Reduced row echelon form

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


Task

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
    lead := 0
    rowCount := the number of rows in M
    columnCount := the number of columns in M
    for 0 ≤ r < rowCount do
        if columnCountlead then
            stop
        end if
        i = r
        while M[i, lead] = 0 do
            i = i + 1
            if rowCount = i then
                i = r
                lead = lead + 1
                if columnCount = lead then
                    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 ir do
                Subtract M[i, lead] multiplied by row r from row i
            end if
        end for
        lead = lead + 1
    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[edit]

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

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[edit]

Translation of: BBC BASIC
*        reduced row echelon form  27/08/2015
RREF     CSECT
         USING  RREF,R12
         LR     R12,R15
         LA     R10,1              lead=1
         LA     R7,1
LOOPR    CH     R7,NROWS           do r=1 to nrows
         BH     ELOOPR
         CH     R10,NCOLS          if lead>=ncols
         BNL    ELOOPR
         LR     R8,R7              i=r
WHILE    LR     R1,R8              do while m(i,lead)=0
         BCTR   R1,0
         MH     R1,NCOLS
         LR     R6,R10             lead
         BCTR   R6,0
         AR     R1,R6
         SLA    R1,2
         L      R6,M(R1)           m(i,lead)
         LTR    R6,R6
         BNZ    EWHILE             m(i,lead)<>0           
         LA     R8,1(R8)           i=i+1
         CH     R8,NROWS           if i=nrows
         BNE    EIF
         LR     R8,R7              i=r
         LA     R10,1(R10)         lead=lead+1
         CH     R10,NCOLS          if lead=ncols
         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
         LR     R6,R10             lead
         BCTR   R6,0
         AR     R1,R6
         SLA    R1,2
         L      R11,M(R1)          n=m(r,lead)
         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
         LR     R6,R10             lead
         BCTR   R6,0
         AR     R1,R6
         SLA    R1,2
         L      R11,M(R1)          n=m(i,lead)
         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
ELOOPI3  LA     R10,1(R10)         lead=lead+1
         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[edit]

_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++) {
      if(columns <= lead)
         break;
      i = r;

      while(_m[i][lead] == 0) {
         i++;

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

            if(columns == lead)
               return this;
         }
      }
      rowSwitch(i, r);
      var val:Number = _m[r][lead];

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

      for(i = 0; i < rows; i++) {
         if(i == r)
            continue;
         val = _m[i][lead];

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

Ada[edit]

matrices.ads:

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;

matrices.adb:

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;
               Lead := Lead + 1;
               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,
            Divisor => Result (Row, Lead));
         for Other_Row in Result'Range (1) loop
            if Other_Row /= Row then
               Subtract_Rows
                 (From       => Result,
                  Subtrahend => Row,
                  Minuend    => Other_Row,
                  Factor     => Result (Other_Row, Lead));
            end if;
         end loop;
         Lead := Lead + 1;
      end loop Rows;
      return Result;
   end Reduced_Row_Echelon_form;
end Matrices;

Example use: main.adb:

with Matrices;
with Ada.Text_IO;
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);
            Ada.Text_IO.Put (' ');
         end loop;
         Ada.Text_IO.New_Line;
      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);
   Ada.Text_IO.Put_Line ("reduced to:");
   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[edit]

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

    lead = r = 0;
    while (r < rows && lead < columns) {
        i = r;
        while (!l.q_list(i)[lead]) {
            i += 1;
            if (i == rows) {
                i = r;
                lead += 1;
                if (lead == columns) {
                    break;
                }
            }
        }
        if (lead == columns) {
            break;
        }

        u = l[i];

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

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

        lead += 1;

        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[edit]

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

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
        integer lead;
        lead := 0;
        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;
                    lead := lead + 1;
                    if cMax = lead then goto done
                end if_rowCount_eq_i
            end while_M_i_lead_eq_0 ;
            % 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] %
                real rLead;
                rLead := M( r, lead );
                for c := 0 until cMax do M( r, c ) := M( r, c ) / rLead
            end if_M_r_lead_ne_0 ;
            for i := 0 until rMax do begin
                if i not = r then begin
                    % Subtract M[i, lead] multiplied by row r from row i %
                    real iLead;
                    iLead := M( i, lead );
                    for c := 0 until cMax do M( i, c ) := M( i, c ) - ( iLead * M( r, c ) )
                end if_i_ne_r
            end for_i ;
            lead := lead + 1
        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

AutoHotkey[edit]

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[edit]

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]

BASIC256[edit]

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[edit]

      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[edit]

#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#[edit]

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++[edit]

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[edit]

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[edit]

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

Euphoria[edit]

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[edit]

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[edit]

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[edit]

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[edit]

2D representation[edit]

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[edit]

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[edit]

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]

Haskell[edit]

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[edit]

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)
    lead := 1
    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]
            }
        lead +:= 1
        }
    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[edit]

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
pivot=: dyad define
  '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[edit]

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[edit]

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[edit]

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("  ");

The Task

# 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 )
        | .lead += 1
        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[edit]

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[edit]

// version 1.1.51

typealias Matrix = Array<DoubleArray>

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

        while (this[i][lead] == 0.0) {
            i++
            if (rowCount == i) {
                i = r
                lead++
                if (colCount == lead) return
            }
        }

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

        if (this[r][lead] != 0.0) {
           val div = this[r][lead]
           for (j in 0 until colCount) this[r][j] /= div
        }

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

        lead++
    }
}

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

rref([1, 2, -1, -4; 2, 3, -1, -11; -2, 0, -3, 22])

Maxima[edit]

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[edit]

Using rationals[edit]

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:
      line.addSep(" ", 0)
      line.add val.toFloat.formatFloat(ffDecimal, 2).align(7)
    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[edit]

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.

  var lead = 0

  for r in 0..<mat.M:

    if lead >= mat.N: return

    var i = r
    while mat[i][lead] == 0:
      inc i
      if i == mat.M:
        i = r
        inc lead
        if lead == mat.N: return
    swap mat[i], mat[r]

    let d = mat[r][lead]
    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:
        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:
      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[edit]

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[edit]

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[edit]

A = [ 1, 2, -1, -4; 2, 3, -1, -11; -2, 0, -3, 22];
refA = rref(A);
disp(refA);

PARI/GP[edit]

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[edit]

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[edit]

Translation of: Euphoria
with javascript_semantics
function ToReducedRowEchelonForm(sequence M)
integer lead = 1,
        rowCount = length(M),
        columnCount = length(M[1]),
        i
    for r=1 to rowCount do
        if lead>=columnCount then exit end if
        i = r
        while M[i][lead]=0 do
            i += 1
            if i=rowCount then
                i = r
                lead += 1
                if lead=columnCount then exit end if
            end if
        end while
        object mr = sq_div(M[i],M[i][lead])
        M[i] = M[r]
        M[r] = mr
        for j=1 to rowCount do
            if j!=r then
                M[j] = sq_sub(M[j],sq_mul(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}}

PHP[edit]

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[edit]

(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[edit]

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


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[edit]

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[edit]

#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[edit]

(formerly Perl 6)

Following pseudocode[edit]

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[edit]

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[edit]

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[edit]

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[edit]

# 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
lead = 1
for r = 1 to nrows
      if lead >= ncols
         exit
      ok
      i = r
      while m[i][lead] = 0
                i = i + 1
                if i = nrows
                   i = r
                   lead = lead + 1
                   if lead = ncols
                      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
      n = m[r][lead]
      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 
              n = m[i][lead]
              for j = 1 to ncols
                   m[i][j] = m[i][j] - m[r][j] * n
              next
           ok
      next
     lead = lead + 1
next

Output:

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

Ruby[edit]

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

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[edit]

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[edit]

Works with: Scheme version RRS
(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[edit]

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[edit]

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[edit]

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

Tcl[edit]

Using utility procs defined at Matrix Transpose#Tcl

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

proc toRREF {m} {
    set lead 0
    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 $r
                incr lead
                if {$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]]]
                }
            }
        }
        incr lead
    }
    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[edit]

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[edit]

rref([1,2,–1,–4; 2,3,–1,–11; –2,0,–3,22])

Output (in prettyprint mode):

Matrices can also be stored in variables, and entered interactively using the Data/Matrix Editor.

Ursala[edit]

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[edit]

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[edit]

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[edit]

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|

Yabasic[edit]

// 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
    lead = 1
    for r = 1 to nrows
        if lead >= ncols  break
        i = r
        while matrix(i, lead) = 0
            i = i + 1
            if i = nrows then
                i = r
                lead = lead + 1
                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
        n = matrix(r, lead)
        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 
                n = matrix(i, lead)
                for j = 1 to ncols
                    matrix(i, j) = matrix(i, j) - matrix(r, j) * n
                next
            end if
        next
        lead = lead + 1
    next
end sub

zkl[edit]

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

var [const] GSL=Import("zklGSL");	// libGSL (GNU Scientific Library)
fcn toReducedRowEchelonForm(M){  // in place
   lead,rows,columns := 0,M.rows,M.cols;
   foreach r in (rows){
      if (columns<=lead) return(M);
      i:=r;
      while(M[i,lead]==0){  // not a great check to use with real numbers
	 i+=1;
	 if(i==rows){
	    i=r; lead+=1;
	    if(lead==columns) return(M);
	 }
      }
      M.swapRows(i,r);
      if(x:=M[r,lead]) M[r]/=x;
      foreach i in (rows){ if(i!=r) M[i]-=M[r]*M[i,lead] }
      lead+=1;
   }
   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
   lead,rowCount,columnCount := 0,m.len(),m[1].len();
   foreach r in (rowCount){
      if(columnCount<=lead) break;
      i:=r;
      while(m[i][lead]==0){
	 i+=1;
	 if(rowCount==i){
	    i=r; lead+=1;
	    if(columnCount==lead) break;
	 }
      }//while
      m.swap(i,r); // Swap rows i and r
      if(n:=m[r][lead]) m[r]=m[r].apply('/(n)); //divide row r by M[r,lead]
      foreach i in (rowCount){
         if(i!=r) // Subtract M[i, lead] multiplied by row r from row i
	    m[i]=m[i].zipWith('-,m[r].apply('*(m[i][lead])))
      }//foreach
      lead+=1;
   }//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 

References[edit]