), Next j if pos>0 then print Next i End sub } Base1 Low bound 0 for array Module base0 { dim base 0, A(3, 4) A(0, 0)= 1, 2, -1, -4, 2 , 3, -1, -11, -2 , 0 , -3, 22 lead=0 rowcount=3 columncount=4 gosub disp() for r=0 to rowcount-1 { if columncount<=lead then exit i=r while A(i,lead)=0 { i++ if rowcount=i then i=r : lead++ : if columncount<lead then exit } for c =0 to columncount-1 { swap A(i, c), A(r, c) } if A(r, lead)<>0 then { div1=A(r,lead) For c =0 to columncount-1 { A( r, c)/=div1 } } for i=0 to rowcount-1 { if i<>r then { mult=A(i,lead) for j=0 to columncount-1 { A(i,j)-=A(r,j)*mult } } } lead=lead+1 } disp() sub disp() local i, j for i=0 to rowcount-1 for j=0 to columncount-1 Print A(i, j), Next j if pos>0 then print Next i End sub } base0 ## Maple with(LinearAlgebra): ReducedRowEchelonForm(<<1,2,-2>|<2,3,0>|<-1,-1,-3>|<-4,-11,22>>); Output:  [1 0 0 -8] [ ] [0 1 0 1] [ ] [0 0 1 -2]  ## Mathematica/Wolfram Language RowReduce[{{1, 2, -1, -4}, {2, 3, -1, -11}, {-2, 0, -3, 22}}]  Output: {{1, 0, 0, -8}, {0, 1, 0, 1}, {0, 0, 1, -2}} ## MATLAB rref([1, 2, -1, -4; 2, 3, -1, -11; -2, 0, -3, 22])  ## Maxima rref(a):=block([p,q,k],[p,q]:matrix_size(a),a:echelon(a), k:min(p,q), for i thru min(p,q) do (if a[i,i]=0 then (k:i-1,return())), for i:k thru 2 step -1 do (for j from i-1 thru 1 step -1 do a:rowop(a,j,i,a[j,i])), a)$

a: matrix([12,-27,36,44,59],
[26,41,-54,24,23],
[33,70,59,15,-68],
[43,16,29,-52,-61],
[-43,20,71,88,11])$rref(a); matrix([1,0,0,0,1/2],[0,1,0,0,-1],[0,0,1,0,-1/2],[0,0,0,1,1],[0,0,0,0,0])  ## Nim ### Using rationals To avoid rounding issues, we can use rationals and convert to floats only at the end. import rationals, strutils type Fraction = Rational[int] const Zero: Fraction = 0 // 1 type Matrix[M, N: static Positive] = array[M, array[N, Fraction]] func toMatrix[M, N: static Positive](a: array[M, array[N, int]]): Matrix[M, N] = ## Convert a matrix of integers to a matrix of integer fractions. for i in 0..<M: for j in 0..<N: result[i][j] = a[i][j] // 1 func transformToRref(mat: var Matrix) = ## Transform the given matrix to reduced row echelon form. var lead = 0 for r in 0..<mat.M: if lead >= mat.N: return var i = r while mat[i][lead] == Zero: inc i if i == mat.M: i = r inc lead if lead == mat.N: return swap mat[i], mat[r] if (let d = mat[r][lead]; d) != Zero: for item in mat[r].mitems: item /= d for i in 0..<mat.M: if i != r: let m = mat[i][lead] for c in 0..<mat.N: mat[i][c] -= mat[r][c] * m inc lead proc $(mat: Matrix): string =
## Display a matrix.

for row in mat:
var line = ""
for val in row:
echo line

#———————————————————————————————————————————————————————————————————————————————————————————————————

template runTest(mat: Matrix) =
## Run a test using matrix "mat".

echo "Original matrix:"
echo mat
echo "Reduced row echelon form:"
mat.transformToRref()
echo mat
echo ""

var m1 = [[ 1, 2, -1,  -4],
[ 2, 3, -1, -11],
[-2, 0, -3,  22]].toMatrix()

var m2 = [[2, 0, -1,  0,  0],
[1, 0,  0, -1,  0],
[3, 0,  0, -2, -1],
[0, 1,  0,  0, -2],
[0, 1, -1,  0,  0]].toMatrix()

var m3 = [[1,  2,  3,  4,  3,  1],
[2,  4,  6,  2,  6,  2],
[3,  6, 18,  9,  9, -6],
[4,  8, 12, 10, 12,  4],
[5, 10, 24, 11, 15, -4]].toMatrix()

var m4 = [[0, 1],
[1, 2],
[0, 5]].toMatrix()

runTest(m1)
runTest(m2)
runTest(m3)
runTest(m4)

Output:
Original matrix:
1.00    2.00   -1.00   -4.00
2.00    3.00   -1.00  -11.00
-2.00    0.00   -3.00   22.00

Reduced row echelon form:
1.00    0.00    0.00   -8.00
0.00    1.00    0.00    1.00
0.00    0.00    1.00   -2.00

Original matrix:
2.00    0.00   -1.00    0.00    0.00
1.00    0.00    0.00   -1.00    0.00
3.00    0.00    0.00   -2.00   -1.00
0.00    1.00    0.00    0.00   -2.00
0.00    1.00   -1.00    0.00    0.00

Reduced row echelon form:
1.00    0.00    0.00    0.00   -1.00
0.00    1.00    0.00    0.00   -2.00
0.00    0.00    1.00    0.00   -2.00
0.00    0.00    0.00    1.00   -1.00
0.00    0.00    0.00    0.00    0.00

Original matrix:
1.00    2.00    3.00    4.00    3.00    1.00
2.00    4.00    6.00    2.00    6.00    2.00
3.00    6.00   18.00    9.00    9.00   -6.00
4.00    8.00   12.00   10.00   12.00    4.00
5.00   10.00   24.00   11.00   15.00   -4.00

Reduced row echelon form:
1.00    2.00    0.00    0.00    3.00    4.00
0.00    0.00    1.00    0.00    0.00   -1.00
0.00    0.00    0.00    1.00    0.00    0.00
0.00    0.00    0.00    0.00    0.00    0.00
0.00    0.00    0.00    0.00    0.00    0.00

Original matrix:
0.00    1.00
1.00    2.00
0.00    5.00

Reduced row echelon form:
1.00    0.00
0.00    1.00
0.00    0.00

### Using floats

When using floats, we have to be careful when doing comparisons. The previous program adapted to use floats instead of rationals may give wrong results. This would be the case with the second matrix. To get the right result, we have to do a comparison to an epsilon rather than zero. Here is the program modified to work with floats:

import strutils, strformat

const Eps = 1e-10

type Matrix[M, N: static Positive] = array[M, array[N, float]]

func toMatrix[M, N: static Positive](a: array[M, array[N, int]]): Matrix[M, N] =
## Convert a matrix of integers to a matrix of floats.
for i in 0..<M:
for j in 0..<N:
result[i][j] = a[i][j].toFloat

func transformToRref(mat: var Matrix) =
## Transform the given matrix to reduced row echelon form.

for r in 0..<mat.M:

var i = r
inc i
if i == mat.M:
i = r
swap mat[i], mat[r]

if abs(d) > Eps:    # Checking "d != 0" will give wrong results in some cases.
for item in mat[r].mitems:
item /= d

for i in 0..<mat.M:
if i != r:
for c in 0..<mat.N:
mat[i][c] -= mat[r][c] * m

proc $(mat: Matrix): string = ## Display a matrix. for row in mat: var line = "" for val in row: line.addSep(" ", 0) line.add &"{val:7.2f}" echo line #——————————————————————————————————————————————————————————————————————————————————————————————————— template runTest(mat: Matrix) = ## Run a test using matrix "mat". echo "Original matrix:" echo mat echo "Reduced row echelon form:" mat.transformToRref() echo mat echo "" var m1 = [[ 1, 2, -1, -4], [ 2, 3, -1, -11], [-2, 0, -3, 22]].toMatrix() var m2 = [[2, 0, -1, 0, 0], [1, 0, 0, -1, 0], [3, 0, 0, -2, -1], [0, 1, 0, 0, -2], [0, 1, -1, 0, 0]].toMatrix() var m3 = [[1, 2, 3, 4, 3, 1], [2, 4, 6, 2, 6, 2], [3, 6, 18, 9, 9, -6], [4, 8, 12, 10, 12, 4], [5, 10, 24, 11, 15, -4]].toMatrix() var m4 = [[0, 1], [1, 2], [0, 5]].toMatrix() runTest(m1) runTest(m2) runTest(m3) runTest(m4)  Output: Same result as that of the program working with rationals (at least for the matrices used here). ## Objeck class RowEchelon { function : Main(args : String[]) ~ Nil { matrix := [ [1, 2, -1, -4 ] [2, 3, -1, -11 ] [-2, 0, -3, 22] ]; matrix := Rref(matrix); sizes := matrix->Size(); for(i := 0; i < sizes[0]; i += 1;) { for(j := 0; j < sizes[1]; j += 1;) { IO.Console->Print(matrix[i,j])->Print(","); }; IO.Console->PrintLine(); }; } function : native : Rref(matrix : Int[,]) ~ Int[,] { lead := 0; sizes := matrix->Size(); rowCount := sizes[0]; columnCount := sizes[1]; for(r := 0; r < rowCount; r+=1;) { if (columnCount <= lead) { break; }; i := r; while(matrix[i, lead] = 0) { i+=1; if (i = rowCount) { i := r; lead += 1; if (columnCount = lead) { lead-=1; break; }; }; }; for (j := 0; j < columnCount; j+=1;) { temp := matrix[r, j]; matrix[r, j] := matrix[i, j]; matrix[i, j] := temp; }; div := matrix[r, lead]; for(j := 0; j < columnCount; j+=1;) { matrix[r, j] /= div; }; for(j := 0; j < rowCount; j+=1;) { if (j <> r) { sub := matrix[j, lead]; for (k := 0; k < columnCount; k+=1;) { matrix[j, k] -= sub * matrix[r, k]; }; }; }; lead+=1; }; return matrix; } } ## OCaml let swap_rows m i j = let tmp = m.(i) in m.(i) <- m.(j); m.(j) <- tmp; ;; let rref m = try let lead = ref 0 and rows = Array.length m and cols = Array.length m.(0) in for r = 0 to pred rows do if cols <= !lead then raise Exit; let i = ref r in while m.(!i).(!lead) = 0 do incr i; if rows = !i then begin i := r; incr lead; if cols = !lead then raise Exit; end done; swap_rows m !i r; let lv = m.(r).(!lead) in m.(r) <- Array.map (fun v -> v / lv) m.(r); for i = 0 to pred rows do if i <> r then let lv = m.(i).(!lead) in m.(i) <- Array.mapi (fun i iv -> iv - lv * m.(r).(i)) m.(i); done; incr lead; done with Exit -> () ;; let () = let m = [| [| 1; 2; -1; -4 |]; [| 2; 3; -1; -11 |]; [| -2; 0; -3; 22 |]; |] in rref m; Array.iter (fun row -> Array.iter (fun v -> Printf.printf " %d" v ) row; print_newline() ) m  Another implementation: let rref m = let nr, nc = Array.length m, Array.length m.(0) in let add r s k = for i = 0 to nc-1 do m.(r).(i) <- m.(r).(i) +. m.(s).(i)*.k done in for c = 0 to min (nc-1) (nr-1) do for r = c+1 to nr-1 do if abs_float m.(c).(c) < abs_float m.(r).(c) then let v = m.(r) in (m.(r) <- m.(c); m.(c) <- v) done; let t = m.(c).(c) in if t <> 0.0 then begin for r = 0 to nr-1 do if r <> c then add r c (-.m.(r).(c)/.t) done; for i = 0 to nc-1 do m.(c).(i) <- m.(c).(i)/.t done end done;; let mat = [| [| 1.0; 2.0; -.1.0; -.4.0;|]; [| 2.0; 3.0; -.1.0; -.11.0;|]; [|-.2.0; 0.0; -.3.0; 22.0;|] |] in let pr v = Array.iter (Printf.printf " %9.4f") v; print_newline() in let show = Array.iter pr in show mat; print_newline(); rref mat; show mat  ## Octave A = [ 1, 2, -1, -4; 2, 3, -1, -11; -2, 0, -3, 22]; refA = rref(A); disp(refA);  ## PARI/GP PARI has a built-in matrix type, but no commands for row-echelon form. This is a basic one implementing Gauss-Jordan reduction. matrref(M)= { my(s=matsize(M),t=s[1]); for(i=1,s[2], if(M[t,i]==0, next); M[t,] /= M[t,i]; for(j=1,t-1, M[j,] -= M[j,i]*M[t,] ); for(j=t+1,s[1], M[j,] -= M[j,i]*M[t,] ); if(t--<1,break) ); M; } addhelp(matrref, "matrref(M): Returns the reduced row-echelon form of the matrix M."); A faster, dimension-limited one can be constructed from the built-in matsolve command: rref(M)={ my(d=matsize(M)); if(d[1]+1 != d[2], error("Bad size in rref"), d=d[1]); concat(matid(d), matsolve(matrix(d,d,x,y,M[x,y]), M[,d+1])) }; Example: rref([1,2,-1,-4;2,3,-1,-11;-2,0,-3,22]) Output: %1 = [1 0 0 -8] [0 1 0 1] [0 0 1 -2] ## Perl Translation of: Python Note that the function defined here takes an array reference, which is modified in place. sub rref {our @m; local *m = shift; @m or return; my ($lead, $rows,$cols) = (0, scalar(@m), scalar(@{$m[0]})); foreach my$r (0 .. $rows - 1) {$lead < $cols or return; my$i = $r; until ($m[$i][$lead])
{++$i ==$rows or next;
$i =$r;
++$lead ==$cols and return;}

@m[$i,$r] = @m[$r,$i];
my $lv =$m[$r][$lead];
$_ /=$lv foreach @{ $m[$r] };

my @mr = @{ $m[$r] };
foreach my $i (0 ..$rows - 1)
{$i ==$r and next;
($lv, my$n) = ($m[$i][$lead], -1);$_ -= $lv *$mr[++$n] foreach @{$m[$i] };} ++$lead;}}

sub display { join("\n" => map join(" " => map(sprintf("%4d", $_), @$_)), @{+shift})."\n" }

@m =
(
[  1,  2,  -1,  -4 ],
[  2,  3,  -1, -11 ],
[ -2,  0,  -3,  22 ]
);

rref(\@m);
print display(\@m);

Output:
   1    0    0   -8
0    1    0    1
0    0    1   -2

## Phix

Translation of: Euphoria
with javascript_semantics
function ToReducedRowEchelonForm(sequence M)
rowCount = length(M),
columnCount = length(M[1]),
i
for r=1 to rowCount do
if lead>=columnCount then exit end if
i = r
i += 1
if i=rowCount then
i = r
if lead=columnCount then exit end if
end if
end while
M[i] = M[r]
M[r] = mr
for j=1 to rowCount do
if j!=r then
end if
end for
end for
return M
end function

? ToReducedRowEchelonForm(
{ { 1, 2, -1, -4 },
{ 2, 3, -1, -11 },
{ -2, 0, -3, 22 } })

Output:
{{1,0,0,-8},{0,1,0,1},{0,0,1,-2}}


## PHP

Works with: PHP version 5.x
Translation of: Java
<?php

function rref($matrix) {$lead = 0;
$rowCount = count($matrix);
if ($rowCount == 0) return$matrix;
$columnCount = 0; if (isset($matrix[0])) {
$columnCount = count($matrix[0]);
}
for ($r = 0;$r < $rowCount;$r++) {
if ($lead >=$columnCount)
break;        {
$i =$r;
while ($matrix[$i][$lead] == 0) {$i++;
if ($i ==$rowCount) {
$i =$r;
$lead++; if ($lead == $columnCount) return$matrix;
}
}
$temp =$matrix[$r];$matrix[$r] =$matrix[$i];$matrix[$i] =$temp;
}        {
$lv =$matrix[$r][$lead];
for ($j = 0;$j < $columnCount;$j++) {
$matrix[$r][$j] =$matrix[$r][$j] / $lv; } } for ($i = 0; $i <$rowCount; $i++) { if ($i != $r) {$lv = $matrix[$i][$lead]; for ($j = 0; $j <$columnCount; $j++) {$matrix[$i][$j] -= $lv *$matrix[$r][$j];
}
}
}
$lead++; } return$matrix;
}
?>


## PicoLisp

(de reducedRowEchelonForm (Mat)
(let (Lead 1  Cols (length (car Mat)))
(for (X Mat X (cdr X))
(NIL
(loop
(T (seek '((R) (n0 (get R 1 Lead))) X)
@ )
(T (> (inc 'Lead) Cols)) ) )
(xchg @ X)
(let D (get X 1 Lead)
(map
'((R) (set R (/ (car R) D)))
(car X) ) )
(for Y Mat
(unless (== Y (car X))
(let N (- (get Y Lead))
(map
'((Dst Src)
(inc Dst (* N (car Src))) )
Y
(car X) ) ) ) )
(T (> (inc 'Lead) Cols)) ) )
Mat )
Output:
(reducedRowEchelonForm
'(( 1  2  -1   -4) ( 2  3  -1  -11) (-2  0  -3   22)) )
-> ((1 0 0 -8) (0 1 0 1) (0 0 1 -2))

## Python

def ToReducedRowEchelonForm( M):
if not M: return
rowCount = len(M)
columnCount = len(M[0])
for r in range(rowCount):
return
i = r
i += 1
if i == rowCount:
i = r
return
M[i],M[r] = M[r],M[i]
M[r] = [ mrx / float(lv) for mrx in M[r]]
for i in range(rowCount):
if i != r:
M[i] = [ iv - lv*rv for rv,iv in zip(M[r],M[i])]

mtx = [
[ 1, 2, -1, -4],
[ 2, 3, -1, -11],
[-2, 0, -3, 22],]

ToReducedRowEchelonForm( mtx )

for rw in mtx:
print ', '.join( (str(rv) for rv in rw) )


## R

Translation of: Fortran
rref <- function(m) {
pivot <- 1
norow <- nrow(m)
nocolumn <- ncol(m)
for(r in 1:norow) {
if ( nocolumn <= pivot ) break;
i <- r
while( m[i,pivot] == 0 ) {
i <- i + 1
if ( norow == i ) {
i <- r
pivot <- pivot + 1
if ( nocolumn == pivot ) return(m)
}
}
trow <- m[i, ]
m[i, ] <- m[r, ]
m[r, ] <- trow
m[r, ] <- m[r, ] / m[r, pivot]
for(i in 1:norow) {
if ( i != r )
m[i, ] <- m[i, ] - m[r, ] * m[i, pivot]
}
pivot <- pivot + 1
}
return(m)
}

m <- matrix(c(1, 2, -1, -4,
2, 3, -1, -11,
-2, 0, -3, 22), 3, 4, byrow=TRUE)
print(m)
print(rref(m))

## Racket

#lang racket
(require math)
(define (reduced-echelon M)
(matrix-row-echelon M #t #t))

(reduced-echelon
(matrix [[1 2 -1 -4]
[2 3 -1 -11]
[-2 0 -3 22]]))

Output:
(mutable-array
#[#[1 0 0 -8]
#[0 1 0 1]
#[0 0 1 -2]])


## Raku

(formerly Perl 6)

### Following pseudocode

Translation of: Perl
sub rref (@m) {
my ($lead,$rows, $cols) = 0, @m, @m[0]; for ^$rows -> $r { return @m unless$lead < $cols; my$i = $r; until @m[$i;$lead] { next unless ++$i == $rows;$i = $r; return @m if ++$lead == $cols; } @m[$i, $r] = @m[$r, $i] if$r != $i; @m[$r] »/=» $= @m[$r;$lead]; for ^$rows -> $n { next if$n == $r; @m[$n] »-=» @m[$r] »×» (@m[$n;$lead] // 0); } ++$lead;
}
@m
}

sub rat-or-int ($num) { return$num unless $num ~~ Rat; return$num.narrow if $num.narrow ~~ Int;$num.nude.join: '/';
}

sub say_it ($message, @array) { say "\n$message";
$_».&rat-or-int.fmt(" %5s").say for @array; } my @M = ( [ # base test case [ 1, 2, -1, -4 ], [ 2, 3, -1, -11 ], [ -2, 0, -3, 22 ], ], [ # mix of number styles [ 3, 0, -3, 1 ], [ .5, 3/2, -3, -2 ], [ .2, 4/5, -1.6, .3 ], ], [ # degenerate case [ 1, 2, 3, 4, 3, 1], [ 2, 4, 6, 2, 6, 2], [ 3, 6, 18, 9, 9, -6], [ 4, 8, 12, 10, 12, 4], [ 5, 10, 24, 11, 15, -4], ], [ # larger matrix [1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0], [0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0], ] ); for @M -> @matrix { say_it( 'Original Matrix', @matrix ); say_it( 'Reduced Row Echelon Form Matrix', rref(@matrix) ); say "\n"; }  Raku handles rational numbers internally as a ratio of two integers to maintain precision. For some situations it is useful to return the ratio rather than the floating point result. Output: Original Matrix 1 2 -1 -4 2 3 -1 -11 -2 0 -3 22 Reduced Row Echelon Form Matrix 1 0 0 -8 0 1 0 1 0 0 1 -2 Original Matrix 3 0 -3 1 1/2 3/2 -3 -2 1/5 4/5 -8/5 3/10 Reduced Row Echelon Form Matrix 1 0 0 -41/2 0 1 0 -217/6 0 0 1 -125/6 Original Matrix 1 2 3 4 3 1 2 4 6 2 6 2 3 6 18 9 9 -6 4 8 12 10 12 4 5 10 24 11 15 -4 Reduced Row Echelon Form Matrix 1 2 0 0 3 4 0 0 1 0 0 -1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Original Matrix 1 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 1 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 -1 0 0 0 Reduced Row Echelon Form Matrix 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17/39 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4/13 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20/39 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 28/39 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 19/39 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 8/39 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 11/39 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1/3 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 20/39 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 25/39 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 28/39 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 10/13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 20/39 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 32/39  ### Row operations, procedural code Re-implemented as elementary matrix row operations. Follow links for background on row operations and reduced row echelon form sub scale-row ( @M, \scale, \r ) { @M[r] = @M[r] »×» scale } sub shear-row ( @M, \scale, \r1, \r2 ) { @M[r1] = @M[r1] »+» ( @M[r2] »×» scale ) } sub reduce-row ( @M, \r, \c ) { scale-row @M, 1/@M[r;c], r } sub clear-column ( @M, \r, \c ) { shear-row @M, -@M[$_;c], $_, r for @M.keys.grep: * != r } my @M = ( [< 1 2 -1 -4 >], [< 2 3 -1 -11 >], [< -2 0 -3 22 >], ); my$column-count = @M[0];
my $col = 0; for @M.keys ->$row {
reduce-row( @M, $row,$col );
clear-column( @M, $row,$col );
last if ++$col ==$column-count;
}

say @$_».fmt(' %4g') for @M;  Output: [ 1 0 0 -8] [ 0 1 0 1] [ 0 0 1 -2] ### Row operations, object-oriented code The same code as previous section, recast into OO. Also, scale and shear are recast as unscale and unshear, which fit the problem better. class Matrix is Array { method unscale-row ( @M: \scale, \row ) { @M[row] = @M[row] »/» scale } method unshear-row ( @M: \scale, \r1, \r2 ) { @M[r1] = @M[r1] »-» @M[r2] »×» scale } method reduce-row ( @M: \row, \col ) { @M.unscale-row( @M[row;col], row ) } method clear-column ( @M: \row, \col ) { @M.unshear-row( @M[$_;col], $_, row ) for @M.keys.grep: * != row } method reduced-row-echelon-form ( @M: ) { my$column-count =  @M[0];
my $col = 0; for @M.keys ->$row {
@M.reduce-row(   $row,$col );
@M.clear-column( $row,$col );
return if ++$col ==$column-count;
}
}
}

my $M = Matrix.new( [< 1 2 -1 -4 >], [< 2 3 -1 -11 >], [< -2 0 -3 22 >], );$M.reduced-row-echelon-form;
say @$_».fmt(' %4g') for @$M;

Output:
[    1     0     0    -8]
[    0     1     0     1]
[    0     0     1    -2]

## REXX

Reduced Row Echelon Form   (a.k.a.   row canonical form)   of a matrix, with optimization added.

/*REXX pgm performs Reduced Row Echelon Form (RREF), AKA row canonical form on a matrix)*/
cols= 0;  w= 0;   @. =0                          /*max cols in a row; max width; matrix.*/
mat.=;                  mat.1=  '    1   2   -1      -4   '
mat.2=  '    2   3   -1     -11   '
mat.3=  '   -2   0   -3      22   '
do r=1  until mat.r=='';      _=mat.r  /*build  @.row.col  from (matrix) mat.X*/
do c=1  until _='';       parse  var   _    @.r.c  _
w= max(w, length(@.r.c) + 1) /*find the maximum width of an element.*/
end   /*c*/
cols= max(cols, c)                     /*save the maximum number of columns.  */
end   /*r*/
rows= r-1                                        /*adjust the row count (from DO loop). */
call showMat  'original matrix'                  /*display the original matrix──►screen.*/
!= 1                                             /*set the working column pointer to  1.*/
/* ┌──────────────────────◄────────────────◄──── Reduced Row Echelon Form on matrix.*/
do r=1  for rows  while cols>!                 /*begin to perform the heavy lifting.  */
j= r                                           /*use a subsitute index for the DO loop*/
do  while  @.j.!==0;    j= j + 1
if j==rows  then do;    j= r;     != ! + 1;    if cols==!  then leave r;     end
end      /*while*/
/* [↓]  swap rows J,R (but not if same)*/
do _=1  for cols  while j\==r;    parse value   @.r._  @.j._    with    @.j._  @._._
end      /*_*/
?= @.r.!
do d=1  for cols  while ?\=1;     @.r.d= @.r.d / ?
end      /*d*/                             /* [↑] divide row J by @.r.p ──unless≡1*/
do k=1  for rows;             ?= @.k.! /*subtract (row K)   @.r.s  from row K.*/
if k==r | ?=0  then iterate            /*skip  if  row K is the same as row R.*/
do s=1  for cols;          @.k.s= @.k.s   -   ? * @.r.s
end   /*s*/
end      /*k*/                         /* [↑]  for the rest of numbers in row.*/
!= !+1                                         /*bump the working column pointer.     */
end          /*r*/

call showMat  'matrix RREF'                      /*display the reduced row echelon form.*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
showMat: parse arg title;          say;  say center(title, 3 + (cols+1) * w, '─');    say
do      r=1  for rows;   _=
do c=1  for cols
if @.r.c==''  then do;   say "***error*** matrix element isn't defined:"
say 'row'    r",  column"    c'.';        exit 13
end
_= _  right(@.r.c, w)
end   /*c*/
say _                                 /*display a matrix row to the terminal.*/
end        /*r*/;       return

output   when using the default (internal) input:
────original matrix────

1    2   -1   -4
2    3   -1  -11
-2    0   -3   22

──────matrix RREF──────

1    0    0   -8
0    1    0    1
0    0    1   -2


## Ring

# Project : Reduced row echelon form

matrix = [[1, 2, -1, -4],
[2, 3, -1, -11],
[ -2, 0, -3, 22]]
ref(matrix)
for row = 1 to 3
for col = 1 to 4
if matrix[row][col] = -0
see "0 "
else
see "" + matrix[row][col] + " "
ok
next
see nl
next

func ref(m)
nrows = 3
ncols = 4
for r = 1 to nrows
exit
ok
i = r
i = i + 1
if i = nrows
i = r
exit 2
ok
ok
end
for j = 1 to ncols
temp = m[i][j]
m[i][j] = m[r][j]
m[r][j] = temp
next
if n != 0
for j = 1 to ncols
m[r][j] = m[r][j] / n
next
ok
for i = 1 to nrows
if i != r
for j = 1 to ncols
m[i][j] = m[i][j] - m[r][j] * n
next
ok
next
next

Output:

1 0 0 -8
0 1 0  1
0 0 1 -2


## RPL

The RREF built-in intruction is available for HP-48G and newer models.

[[1 2 -1 -4] [2 3 -1 -11] [-2 0 -3 22]] RREF

Output:
1: [[ 1 0 0 -8 ]
[ 0 1 0 1 ]
[ 0 0 1 -2 ]]


## Ruby

Works with: Ruby version 1.9.3
# returns an 2-D array where each element is a Rational
def reduced_row_echelon_form(ary)
rows = ary.size
cols = ary[0].size
rary = convert_to(ary, :to_r)  # use rational arithmetic
catch :done  do
rows.times do |r|
throw :done  if cols <= lead
i = r
i += 1
if rows == i
i = r
throw :done  if cols == lead
end
end
# swap rows i and r
rary[i], rary[r] = rary[r], rary[i]
# normalize row r
rary[r].collect! {|x| x / v}
# reduce other rows
rows.times do |i|
next if i == r
rary[i].each_index {|j| rary[i][j] -= v * rary[r][j]}
end
end
end
rary
end

# type should be one of :to_s, :to_i, :to_f, :to_r
def convert_to(ary, type)
ary.each_with_object([]) do |row, new|
new << row.collect {|elem| elem.send(type)}
end
end

class Rational
alias _to_s to_s
def to_s
denominator==1 ? numerator.to_s : _to_s
end
end

def print_matrix(m)
max = m[0].collect {-1}
m.each {|row| row.each_index {|i| max[i] = [max[i], row[i].to_s.length].max}}
m.each {|row| row.each_index {|i| print "%#{max[i]}s " % row[i]}; puts}
end

mtx = [
[ 1, 2, -1, -4],
[ 2, 3, -1,-11],
[-2, 0, -3, 22]
]
print_matrix reduced_row_echelon_form(mtx)
puts

mtx = [
[ 1, 2, 3, 7],
[-4, 7,-2, 7],
[ 3, 3, 0, 7]
]
reduced = reduced_row_echelon_form(mtx)
print_matrix reduced
print_matrix convert_to(reduced, :to_f)

Output:
1 0 0 -8
0 1 0  1
0 0 1 -2

1 0 0 2/3
0 1 0 5/3
0 0 1   1
1.0 0.0 0.0 0.6666666666666666
0.0 1.0 0.0 1.6666666666666667
0.0 0.0 1.0                1.0


## Rust

Translation of: FORTRAN

I have tried to avoid state mutation with respect to the input matrix and adopt as functional a style as possible in this translation, so for larger matrices one may want to consider memory usage implications.

fn main() {
let mut matrix_to_reduce: Vec<Vec<f64>> = vec![vec![1.0, 2.0 , -1.0, -4.0],
vec![2.0, 3.0, -1.0, -11.0],
vec![-2.0, 0.0, -3.0, 22.0]];
let mut r_mat_to_red = &mut matrix_to_reduce;
let rr_mat_to_red = &mut r_mat_to_red;

println!("Matrix to reduce:\n{:?}", rr_mat_to_red);
let reduced_matrix = reduced_row_echelon_form(rr_mat_to_red);
println!("Reduced matrix:\n{:?}", reduced_matrix);
}

fn reduced_row_echelon_form(matrix: &mut Vec<Vec<f64>>) -> Vec<Vec<f64>> {
let mut matrix_out: Vec<Vec<f64>> = matrix.to_vec();
let mut pivot = 0;
let row_count = matrix_out.len();
let column_count = matrix_out[0].len();

'outer: for r in 0..row_count {
if column_count <= pivot {
break;
}
let mut i = r;
while matrix_out[i][pivot] == 0.0 {
i = i+1;
if i == row_count {
i = r;
pivot = pivot + 1;
if column_count == pivot {
pivot = pivot - 1;
break 'outer;
}
}
}
for j in 0..row_count {
let temp = matrix_out[r][j];
matrix_out[r][j] = matrix_out[i][j];
matrix_out[i][j] = temp;
}
let divisor = matrix_out[r][pivot];
if divisor != 0.0 {
for j in 0..column_count {
matrix_out[r][j] = matrix_out[r][j] / divisor;
}
}
for j in 0..row_count {
if j != r {
let hold = matrix_out[j][pivot];
for k in 0..column_count {
matrix_out[j][k] = matrix_out[j][k] - ( hold * matrix_out[r][k]);
}
}
}
pivot = pivot + 1;
}
matrix_out
}


Output:

Matrix to reduce:
[[1.0, 2.0, -1.0, -4.0], [2.0, 3.0, -1.0, -11.0], [-2.0, 0.0, -3.0, 22.0]]
Reduced matrix:
[[1.0, 0.0, 0.0, -8.0], [-0.0, 1.0, 0.0, 1.0], [-0.0, -0.0, 1.0, -2.0]]


## Sage

Works with: Sage version 4.6.2
sage: m = matrix(ZZ, [[1,2,-1,-4],[2,3,-1,-11],[-2,0,-3,22]])
sage: m.rref()
[ 1  0  0 -8]
[ 0  1  0  1]
[ 0  0  1 -2]


## Scheme

Works with: Scheme version R${\displaystyle ^{5}}$RS
(define (reduced-row-echelon-form matrix)
(define (clean-down matrix from-row column)
(cons (car matrix)
(if (zero? from-row)
(map (lambda (row)
(map -
row
(map (lambda (element)
(/ (* element (list-ref row column))
(list-ref (car matrix) column)))
(car matrix))))
(cdr matrix))
(clean-down (cdr matrix) (- from-row 1) column))))
(define (clean-up matrix until-row column)
(if (zero? until-row)
matrix
(cons (map -
(car matrix)
(map (lambda (element)
(/ (* element (list-ref (car matrix) column))
(list-ref (list-ref matrix until-row) column)))
(list-ref matrix until-row)))
(clean-up (cdr matrix) (- until-row 1) column))))
(define (normalise matrix row with-column)
(if (zero? row)
(cons (map (lambda (element)
(/ element (list-ref (car matrix) with-column)))
(car matrix))
(cdr matrix))
(cons (car matrix) (normalise (cdr matrix) (- row 1) with-column))))
(define (repeat procedure matrix indices)
(if (null? indices)
matrix
(repeat procedure
(procedure matrix (car indices) (car indices))
(cdr indices))))
(define (iota start stop)
(if (> start stop)
(list)
(cons start (iota (+ start 1) stop))))
(let ((indices (iota 0 (- (length matrix) 1))))
(repeat normalise
(repeat clean-up
(repeat clean-down
matrix
indices)
indices)
indices)))


Example:

(define matrix
(list (list 1 2 -1 -4) (list 2 3 -1 -11) (list -2 0 -3 22)))

(display (reduced-row-echelon-form matrix))
(newline)

Output:
((1 0 0 -8) (0 1 0 1) (0 0 1 -2))


## Seed7

const type: matrix is array array float;

const proc: toReducedRowEchelonForm (inout matrix: mat) is func
local
var integer: numRows is 0;
var integer: numColumns is 0;
var integer: row is 0;
var integer: column is 0;
var integer: pivot is 0;
var float: factor is 0.0;
begin
numRows := length(mat);
numColumns := length(mat[1]);
for row range numRows downto 1 do
column := 1;
while column <= numColumns and mat[row][column] = 0.0 do
incr(column);
end while;
if column > numColumns then
# Empty rows are moved to the bottom
mat := mat[.. pred(row)] & mat[succ(row) ..] & [] (mat[row]);
decr(numRows);
end if;
end for;
for pivot range 1 to numRows do
if mat[pivot][pivot] = 0.0 then
# Find a row were the pivot column is not zero
row := 1;
while row <= numRows and mat[row][pivot] = 0.0 do
incr(row);
end while;
# Add row were the pivot column is not zero
for column range 1 to numColumns do
mat[pivot][column] +:= mat[row][column];
end for;
end if;
if mat[pivot][pivot] <> 1.0 then
# Make sure that the pivot element is 1.0
factor := 1.0 / mat[pivot][pivot];
for column range pivot to numColumns do
mat[pivot][column] := mat[pivot][column] * factor;
end for;
end if;
for row range 1 to numRows do
if row <> pivot and mat[row][pivot] <> 0.0 then
# Make sure that in all other rows the pivot column contains zero
factor := -mat[row][pivot];
for column range pivot to numColumns do
mat[row][column] +:= mat[pivot][column] * factor;
end for;
end if;
end for;
end for;
end func;

Original source: [1]

## Sidef

Translation of: Raku
func rref (M) {
var (j, rows, cols) = (0, M.len, M[0].len)

for r in (^rows) {
j < cols || return M

var i = r
while (!M[i][j]) {
++i == rows || next
i = r
++j == cols && return M
}

M[i, r] = M[r, i] if (r != i)
M[r] = (M[r] »/» M[r][j])

for n in (^rows) {
next if (n == r)
M[n] = (M[n] »-« (M[r] »*» M[n][j]))
}
++j
}

return M
}

func say_it (message, array) {
say "\n#{message}";
array.each { |row|
say row.map { |n| " %5s" % n.as_rat }.join
}
}

var M = [
[ # base test case
[  1,  2,  -1,  -4 ],
[  2,  3,  -1, -11 ],
[ -2,  0,  -3,  22 ],
],
[ # mix of number styles
[  3,   0,  -3,    1 ],
[ .5, 3/2,  -3,   -2 ],
[ .2, 4/5,  -1.6, .3 ],
],
[ # degenerate case
[ 1,  2,  3,  4,  3,  1],
[ 2,  4,  6,  2,  6,  2],
[ 3,  6, 18,  9,  9, -6],
[ 4,  8, 12, 10, 12,  4],
[ 5, 10, 24, 11, 15, -4],
],
];

M.each { |matrix|
say_it('Original Matrix', matrix);
say_it('Reduced Row Echelon Form Matrix', rref(matrix));
say '';
}

Output:
Original Matrix
1     2    -1    -4
2     3    -1   -11
-2     0    -3    22

Reduced Row Echelon Form Matrix
1     0     0    -8
0     1     0     1
0     0     1    -2

Original Matrix
3     0    -3     1
1/2   3/2    -3    -2
1/5   4/5  -8/5  3/10

Reduced Row Echelon Form Matrix
1     0     0 -41/2
0     1     0 -217/6
0     0     1 -125/6

Original Matrix
1     2     3     4     3     1
2     4     6     2     6     2
3     6    18     9     9    -6
4     8    12    10    12     4
5    10    24    11    15    -4

Reduced Row Echelon Form Matrix
1     2     0     0     3     4
0     0     1     0     0    -1
0     0     0     1     0     0
0     0     0     0     0     0
0     0     0     0     0     0


## Swift

        var lead = 0
for r in 0..<rows {
if (cols <= lead) { break }
var i = r
i += 1
if (i == rows) {
i = r
break
}
}
}
for j in 0..<cols {
let temp = m[r][j]
m[r][j] = m[i][j]
m[i][j] = temp
}
if (div != 0) {
for j in 0..<cols {
m[r][j] /= div
}
}
for j in 0..<rows {
if (j != r) {
for k in 0..<cols {
m[j][k] -= (sub * m[r][k])
}
}
}
}


## Tcl

Using utility procs defined at Matrix Transpose#Tcl

package require Tcl 8.5
namespace path {::tcl::mathop ::tcl::mathfunc}

proc toRREF {m} {
lassign [size $m] rows cols for {set r 0} {$r < $rows} {incr r} { if {$cols <= $lead} { break } set i$r
while {[lindex $m$i $lead] == 0} { incr i if {$rows == $i} { set i$r
if {$cols ==$lead} {
# Tcl can't break out of nested loops
return $m } } } # swap rows i and r foreach idx [list$i $r] row [list [lindex$m $r] [lindex$m $i]] { lset m$idx $row } # divide row r by m(r,lead) set val [lindex$m $r$lead]
for {set j 0} {$j <$cols} {incr j} {
lset m $r$j [/ [double [lindex $m$r $j]]$val]
}

for {set i 0} {$i <$rows} {incr i} {
if {$i !=$r} {
# subtract m(i,lead) multiplied by row r from row i
set val [lindex $m$i $lead] for {set j 0} {$j < $cols} {incr j} { lset m$i $j [- [lindex$m $i$j] [* $val [lindex$m $r$j]]]
}
}
}
}
return $m } set m {{1 2 -1 -4} {2 3 -1 -11} {-2 0 -3 22}} print_matrix$m
print_matrix [toRREF $m]  Output:  1 2 -1 -4 2 3 -1 -11 -2 0 -3 22 1.0 0.0 0.0 -8.0 -0.0 1.0 0.0 1.0 -0.0 -0.0 1.0 -2.0  ## TI-83 BASIC Builtin function: rref() rref([[1,2,-1,-4][2,3,-1,-11][-2,0,-3,22]]) Output:  [[1 0 0 -8] [0 1 0 1] [0 0 1 -2]]  ## TI-89 BASIC rref([1,2,–1,–4; 2,3,–1,–11; –2,0,–3,22]) Output (in prettyprint mode): ${\displaystyle {\begin{bmatrix}1&0&0&-8\\0&1&0&1\\0&0&1&-2\end{bmatrix}}}$ Matrices can also be stored in variables, and entered interactively using the Data/Matrix Editor. ## Ursala The most convenient representation for a matrix in Ursala is as a list of lists. Several auxiliary functions are defined to make this task more manageable. The pivot function reorders the rows to position the first column entry with maximum magnitude in the first row. The descending function is a second order function abstracting the pattern of recursion down the major diagonal of a matrix. The reflect function allows the code for the first phase in the reduction to be reused during the upward traversal by appropriately permuting the rows and columns. The row_reduce function adds a multiple of the top row to each subsequent row so as to cancel the first column. These are all combined in the main rref function. #import std #import flo pivot = -<x fleq+ abs~~bh descending = ~&a^&+ ^|ahPathS2fattS2RpC/~& reflect = ~&lxPrTSx+ *iiD ~&l-~brS+ zipp0 row_reduce = ^C/vid*hhiD *htD minus^*p/~&r times^*D/vid@bh ~&l rref = reflect+ (descending row_reduce)+ reflect+ descending row_reduce+ pivot #show+ test = printf/*=*'%8.4f' rref < <1.,2.,-1.,-4.>, <2.,3.,-1.,-11.>, <-2.,0.,-3.,22.>> Output:  1.0000 0.0000 0.0000 -8.0000 0.0000 1.0000 0.0000 1.0000 0.0000 0.0000 1.0000 -2.0000 An alternative and more efficient solution is to use the msolve library function as shown, which interfaces with the lapack library if available. This solution is applicable only if the input is a non-singular augmented square matrix. #import lin rref = @ySzSX msolve; ^plrNCTS\~& ~&iiDlSzyCK9+ :/1.+ 0.!*t ## VBA Translation of: Phix Private Function ToReducedRowEchelonForm(M As Variant) As Variant Dim lead As Integer: lead = 0 Dim rowCount As Integer: rowCount = UBound(M) Dim columnCount As Integer: columnCount = UBound(M(0)) Dim i As Integer For r = 0 To rowCount If lead >= columnCount Then Exit For End If i = r Do While M(i)(lead) = 0 i = i + 1 If i = rowCount Then i = r lead = lead + 1 If lead = columnCount Then Exit For End If End If Loop Dim tmp As Variant tmp = M(r) M(r) = M(i) M(i) = tmp If M(r)(lead) <> 0 Then div = M(r)(lead) For t = LBound(M(r)) To UBound(M(r)) M(r)(t) = M(r)(t) / div Next t End If For j = 0 To rowCount If j <> r Then subt = M(j)(lead) For t = LBound(M(j)) To UBound(M(j)) M(j)(t) = M(j)(t) - subt * M(r)(t) Next t End If Next j lead = lead + 1 Next r ToReducedRowEchelonForm = M End Function Public Sub main() r = ToReducedRowEchelonForm(Array( _ Array(1, 2, -1, -4), _ Array(2, 3, -1, -11), _ Array(-2, 0, -3, 22))) For i = LBound(r) To UBound(r) Debug.Print Join(r(i), vbTab) Next i End Sub Output: 1 0 0 -8 0 1 0 1 0 0 1 -2 ## Visual FoxPro Translation of Fortran. CLOSE DATABASES ALL LOCAL lnRows As Integer, lnCols As Integer, lcSafety As String LOCAL ARRAY matrix[1] lcSafety = SET("Safety") SET SAFETY OFF CLEAR CREATE CURSOR results (c1 B(6), c2 B(6), c3 B(6), c4 B(6)) CREATE CURSOR curs1(c1 I, c2 I, c3 I, c4 I) INSERT INTO curs1 VALUES (1,2,-1,-4) INSERT INTO curs1 VALUES (2,3,-1,-11) INSERT INTO curs1 VALUES (-2,0,-3,22) lnRows = RECCOUNT() && 3 lnCols = FCOUNT() && 4 SELECT * FROM curs1 INTO ARRAY matrix IF RREF(@matrix, lnRows, lnCols) SELECT results APPEND FROM ARRAY matrix BROWSE NORMAL IN SCREEN ENDIF SET SAFETY &lcSafety FUNCTION RREF(mat, tnRows As Integer, tnCols As Integer) As Boolean LOCAL lnPivot As Integer, i As Integer, r As Integer, j As Integer, ; p As Double. llResult As Boolean, llExit As Boolean llResult = .T. llExit = .F. lnPivot = 1 FOR r = 1 TO tnRows IF lnPivot > tnCols EXIT ENDIF i = r DO WHILE mat[i,lnPivot] = 0 i = i + 1 IF i = tnRows i = r lnPivot = lnPivot + 1 IF lnPivot > tnCols llExit = .T. EXIT ENDIF ENDIF ENDDO IF llExit EXIT ENDIF ASwapRows(@mat, i, r) p = mat[r,lnPivot] IF p # 0 FOR j = 1 TO tnCols mat[r,j] = mat[r,j]/p ENDFOR ELSE ? "Divison by zero." llResult = .F. EXIT ENDIF FOR i = 1 TO tnRows IF i # r p = mat[i,lnPivot] FOR j = 1 TO tnCols mat[i,j] = mat[i,j] - mat[r,j]*p ENDFOR ENDIF ENDFOR lnPivot = lnPivot + 1 ENDFOR RETURN llResult ENDFUNC PROCEDURE ASwapRows(arr, tnRow1 As Integer, tnRow2 As Integer) *!* Interchange rows tnRow1 and tnRow2 of array arr. LOCAL n As Integer n = ALEN(arr,2) LOCAL ARRAY tmp[1,n] STORE 0 TO tmp ACPY2(@arr, @tmp, tnRow1, 1) ACPY2(@arr, @arr, tnRow2, tnRow1) ACPY2(@tmp, @arr, 1, tnRow2) ENDPROC PROCEDURE ACPY2(m1, m2, tnSrcRow As Integer, tnDestRow As Integer) *!* Copy m1[tnSrcRow,*] to m2[tnDestRow,*] *!* m1 and m2 must have the same number of columns. LOCAL n As Integer, e1 As Integer, e2 As Integer n = ALEN(m1,2) e1 = AELEMENT(m1,tnSrcRow,1) e2 = AELEMENT(m2,tnDestRow,1) ACOPY(m1, m2, e1, n, e2) ENDPROC  Output:  C1 C2 C3 C4 1.000000 0.000000 0.000000 -8.000000 0.000000 1.000000 0.000000 1.000000 0.000000 0.000000 1.000000 -2.000000  ## Wren Library: Wren-fmt Library: Wren-matrix The above module has a method for this built in as it's needed to implement matrix inversion using the Gauss-Jordan method. However, as in the example here, it's not just restricted to square matrices. import "./matrix" for Matrix import "./fmt" for Fmt var m = Matrix.new([ [ 1, 2, -1, -4], [ 2, 3, -1, -11], [-2, 0, -3, 22] ]) System.print("Original:\n") Fmt.mprint(m, 3, 0) System.print("\nRREF:\n") m.toReducedRowEchelonForm Fmt.mprint(m, 3, 0)  Output: Original: | 1 2 -1 -4| | 2 3 -1 -11| | -2 0 -3 22| RREF: | 1 0 0 -8| | 0 1 0 1| | 0 0 1 -2|  ## XPL0 proc ReducedRowEchelonForm(M, Rows, Cols); \Replace M with its reduced row echelon form real M; int Rows, Cols; int Lead, R, C, I; real RLead, ILead, T; [Lead:= 0; for R:= 0 to Rows-1 do [if Lead >= Cols then return; I:= R; while M(I, Lead) = 0. do [I:= I+1; if I = Rows-1 then [I:= R; Lead:= Lead+1; if Lead = Cols-1 then return; ]; ]; \Swap rows I and R T:= M(I); M(I):= M(R); M(R):= T; if M(R, Lead) # 0. then \Divide row R by M[R, Lead] [RLead:= M(R, Lead); for C:= 0 to Cols-1 do M(R, C):= M(R, C) / RLead; ]; for I:= 0 to Rows-1 do [if I # R then \Subtract M[I, Lead] multiplied by row R from row I [ILead:= M(I, Lead); for C:= 0 to Cols-1 do M(I, C):= M(I, C) - ILead * M(R, C); ]; ]; Lead:= Lead+1; ]; ]; real M; int R, C; [M:= [ [ 1., 2., -1., -4.], [ 2., 3., -1.,-11.], [-2., 0., -3., 22.] ]; ReducedRowEchelonForm(M, 3, 4); Format(4,1); for R:= 0 to 3-1 do [for C:= 0 to 4-1 do RlOut(0, M(R,C)); CrLf(0); ]; ] Output:  1.0 0.0 0.0 -8.0 0.0 1.0 0.0 1.0 0.0 0.0 1.0 -2.0  ## Yabasic // Rosetta Code problem: https://rosettacode.org/wiki/Reduced_row_echelon_form // by Jjuanhdez, 06/2022 dim matrix (3, 4) matrix(1, 1) = 1 : matrix(1, 2) = 2 : matrix(1, 3) = -1 : matrix(1, 4) = -4 matrix(2, 1) = 2 : matrix(2, 2) = 3 : matrix(2, 3) = -1 : matrix(2, 4) = -11 matrix(3, 1) = -2 : matrix(3, 2) = 0 : matrix(3, 3) = -3 : matrix(3, 4) = 22 RREF (matrix()) for row = 1 to 3 for col = 1 to 4 if matrix(row, col) = 0 then print "0", chr$(9);
else
print matrix(row, col), chr\$(9);
end if
next
print
next
end

sub RREF(x())
local nrows, ncols, lead, r, i, j, n
nrows = arraysize(matrix(), 1)  //3
ncols = arraysize(matrix(), 2)  //4
for r = 1 to nrows
i = r
i = i + 1
if i = nrows then
i = r
if lead = ncols break 2
end if
wend
for j = 1 to ncols
temp = matrix(i, j)
matrix(i, j) = matrix(r, j)
matrix(r, j) = temp
next
if n <> 0 then
for j = 1 to ncols
matrix(r, j) = matrix(r, j) / n
next
end if
for i = 1 to nrows
if i <> r then
for j = 1 to ncols
matrix(i, j) = matrix(i, j) - matrix(r, j) * n
next
end if
next
next
end sub

## zkl

The "best" way is to use the GNU Scientific Library:

var [const] GSL=Import("zklGSL");	// libGSL (GNU Scientific Library)
fcn toReducedRowEchelonForm(M){  // in place
foreach r in (rows){
i:=r;
while(M[i,lead]==0){  // not a great check to use with real numbers
i+=1;
if(i==rows){
}
}
M.swapRows(i,r);
foreach i in (rows){ if(i!=r) M[i]-=M[r]*M[i,lead] }
}
M
}
A:=GSL.Matrix(3,4).set( 1, 2, -1,  -4,
2, 3, -1, -11,
-2, 0, -3,  22);
toReducedRowEchelonForm(A).format(5,1).println();
Output:
  1.0,  0.0,  0.0, -8.0
0.0,  1.0,  0.0,  1.0
0.0,  0.0,  1.0, -2.0


Or, using lists of lists and direct implementation of the pseudo-code given, lots of generating new rows rather than modifying the rows themselves.

fcn toReducedRowEchelonForm(m){ // m is modified, the rows are not
foreach r in (rowCount){
i:=r;
i+=1;
if(rowCount==i){
}
}//while
m.swap(i,r); // Swap rows i and r
foreach i in (rowCount){
if(i!=r) // Subtract M[i, lead] multiplied by row r from row i
}//foreach
}//foreach
m
}
m:=List( T( 1, 2, -1, -4,),  // T is read only list
T( 2, 3, -1, -11,),
T(-2, 0, -3,  22,));
printM(m);
println("-->");
printM(toReducedRowEchelonForm(m));

fcn printM(m){ m.pump(Console.println,rowFmt) }
fcn rowFmt(row){ ("%4d "*row.len()).fmt(row.xplode()) }
Output:
   1    2   -1   -4
2    3   -1  -11
-2    0   -3   22
-->
1    0    0   -8
0    1    0    1
0    0    1   -2