Reduced row echelon form
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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 columnCount ≤ lead 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 i ≠ r 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
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
* 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
_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
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
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
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
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
ATS
This program was made by modifying Gauss-Jordan_matrix_inversion#ATS. (The latter program is equivalent to finding the RREF of a particular matrix.)
%{^
#include <math.h>
#include <float.h>
%}
#include "share/atspre_staload.hats"
macdef NAN = g0f2f ($extval (float, "NAN"))
macdef Zero = g0i2f 0
macdef One = g0i2f 1
macdef Two = g0i2f 2
(* The following is often done by a single machine instruction. *)
macdef multiply_and_add (x, y, z) = (,(x) * ,(y)) + ,(z)
(*------------------------------------------------------------------*)
(* A "little matrix library" *)
typedef Matrix_Index_Map (m1 : int, n1 : int, m0 : int, n0 : int) =
{i1, j1 : pos | i1 <= m1; j1 <= n1}
(int i1, int j1) -<cloref0>
[i0, j0 : pos | i0 <= m0; j0 <= n0]
@(int i0, int j0)
datatype Real_Matrix (tk : tkind,
m1 : int, n1 : int,
m0 : int, n0 : int) =
| Real_Matrix of (matrixref (g0float tk, m0, n0),
int m1, int n1, int m0, int n0,
Matrix_Index_Map (m1, n1, m0, n0))
typedef Real_Matrix (tk : tkind, m1 : int, n1 : int) =
[m0, n0 : pos] Real_Matrix (tk, m1, n1, m0, n0)
typedef Real_Vector (tk : tkind, m1 : int, n1 : int) =
[m1 == 1 || n1 == 1] Real_Matrix (tk, m1, n1)
typedef Real_Row (tk : tkind, n1 : int) = Real_Vector (tk, 1, n1)
typedef Real_Column (tk : tkind, m1 : int) = Real_Vector (tk, m1, 1)
extern fn {tk : tkind}
Real_Matrix_make_elt :
{m0, n0 : pos}
(int m0, int n0, g0float tk) -< !wrt >
Real_Matrix (tk, m0, n0, m0, n0)
extern fn {tk : tkind}
Real_Matrix_copy :
{m1, n1 : pos}
Real_Matrix (tk, m1, n1) -< !refwrt > Real_Matrix (tk, m1, n1)
extern fn {tk : tkind}
Real_Matrix_copy_to :
{m1, n1 : pos}
(Real_Matrix (tk, m1, n1), (* destination *)
Real_Matrix (tk, m1, n1)) -< !refwrt >
void
extern fn {tk : tkind}
Real_Matrix_fill_with_elt :
{m1, n1 : pos}
(Real_Matrix (tk, m1, n1), g0float tk) -< !refwrt > void
extern fn {}
Real_Matrix_dimension :
{tk : tkind}
{m1, n1 : pos}
Real_Matrix (tk, m1, n1) -<> @(int m1, int n1)
extern fn {tk : tkind}
Real_Matrix_get_at :
{m1, n1 : pos}
{i1, j1 : pos | i1 <= m1; j1 <= n1}
(Real_Matrix (tk, m1, n1), int i1, int j1) -< !ref > g0float tk
extern fn {tk : tkind}
Real_Matrix_set_at :
{m1, n1 : pos}
{i1, j1 : pos | i1 <= m1; j1 <= n1}
(Real_Matrix (tk, m1, n1), int i1, int j1, g0float tk) -< !refwrt >
void
extern fn {}
Real_Matrix_apply_index_map :
{tk : tkind}
{m1, n1 : pos}
{m0, n0 : pos}
(Real_Matrix (tk, m0, n0), int m1, int n1,
Matrix_Index_Map (m1, n1, m0, n0)) -<>
Real_Matrix (tk, m1, n1)
extern fn {}
Real_Matrix_transpose :
(* This is transposed INDEXING. It does NOT copy the data. *)
{tk : tkind}
{m1, n1 : pos}
{m0, n0 : pos}
Real_Matrix (tk, m1, n1, m0, n0) -<>
Real_Matrix (tk, n1, m1, m0, n0)
extern fn {}
Real_Matrix_block :
(* This is block (submatrix) INDEXING. It does NOT copy the data. *)
{tk : tkind}
{p0, p1 : pos | p0 <= p1}
{q0, q1 : pos | q0 <= q1}
{m1, n1 : pos | p1 <= m1; q1 <= n1}
{m0, n0 : pos}
(Real_Matrix (tk, m1, n1, m0, n0),
int p0, int p1, int q0, int q1) -<>
Real_Matrix (tk, p1 - p0 + 1, q1 - q0 + 1, m0, n0)
extern fn {tk : tkind}
Real_Matrix_unit_matrix :
{m : pos}
int m -< !refwrt > Real_Matrix (tk, m, m)
extern fn {tk : tkind}
Real_Matrix_unit_matrix_to :
{m : pos}
Real_Matrix (tk, m, m) -< !refwrt > void
extern fn {tk : tkind}
Real_Matrix_matrix_sum :
{m, n : pos}
(Real_Matrix (tk, m, n), Real_Matrix (tk, m, n)) -< !refwrt >
Real_Matrix (tk, m, n)
extern fn {tk : tkind}
Real_Matrix_matrix_sum_to :
{m, n : pos}
(Real_Matrix (tk, m, n), (* destination*)
Real_Matrix (tk, m, n),
Real_Matrix (tk, m, n)) -< !refwrt >
void
extern fn {tk : tkind}
Real_Matrix_matrix_difference :
{m, n : pos}
(Real_Matrix (tk, m, n), Real_Matrix (tk, m, n)) -< !refwrt >
Real_Matrix (tk, m, n)
extern fn {tk : tkind}
Real_Matrix_matrix_difference_to :
{m, n : pos}
(Real_Matrix (tk, m, n), (* destination*)
Real_Matrix (tk, m, n),
Real_Matrix (tk, m, n)) -< !refwrt >
void
extern fn {tk : tkind}
Real_Matrix_matrix_product :
{m, n, p : pos}
(Real_Matrix (tk, m, n), Real_Matrix (tk, n, p)) -< !refwrt >
Real_Matrix (tk, m, p)
extern fn {tk : tkind}
Real_Matrix_matrix_product_to :
{m, n, p : pos}
(Real_Matrix (tk, m, p), (* destination*)
Real_Matrix (tk, m, n),
Real_Matrix (tk, n, p)) -< !refwrt >
void
extern fn {tk : tkind}
Real_Matrix_scalar_product :
{m, n : pos}
(Real_Matrix (tk, m, n), g0float tk) -< !refwrt >
Real_Matrix (tk, m, n)
extern fn {tk : tkind}
Real_Matrix_scalar_product_2 :
{m, n : pos}
(g0float tk, Real_Matrix (tk, m, n)) -< !refwrt >
Real_Matrix (tk, m, n)
extern fn {tk : tkind}
Real_Matrix_scalar_product_to :
{m, n : pos}
(Real_Matrix (tk, m, n), (* destination*)
Real_Matrix (tk, m, n), g0float tk) -< !refwrt > void
extern fn {tk : tkind} (* Useful for debugging. *)
Real_Matrix_fprint :
{m, n : pos}
(FILEref, Real_Matrix (tk, m, n)) -<1> void
overload copy with Real_Matrix_copy
overload copy_to with Real_Matrix_copy_to
overload fill_with_elt with Real_Matrix_fill_with_elt
overload dimension with Real_Matrix_dimension
overload [] with Real_Matrix_get_at
overload [] with Real_Matrix_set_at
overload apply_index_map with Real_Matrix_apply_index_map
overload transpose with Real_Matrix_transpose
overload block with Real_Matrix_block
overload unit_matrix with Real_Matrix_unit_matrix
overload unit_matrix_to with Real_Matrix_unit_matrix_to
overload matrix_sum with Real_Matrix_matrix_sum
overload matrix_sum_to with Real_Matrix_matrix_sum_to
overload matrix_difference with Real_Matrix_matrix_difference
overload matrix_difference_to with Real_Matrix_matrix_difference_to
overload matrix_product with Real_Matrix_matrix_product
overload matrix_product_to with Real_Matrix_matrix_product_to
overload scalar_product with Real_Matrix_scalar_product
overload scalar_product with Real_Matrix_scalar_product_2
overload scalar_product_to with Real_Matrix_scalar_product_to
overload + with matrix_sum
overload - with matrix_difference
overload * with matrix_product
overload * with scalar_product
(*------------------------------------------------------------------*)
(* Implementation of the "little matrix library" *)
implement {tk}
Real_Matrix_make_elt (m0, n0, elt) =
Real_Matrix (matrixref_make_elt<g0float tk> (i2sz m0, i2sz n0, elt),
m0, n0, m0, n0, lam (i1, j1) => @(i1, j1))
implement {}
Real_Matrix_dimension A =
case+ A of Real_Matrix (_, m1, n1, _, _, _) => @(m1, n1)
implement {tk}
Real_Matrix_get_at (A, i1, j1) =
let
val+ Real_Matrix (storage, _, _, _, n0, index_map) = A
val @(i0, j0) = index_map (i1, j1)
in
matrixref_get_at<g0float tk> (storage, pred i0, n0, pred j0)
end
implement {tk}
Real_Matrix_set_at (A, i1, j1, x) =
let
val+ Real_Matrix (storage, _, _, _, n0, index_map) = A
val @(i0, j0) = index_map (i1, j1)
in
matrixref_set_at<g0float tk> (storage, pred i0, n0, pred j0, x)
end
implement {}
Real_Matrix_apply_index_map (A, m1, n1, index_map) =
(* This is not the most efficient way to acquire new indexing, but
it will work. It requires three closures, instead of the two
needed by our implementations of "transpose" and "block". *)
let
val+ Real_Matrix (storage, m1a, n1a, m0, n0, index_map_1a) = A
in
Real_Matrix (storage, m1, n1, m0, n0,
lam (i1, j1) =>
index_map_1a (i1a, j1a) where
{ val @(i1a, j1a) = index_map (i1, j1) })
end
implement {}
Real_Matrix_transpose A =
let
val+ Real_Matrix (storage, m1, n1, m0, n0, index_map) = A
in
Real_Matrix (storage, n1, m1, m0, n0,
lam (i1, j1) => index_map (j1, i1))
end
implement {}
Real_Matrix_block (A, p0, p1, q0, q1) =
let
val+ Real_Matrix (storage, m1, n1, m0, n0, index_map) = A
in
Real_Matrix (storage, succ (p1 - p0), succ (q1 - q0), m0, n0,
lam (i1, j1) =>
index_map (p0 + pred i1, q0 + pred j1))
end
implement {tk}
Real_Matrix_copy A =
let
val @(m1, n1) = dimension A
val C = Real_Matrix_make_elt<tk> (m1, n1, A[1, 1])
val () = copy_to<tk> (C, A)
in
C
end
implement {tk}
Real_Matrix_copy_to (Dst, Src) =
let
val @(m1, n1) = dimension Src
prval [m1 : int] EQINT () = eqint_make_gint m1
prval [n1 : int] EQINT () = eqint_make_gint n1
var i : intGte 1
in
for* {i : pos | i <= m1 + 1} .<(m1 + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m1; i := succ i)
let
var j : intGte 1
in
for* {j : pos | j <= n1 + 1} .<(n1 + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n1; j := succ j)
Dst[i, j] := Src[i, j]
end
end
implement {tk}
Real_Matrix_fill_with_elt (A, elt) =
let
val @(m1, n1) = dimension A
prval [m1 : int] EQINT () = eqint_make_gint m1
prval [n1 : int] EQINT () = eqint_make_gint n1
var i : intGte 1
in
for* {i : pos | i <= m1 + 1} .<(m1 + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m1; i := succ i)
let
var j : intGte 1
in
for* {j : pos | j <= n1 + 1} .<(n1 + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n1; j := succ j)
A[i, j] := elt
end
end
implement {tk}
Real_Matrix_unit_matrix {m} m =
let
val A = Real_Matrix_make_elt<tk> (m, m, Zero)
var i : intGte 1
in
for* {i : pos | i <= m + 1} .<(m + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m; i := succ i)
A[i, i] := One;
A
end
implement {tk}
Real_Matrix_unit_matrix_to A =
let
val @(m, _) = dimension A
prval [m : int] EQINT () = eqint_make_gint m
var i : intGte 1
in
for* {i : pos | i <= m + 1} .<(m + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m; i := succ i)
let
var j : intGte 1
in
for* {j : pos | j <= m + 1} .<(m + 1) - j>.
(j : int j) =>
(j := 1; j <> succ m; j := succ j)
A[i, j] := (if i = j then One else Zero)
end
end
implement {tk}
Real_Matrix_matrix_sum (A, B) =
let
val @(m, n) = dimension A
val C = Real_Matrix_make_elt<tk> (m, n, NAN)
val () = matrix_sum_to<tk> (C, A, B)
in
C
end
implement {tk}
Real_Matrix_matrix_sum_to (C, A, B) =
let
val @(m, n) = dimension A
prval [m : int] EQINT () = eqint_make_gint m
prval [n : int] EQINT () = eqint_make_gint n
var i : intGte 1
in
for* {i : pos | i <= m + 1} .<(m + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m; i := succ i)
let
var j : intGte 1
in
for* {j : pos | j <= n + 1} .<(n + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n; j := succ j)
C[i, j] := A[i, j] + B[i, j]
end
end
implement {tk}
Real_Matrix_matrix_difference (A, B) =
let
val @(m, n) = dimension A
val C = Real_Matrix_make_elt<tk> (m, n, NAN)
val () = matrix_difference_to<tk> (C, A, B)
in
C
end
implement {tk}
Real_Matrix_matrix_difference_to (C, A, B) =
let
val @(m, n) = dimension A
prval [m : int] EQINT () = eqint_make_gint m
prval [n : int] EQINT () = eqint_make_gint n
var i : intGte 1
in
for* {i : pos | i <= m + 1} .<(m + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m; i := succ i)
let
var j : intGte 1
in
for* {j : pos | j <= n + 1} .<(n + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n; j := succ j)
C[i, j] := A[i, j] - B[i, j]
end
end
implement {tk}
Real_Matrix_matrix_product (A, B) =
let
val @(m, n) = dimension A and @(_, p) = dimension B
val C = Real_Matrix_make_elt<tk> (m, p, NAN)
val () = matrix_product_to<tk> (C, A, B)
in
C
end
implement {tk}
Real_Matrix_matrix_product_to (C, A, B) =
let
val @(m, n) = dimension A and @(_, p) = dimension B
prval [m : int] EQINT () = eqint_make_gint m
prval [n : int] EQINT () = eqint_make_gint n
prval [p : int] EQINT () = eqint_make_gint p
var i : intGte 1
in
for* {i : pos | i <= m + 1} .<(m + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m; i := succ i)
let
var k : intGte 1
in
for* {k : pos | k <= p + 1} .<(p + 1) - k>.
(k : int k) =>
(k := 1; k <> succ p; k := succ k)
let
var j : intGte 1
in
C[i, k] := A[i, 1] * B[1, k];
for* {j : pos | j <= n + 1} .<(n + 1) - j>.
(j : int j) =>
(j := 2; j <> succ n; j := succ j)
C[i, k] :=
multiply_and_add (A[i, j], B[j, k], C[i, k])
end
end
end
implement {tk}
Real_Matrix_scalar_product (A, r) =
let
val @(m, n) = dimension A
val C = Real_Matrix_make_elt<tk> (m, n, NAN)
val () = scalar_product_to<tk> (C, A, r)
in
C
end
implement {tk}
Real_Matrix_scalar_product_2 (r, A) =
Real_Matrix_scalar_product<tk> (A, r)
implement {tk}
Real_Matrix_scalar_product_to (C, A, r) =
let
val @(m, n) = dimension A
prval [m : int] EQINT () = eqint_make_gint m
prval [n : int] EQINT () = eqint_make_gint n
var i : intGte 1
in
for* {i : pos | i <= m + 1} .<(m + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m; i := succ i)
let
var j : intGte 1
in
for* {j : pos | j <= n + 1} .<(n + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n; j := succ j)
C[i, j] := A[i, j] * r
end
end
implement {tk}
Real_Matrix_fprint {m, n} (outf, A) =
let
val @(m, n) = dimension A
var i : intGte 1
in
for* {i : pos | i <= m + 1} .<(m + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m; i := succ i)
let
var j : intGte 1
in
for* {j : pos | j <= n + 1} .<(n + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n; j := succ j)
let
typedef FILEstar = $extype"FILE *"
extern castfn FILEref2star : FILEref -<> FILEstar
val _ = $extfcall (int, "fprintf", FILEref2star outf,
"%16.6g", A[i, j])
in
end;
fprintln! (outf)
end
end
(*------------------------------------------------------------------*)
(* Reduced row echelon form, by Gauss-Jordan elimination *)
extern fn {tk : tkind}
Real_Matrix_reduced_row_echelon_form :
{m, n : pos}
Real_Matrix (tk, m, n) -< !refwrt > Real_Matrix (tk, m, n)
implement {tk}
Real_Matrix_reduced_row_echelon_form {m, n} A =
let
val @(m, n) = dimension A
typedef one_to_m = intBtwe (1, m)
typedef one_to_n = intBtwe (1, n)
(* Partial pivoting, to improve the numerical stability. *)
implement
array_tabulate$fopr<one_to_m> i =
let
val i = g1ofg0 (sz2i (succ i))
val () = assertloc ((1 <= i) * (i <= m))
in
i
end
val rows_permutation =
$effmask_all arrayref_tabulate<one_to_m> (i2sz m)
fn
index_map : Matrix_Index_Map (m, n, m, n) =
lam (i1, j1) => $effmask_ref
(@(i0, j1) where { val i0 = rows_permutation[i1 - 1] })
val A = apply_index_map (copy<tk> A, m, n, index_map)
fn {}
exchange_rows (i1 : one_to_m,
i2 : one_to_m) :<!refwrt> void =
if i1 <> i2 then
let
val k1 = rows_permutation[pred i1]
and k2 = rows_permutation[pred i2]
in
rows_permutation[pred i1] := k2;
rows_permutation[pred i2] := k1
end
fn {}
normalize_pivot_row (i : one_to_m,
j : one_to_n) :<!refwrt> void =
let
prval [j : int] EQINT () = eqint_make_gint j
val pivot_val = A[i, j]
var k : intGte 1
in
A[i, j] := One;
for* {k : int | j + 1 <= k; k <= n + 1} .<(n + 1) - k>.
(k : int k) =>
(k := succ j; k <> succ n; k := succ k)
A[i, k] := A[i, k] / pivot_val
end
fn
subtract_normalized_pivot_row (ipiv : one_to_m,
i : one_to_m,
j : one_to_n) :<!refwrt> void =
let
prval [j : int] EQINT () = eqint_make_gint j
val factor = ~A[i, j]
var k : intGte 1
in
A[i, j] := Zero;
for* {k : int | j + 1 <= k; k <= n + 1} .<(n + 1) - k>.
(k : int k) =>
(k := succ j; k <> succ n; k := succ k)
A[i, k] := multiply_and_add (A[ipiv, k], factor, A[i, k])
end
fun
main_loop {i, j : pos | i <= m; i <= j; j <= n + 1}
.<(n + 1) - j>.
(i : int i, j : int j) :<!refwrt> void =
if j <> succ n then
let
fun
select_pivot {k : int | i <= k; k <= m + 1}
.<(m + 1) - k>.
(k : int k,
max_abs : g0float tk,
k_max_abs : intBtwe (i - 1, m))
:<!ref> intBtwe (i - 1, m) =
if k = succ m then
k_max_abs
else
let
val abs_akj = abs A[k, j]
in
if abs_akj > max_abs then
select_pivot (succ k, abs_akj, k)
else
select_pivot (succ k, max_abs, k_max_abs)
end
val i_pivot = select_pivot (i, Zero, pred i)
prval [i_pivot : int] EQINT () = eqint_make_gint i_pivot
in
if i_pivot = pred i then
(* There is no pivot in this column. *)
main_loop (i, succ j)
else
let
var k : intGte 1
in
exchange_rows (i_pivot, i);
normalize_pivot_row (i, j);
for* {k : int | 1 <= k; k <= i} .<i - k>.
(k : int k) =>
(k := 1; k <> i; k := succ k)
subtract_normalized_pivot_row (i, k, j);
for* {k : int | i + 1 <= k; k <= m + 1} .<(m + 1) - k>.
(k : int k) =>
(k := succ i; k <> succ m; k := succ k)
subtract_normalized_pivot_row (i, k, j);
if i <> m then
main_loop (succ i, succ j)
end
end
in
main_loop (1, 1);
A
end
overload reduced_row_echelon_form with
Real_Matrix_reduced_row_echelon_form
(*------------------------------------------------------------------*)
implement
main0 () =
let
val () = println! ()
val () = println! ("Here is the requested solution:")
val () = println! ()
val A = Real_Matrix_make_elt (3, 4, NAN)
val () =
(A[1,1] := 1.0; A[1,2] := 2.0; A[1,3] := ~1.0; A[1,4] := ~4.0;
A[2,1] := 2.0; A[2,2] := 3.0; A[2,3] := ~1.0; A[2,4] := ~11.0;
A[3,1] := ~2.0; A[3,2] := 0.0; A[3,3] := ~3.0; A[3,4] := 22.0)
val B = reduced_row_echelon_form A
val () = Real_Matrix_fprint (stdout_ref, B)
val () = println! ()
val () = println! ("Here is a RREF with a more interesting shape:")
val () = println! ()
val A = Real_Matrix_make_elt (3, 5, NAN)
val () =
(A[1,1] := 0.0; A[1,2] := 0.0; A[1,3] := ~1.0; A[1,4] := 2.0; A[1,5] := 0.0;
A[2,1] := 0.0; A[2,2] := 0.0; A[2,3] := ~1.0; A[2,4] := 1.0; A[2,5] := 1.0;
A[3,1] := 2.0; A[3,2] := 8.0; A[3,3] := 1.0; A[3,4] := ~4.0; A[3,5] := 2.0)
val B = reduced_row_echelon_form A
val () = Real_Matrix_fprint (stdout_ref, B)
val () = println! ()
val () = println! ("It is the RREF of this matrix:")
val () = println! ()
val () = Real_Matrix_fprint (stdout_ref, A)
val () = println! ()
in
end
(*------------------------------------------------------------------*)- Output:
$ patscc -std=gnu2x -g -O2 -DATS_MEMALLOC_GCBDW reduced_row_echelon_task.dats -lgc && ./a.out
Here is the requested solution:
1 0 0 -8
0 1 0 1
0 0 1 -2
Here is a RREF with a more interesting shape:
1 4 0 0 0
0 0 1 0 -2
0 0 0 1 -1
It is the RREF of this matrix:
0 0 -1 2 0
0 0 -1 1 1
2 8 1 -4 2
AutoHotkey
ToReducedRowEchelonForm(M){
rowCount := M.Count() ; the number of rows in M
columnCount := M.1.Count() ; the number of columns in M
r := lead := 1
while (r <= rowCount) {
if (columnCount < lead)
return M
i := r
while (M[i, lead] = 0) {
i++
if (rowCount+1 = i) {
i := r, lead++
if (columnCount+1 = lead)
return M
}
}
if (i<>r)
for col, v in M[i] ; Swap rows i and r
tempVal := M[i, col], M[i, col] := M[r, col], M[r, col] := tempVal
num := M[r, lead]
if (M[r, lead] <> 0)
for col, val in M[r]
M[r, col] /= num ; If M[r, lead] is not 0 divide row r by M[r, lead]
i := 2
while (i <= rowCount) {
num := M[i, lead]
if (i <> r)
for col, val in M[i] ; Subtract M[i, lead] multiplied by row r from row i
M[i, col] -= num * M[r, col]
i++
}
lead++, r++
}
return M
}
Examples:
M := [[1 , 2, -1, -4 ]
, [2 , 3, -1, -11]
, [-2, 0, -3, 22]]
M := ToReducedRowEchelonForm(M)
for row, obj in M
{
for col, v in obj
output .= RegExReplace(v, "\.0+$|0+$") "`t"
output .= "`n"
}
MsgBox % output
return
- Output:
1 0 0 -8 -0 1 0 1 -0 -0 1 -2
AutoIt
Global $ivMatrix[3][4] = [[1, 2, -1, -4],[2, 3, -1, -11],[-2, 0, -3, 22]]
ToReducedRowEchelonForm($ivMatrix)
Func ToReducedRowEchelonForm($matrix)
Local $clonematrix, $i
Local $lead = 0
Local $rowCount = UBound($matrix) - 1
Local $columnCount = UBound($matrix, 2) - 1
For $r = 0 To $rowCount
If $columnCount = $lead Then ExitLoop
$i = $r
While $matrix[$i][$lead] = 0
$i += 1
If $rowCount = $i Then
$i = $r
$lead += 1
If $columnCount = $lead Then ExitLoop
EndIf
WEnd
; There´s no built in Function to swap Rows of a 2-Dimensional Array
; We need to clone our matrix to swap complete lines
$clonematrix = $matrix ; Swap Lines, no
For $s = 0 To $columnCount
$matrix[$r][$s] = $clonematrix[$i][$s]
$matrix[$i][$s] = $clonematrix[$r][$s]
Next
Local $m = $matrix[$r][$lead]
For $k = 0 To $columnCount
$matrix[$r][$k] = $matrix[$r][$k] / $m
Next
For $i = 0 To $rowCount
If $i <> $r Then
Local $m = $matrix[$i][$lead]
For $k = 0 To $columnCount
$matrix[$i][$k] -= $m * $matrix[$r][$k]
Next
EndIf
Next
$lead += 1
Next
; Console Output
For $i = 0 To $rowCount
ConsoleWrite("[")
For $k = 0 To $columnCount
ConsoleWrite($matrix[$i][$k])
If $k <> $columnCount Then ConsoleWrite(",")
Next
ConsoleWrite("]" & @CRLF)
Next
; End of Console Output
Return $matrix
EndFunc ;==>ToReducedRowEchelonForm
- Output:
[1,0,0,-8] [-0,1,0,1] [-0,-0,1,-2]
BASIC
BASIC256
arraybase 1
global matrix
dim matrix = {{1, 2, -1, -4}, {2, 3, -1, -11}, { -2, 0, -3, 22}}
call RREF (matrix)
for row = 1 to 3
for col = 1 to 4
if matrix[row, col] = 0 then
print "0"; chr(9);
else
print matrix[row, col]; chr(9);
end if
next
print
next
end
subroutine RREF(m)
nrows = matrix[?,]
ncols = matrix[,?]
lead = 1
for r = 1 to nrows
if lead >= ncols then exit for
i = r
while matrix[i, lead] = 0
i += 1
if i = nrows then
i = r
lead += 1
if lead = ncols then exit for
end if
end while
for j = 1 to ncols
temp = matrix[i, j]
matrix[i, j] = matrix[r, j]
matrix[r, j] = temp
next
n = matrix[r, lead]
if n <> 1 then
for j = 0 to ncols
matrix[r, j] /= n
next
end if
for i = 1 to nrows
if i <> r then
n = matrix[i, lead]
for j = 1 to ncols
matrix[i, j] -= matrix[r, j] * n
next
end if
next
lead += 1
next
end subroutineBBC BASIC
DIM matrix(2,3)
matrix() = 1, 2, -1, -4, \
\ 2, 3, -1, -11, \
\ -2, 0, -3, 22
PROCrref(matrix())
FOR row% = 0 TO 2
FOR col% = 0 TO 3
PRINT matrix(row%,col%);
NEXT
PRINT
NEXT row%
END
DEF PROCrref(m())
LOCAL lead%, nrows%, ncols%, i%, j%, r%, n
nrows% = DIM(m(),1)+1
ncols% = DIM(m(),2)+1
FOR r% = 0 TO nrows%-1
IF lead% >= ncols% EXIT FOR
i% = r%
WHILE m(i%,lead%) = 0
i% += 1
IF i% = nrows% THEN
i% = r%
lead% += 1
IF lead% = ncols% EXIT FOR
ENDIF
ENDWHILE
FOR j% = 0 TO ncols%-1 : SWAP m(i%,j%),m(r%,j%) : NEXT
n = m(r%,lead%)
IF n <> 0 FOR j% = 0 TO ncols%-1 : m(r%,j%) /= n : NEXT
FOR i% = 0 TO nrows%-1
IF i% <> r% THEN
n = m(i%,lead%)
FOR j% = 0 TO ncols%-1
m(i%,j%) -= m(r%,j%) * n
NEXT
ENDIF
NEXT
lead% += 1
NEXT r%
ENDPROC
- Output:
1 0 0 -8
0 1 0 1
0 0 1 -2
C
#include <stdio.h>
#define TALLOC(n,typ) malloc(n*sizeof(typ))
#define EL_Type int
typedef struct sMtx {
int dim_x, dim_y;
EL_Type *m_stor;
EL_Type **mtx;
} *Matrix, sMatrix;
typedef struct sRvec {
int dim_x;
EL_Type *m_stor;
} *RowVec, sRowVec;
Matrix NewMatrix( int x_dim, int y_dim )
{
int n;
Matrix m;
m = TALLOC( 1, sMatrix);
n = x_dim * y_dim;
m->dim_x = x_dim;
m->dim_y = y_dim;
m->m_stor = TALLOC(n, EL_Type);
m->mtx = TALLOC(m->dim_y, EL_Type *);
for(n=0; n<y_dim; n++) {
m->mtx[n] = m->m_stor+n*x_dim;
}
return m;
}
void MtxSetRow(Matrix m, int irow, EL_Type *v)
{
int ix;
EL_Type *mr;
mr = m->mtx[irow];
for(ix=0; ix<m->dim_x; ix++)
mr[ix] = v[ix];
}
Matrix InitMatrix( int x_dim, int y_dim, EL_Type **v)
{
Matrix m;
int iy;
m = NewMatrix(x_dim, y_dim);
for (iy=0; iy<y_dim; iy++)
MtxSetRow(m, iy, v[iy]);
return m;
}
void MtxDisplay( Matrix m )
{
int iy, ix;
const char *sc;
for (iy=0; iy<m->dim_y; iy++) {
printf(" ");
sc = " ";
for (ix=0; ix<m->dim_x; ix++) {
printf("%s %3d", sc, m->mtx[iy][ix]);
sc = ",";
}
printf("\n");
}
printf("\n");
}
void MtxMulAndAddRows(Matrix m, int ixrdest, int ixrsrc, EL_Type mplr)
{
int ix;
EL_Type *drow, *srow;
drow = m->mtx[ixrdest];
srow = m->mtx[ixrsrc];
for (ix=0; ix<m->dim_x; ix++)
drow[ix] += mplr * srow[ix];
// printf("Mul row %d by %d and add to row %d\n", ixrsrc, mplr, ixrdest);
// MtxDisplay(m);
}
void MtxSwapRows( Matrix m, int rix1, int rix2)
{
EL_Type *r1, *r2, temp;
int ix;
if (rix1 == rix2) return;
r1 = m->mtx[rix1];
r2 = m->mtx[rix2];
for (ix=0; ix<m->dim_x; ix++)
temp = r1[ix]; r1[ix]=r2[ix]; r2[ix]=temp;
// printf("Swap rows %d and %d\n", rix1, rix2);
// MtxDisplay(m);
}
void MtxNormalizeRow( Matrix m, int rix, int lead)
{
int ix;
EL_Type *drow;
EL_Type lv;
drow = m->mtx[rix];
lv = drow[lead];
for (ix=0; ix<m->dim_x; ix++)
drow[ix] /= lv;
// printf("Normalize row %d\n", rix);
// MtxDisplay(m);
}
#define MtxGet( m, rix, cix ) m->mtx[rix][cix]
void MtxToReducedREForm(Matrix m)
{
int lead;
int rix, iix;
EL_Type lv;
int rowCount = m->dim_y;
lead = 0;
for (rix=0; rix<rowCount; rix++) {
if (lead >= m->dim_x)
return;
iix = rix;
while (0 == MtxGet(m, iix,lead)) {
iix++;
if (iix == rowCount) {
iix = rix;
lead++;
if (lead == m->dim_x)
return;
}
}
MtxSwapRows(m, iix, rix );
MtxNormalizeRow(m, rix, lead );
for (iix=0; iix<rowCount; iix++) {
if ( iix != rix ) {
lv = MtxGet(m, iix, lead );
MtxMulAndAddRows(m,iix, rix, -lv) ;
}
}
lead++;
}
}
int main()
{
Matrix m1;
static EL_Type r1[] = {1,2,-1,-4};
static EL_Type r2[] = {2,3,-1,-11};
static EL_Type r3[] = {-2,0,-3,22};
static EL_Type *im[] = { r1, r2, r3 };
m1 = InitMatrix( 4,3, im );
printf("Initial\n");
MtxDisplay(m1);
MtxToReducedREForm(m1);
printf("Reduced R-E form\n");
MtxDisplay(m1);
return 0;
}
C#
using System;
namespace rref
{
class Program
{
static void Main(string[] args)
{
int[,] matrix = new int[3, 4]{
{ 1, 2, -1, -4 },
{ 2, 3, -1, -11 },
{ -2, 0, -3, 22 }
};
matrix = rref(matrix);
}
private static int[,] rref(int[,] matrix)
{
int lead = 0, rowCount = matrix.GetLength(0), columnCount = matrix.GetLength(1);
for (int r = 0; r < rowCount; r++)
{
if (columnCount <= lead) break;
int i = r;
while (matrix[i, lead] == 0)
{
i++;
if (i == rowCount)
{
i = r;
lead++;
if (columnCount == lead)
{
lead--;
break;
}
}
}
for (int j = 0; j < columnCount; j++)
{
int temp = matrix[r, j];
matrix[r, j] = matrix[i, j];
matrix[i, j] = temp;
}
int div = matrix[r, lead];
if(div != 0)
for (int j = 0; j < columnCount; j++) matrix[r, j] /= div;
for (int j = 0; j < rowCount; j++)
{
if (j != r)
{
int sub = matrix[j, lead];
for (int k = 0; k < columnCount; k++) matrix[j, k] -= (sub * matrix[r, k]);
}
}
lead++;
}
return matrix;
}
}
}
C++
Note: This code is written in generic form. While it slightly complicates the code, it allows to use the same code for both built-in arrays and matrix classes. To use it with a matrix class, either program the matrix class to the specifications given in the matrix_traits comment, or specialize matrix_traits for the specific interface of your matrix class.
The test code uses a built-in array for the matrix.
#include <algorithm> // for std::swap
#include <cstddef>
#include <cassert>
// Matrix traits: This describes how a matrix is accessed. By
// externalizing this information into a traits class, the same code
// can be used both with native arrays and matrix classes. To use the
// default implementation of the traits class, a matrix type has to
// provide the following definitions as members:
//
// * typedef ... index_type;
// - The type used for indexing (e.g. size_t)
// * typedef ... value_type;
// - The element type of the matrix (e.g. double)
// * index_type min_row() const;
// - returns the minimal allowed row index
// * index_type max_row() const;
// - returns the maximal allowed row index
// * index_type min_column() const;
// - returns the minimal allowed column index
// * index_type max_column() const;
// - returns the maximal allowed column index
// * value_type& operator()(index_type i, index_type k)
// - returns a reference to the element i,k, where
// min_row() <= i <= max_row()
// min_column() <= k <= max_column()
// * value_type operator()(index_type i, index_type k) const
// - returns the value of element i,k
//
// Note that the functions are all inline and simple, so the compiler
// should completely optimize them away.
template<typename MatrixType> struct matrix_traits
{
typedef typename MatrixType::index_type index_type;
typedef typename MatrixType::value_type value_type;
static index_type min_row(MatrixType const& A)
{ return A.min_row(); }
static index_type max_row(MatrixType const& A)
{ return A.max_row(); }
static index_type min_column(MatrixType const& A)
{ return A.min_column(); }
static index_type max_column(MatrixType const& A)
{ return A.max_column(); }
static value_type& element(MatrixType& A, index_type i, index_type k)
{ return A(i,k); }
static value_type element(MatrixType const& A, index_type i, index_type k)
{ return A(i,k); }
};
// specialization of the matrix traits for built-in two-dimensional
// arrays
template<typename T, std::size_t rows, std::size_t columns>
struct matrix_traits<T[rows][columns]>
{
typedef std::size_t index_type;
typedef T value_type;
static index_type min_row(T const (&)[rows][columns])
{ return 0; }
static index_type max_row(T const (&)[rows][columns])
{ return rows-1; }
static index_type min_column(T const (&)[rows][columns])
{ return 0; }
static index_type max_column(T const (&)[rows][columns])
{ return columns-1; }
static value_type& element(T (&A)[rows][columns],
index_type i, index_type k)
{ return A[i][k]; }
static value_type element(T const (&A)[rows][columns],
index_type i, index_type k)
{ return A[i][k]; }
};
// Swap rows i and k of a matrix A
// Note that due to the reference, both dimensions are preserved for
// built-in arrays
template<typename MatrixType>
void swap_rows(MatrixType& A,
typename matrix_traits<MatrixType>::index_type i,
typename matrix_traits<MatrixType>::index_type k)
{
matrix_traits<MatrixType> mt;
typedef typename matrix_traits<MatrixType>::index_type index_type;
// check indices
assert(mt.min_row(A) <= i);
assert(i <= mt.max_row(A));
assert(mt.min_row(A) <= k);
assert(k <= mt.max_row(A));
for (index_type col = mt.min_column(A); col <= mt.max_column(A); ++col)
std::swap(mt.element(A, i, col), mt.element(A, k, col));
}
// divide row i of matrix A by v
template<typename MatrixType>
void divide_row(MatrixType& A,
typename matrix_traits<MatrixType>::index_type i,
typename matrix_traits<MatrixType>::value_type v)
{
matrix_traits<MatrixType> mt;
typedef typename matrix_traits<MatrixType>::index_type index_type;
assert(mt.min_row(A) <= i);
assert(i <= mt.max_row(A));
assert(v != 0);
for (index_type col = mt.min_column(A); col <= mt.max_column(A); ++col)
mt.element(A, i, col) /= v;
}
// in matrix A, add v times row k to row i
template<typename MatrixType>
void add_multiple_row(MatrixType& A,
typename matrix_traits<MatrixType>::index_type i,
typename matrix_traits<MatrixType>::index_type k,
typename matrix_traits<MatrixType>::value_type v)
{
matrix_traits<MatrixType> mt;
typedef typename matrix_traits<MatrixType>::index_type index_type;
assert(mt.min_row(A) <= i);
assert(i <= mt.max_row(A));
assert(mt.min_row(A) <= k);
assert(k <= mt.max_row(A));
for (index_type col = mt.min_column(A); col <= mt.max_column(A); ++col)
mt.element(A, i, col) += v * mt.element(A, k, col);
}
// convert A to reduced row echelon form
template<typename MatrixType>
void to_reduced_row_echelon_form(MatrixType& A)
{
matrix_traits<MatrixType> mt;
typedef typename matrix_traits<MatrixType>::index_type index_type;
index_type lead = mt.min_row(A);
for (index_type row = mt.min_row(A); row <= mt.max_row(A); ++row)
{
if (lead > mt.max_column(A))
return;
index_type i = row;
while (mt.element(A, i, lead) == 0)
{
++i;
if (i > mt.max_row(A))
{
i = row;
++lead;
if (lead > mt.max_column(A))
return;
}
}
swap_rows(A, i, row);
divide_row(A, row, mt.element(A, row, lead));
for (i = mt.min_row(A); i <= mt.max_row(A); ++i)
{
if (i != row)
add_multiple_row(A, i, row, -mt.element(A, i, lead));
}
}
}
// test code
#include <iostream>
int main()
{
double M[3][4] = { { 1, 2, -1, -4 },
{ 2, 3, -1, -11 },
{ -2, 0, -3, 22 } };
to_reduced_row_echelon_form(M);
for (int i = 0; i < 3; ++i)
{
for (int j = 0; j < 4; ++j)
std::cout << M[i][j] << '\t';
std::cout << "\n";
}
return EXIT_SUCCESS;
}
- Output:
1 0 0 -8 -0 1 0 1 -0 -0 1 -2
Common Lisp
Direct implementation of the pseudo-code given.
(defun convert-to-row-echelon-form (matrix)
(let* ((dimensions (array-dimensions matrix))
(row-count (first dimensions))
(column-count (second dimensions))
(lead 0))
(labels ((find-pivot (start lead)
(let ((i start))
(loop
:while (zerop (aref matrix i lead))
:do (progn
(incf i)
(when (= i row-count)
(setf i start)
(incf lead)
(when (= lead column-count)
(return-from convert-to-row-echelon-form matrix))))
:finally (return (values i lead)))))
(swap-rows (r1 r2)
(loop
:for c :upfrom 0 :below column-count
:do (rotatef (aref matrix r1 c) (aref matrix r2 c))))
(divide-row (r value)
(loop
:for c :upfrom 0 :below column-count
:do (setf (aref matrix r c)
(/ (aref matrix r c) value)))))
(loop
:for r :upfrom 0 :below row-count
:when (<= column-count lead)
:do (return matrix)
:do (multiple-value-bind (i nlead) (find-pivot r lead)
(setf lead nlead)
(swap-rows i r)
(divide-row r (aref matrix r lead))
(loop
:for i :upfrom 0 :below row-count
:when (/= i r)
:do (let ((scale (aref matrix i lead)))
(loop
:for c :upfrom 0 :below column-count
:do (decf (aref matrix i c)
(* scale (aref matrix r c))))))
(incf lead))
:finally (return matrix)))))
D
import std.stdio, std.algorithm, std.array, std.conv;
void toReducedRowEchelonForm(T)(T[][] M) pure nothrow @nogc {
if (M.empty)
return;
immutable nrows = M.length;
immutable ncols = M[0].length;
size_t lead;
foreach (immutable r; 0 .. nrows) {
if (ncols <= lead)
return;
{
size_t i = r;
while (M[i][lead] == 0) {
i++;
if (nrows == i) {
i = r;
lead++;
if (ncols == lead)
return;
}
}
swap(M[i], M[r]);
}
M[r][] /= M[r][lead];
foreach (j, ref mj; M)
if (j != r)
mj[] -= M[r][] * mj[lead];
lead++;
}
}
void main() {
auto A = [[ 1, 2, -1, -4],
[ 2, 3, -1, -11],
[-2, 0, -3, 22]];
A.toReducedRowEchelonForm;
writefln("%(%(%2d %)\n%)", A);
}
- Output:
1 0 0 -8 0 1 0 1 0 0 1 -2
EasyLang
proc rref . m[][] .
nrow = len m[][]
ncol = len m[1][]
lead = 1
for r to nrow
if lead > ncol
return
.
i = r
while m[i][lead] = 0
i += 1
if i > nrow
i = r
lead += 1
if lead > ncol
return
.
.
.
swap m[i][] m[r][]
m = m[r][lead]
for k to ncol
m[r][k] /= m
.
for i to nrow
if i <> r
m = m[i][lead]
for k to ncol
m[i][k] -= m * m[r][k]
.
.
.
lead += 1
.
.
test[][] = [ [ 1 2 -1 -4 ] [ 2 3 -1 -11 ] [ -2 0 -3 22 ] ]
rref test[][]
print test[][]
Euphoria
function ToReducedRowEchelonForm(sequence M)
integer lead,rowCount,columnCount,i
sequence temp
lead = 1
rowCount = length(M)
columnCount = length(M[1])
for r = 1 to rowCount do
if columnCount <= lead then
exit
end if
i = r
while M[i][lead] = 0 do
i += 1
if rowCount = i then
i = r
lead += 1
if columnCount = lead then
exit
end if
end if
end while
temp = M[i]
M[i] = M[r]
M[r] = temp
M[r] /= M[r][lead]
for j = 1 to rowCount do
if j != r then
M[j] -= M[j][lead]*M[r]
end if
end for
lead += 1
end for
return M
end function
? ToReducedRowEchelonForm(
{ { 1, 2, -1, -4 },
{ 2, 3, -1, -11 },
{ -2, 0, -3, 22 } })- Output:
{
{1,0,0,-8},
{0,1,0,1},
{0,0,1,-2}
}
Factor
USE: math.matrices.elimination
{ { 1 2 -1 -4 } { 2 3 -1 -11 } { -2 0 -3 22 } } solution .
- Output:
{ { 1 0 0 -8 } { 0 1 0 1 } { 0 0 1 -2 } }
Fortran
module Rref
implicit none
contains
subroutine to_rref(matrix)
real, dimension(:,:), intent(inout) :: matrix
integer :: pivot, norow, nocolumn
integer :: r, i
real, dimension(:), allocatable :: trow
pivot = 1
norow = size(matrix, 1)
nocolumn = size(matrix, 2)
allocate(trow(nocolumn))
do r = 1, norow
if ( nocolumn <= pivot ) exit
i = r
do while ( matrix(i, pivot) == 0 )
i = i + 1
if ( norow == i ) then
i = r
pivot = pivot + 1
if ( nocolumn == pivot ) return
end if
end do
trow = matrix(i, :)
matrix(i, :) = matrix(r, :)
matrix(r, :) = trow
matrix(r, :) = matrix(r, :) / matrix(r, pivot)
do i = 1, norow
if ( i /= r ) matrix(i, :) = matrix(i, :) - matrix(r, :) * matrix(i, pivot)
end do
pivot = pivot + 1
end do
deallocate(trow)
end subroutine to_rref
end module Rref
program prg_test
use rref
implicit none
real, dimension(3, 4) :: m = reshape( (/ 1, 2, -1, -4, &
2, 3, -1, -11, &
-2, 0, -3, 22 /), &
(/ 3, 4 /), order = (/ 2, 1 /) )
integer :: i
print *, "Original matrix"
do i = 1, size(m,1)
print *, m(i, :)
end do
call to_rref(m)
print *, "Reduced row echelon form"
do i = 1, size(m,1)
print *, m(i, :)
end do
end program prg_test
FreeBASIC
Include the code from Matrix multiplication#FreeBASIC because this function uses the matrix type defined there and I don't want to reproduce it all here.
#include once "matmult.bas"
sub rowswap( byval M as Matrix, i as uinteger, j as uinteger )
dim as integer k
for k = 0 to ubound(M.m, 2)
swap M.m(j, k), M.m(i, k)
next k
end sub
function rowech(byval M as Matrix) as Matrix
dim as uinteger lead = 0, rowCount = 1+ubound(M.m, 1), colCount = 1+ubound(M.m, 2)
dim as uinteger r, i, j
dim as double K
for r = 0 to rowCount-1
if lead >= colCount then exit for
i = r
while M.m(i, lead) = 0
i += 1
if i = rowCount then
i = r
lead += 1
if lead = colCount then exit for
endif
wend
rowswap M, r, i
K = M.m(r,lead)
if K <> 0 then
for j = 0 to colCount-1
M.m(r,j) /= K
next j
endif
for i = 0 to rowCount-1
if i <> r then
K = M.m(i, lead)
for j = 0 to colCount-1
M.m(i,j) -= M.m(r,j) * K
next j
endif
next i
lead += 1
next r
return M
end function
dim as Matrix M = Matrix (3, 4)
dim as Matrix N
M.m(0,0) = 1 : M.m(0,1) = 2 : M.m(0,2) = -1 : M.M(0,3) = -4
M.m(1,0) = 2 : M.m(1,1) = 3 : M.m(1,2) = -1 : M.m(1,3) = -11
M.m(2,0) = -2: M.m(2,1) = 0 : M.m(2,2) = -3 : M.m(2,3) = 22
dim as integer i, j
N = rowech(M)
for i=0 to 2
for j = 0 to 3
print N.m(i, j),
next j
print
next i
- Output:
1 0 0 -8 -0 1 0 1 -0 -0 1 -2
Go
2D representation
From WP pseudocode:
package main
import "fmt"
type matrix [][]float64
func (m matrix) print() {
for _, r := range m {
fmt.Println(r)
}
fmt.Println("")
}
func main() {
m := matrix{
{ 1, 2, -1, -4},
{ 2, 3, -1, -11},
{-2, 0, -3, 22},
}
m.print()
rref(m)
m.print()
}
func rref(m matrix) {
lead := 0
rowCount := len(m)
columnCount := len(m[0])
for r := 0; r < rowCount; r++ {
if lead >= columnCount {
return
}
i := r
for m[i][lead] == 0 {
i++
if rowCount == i {
i = r
lead++
if columnCount == lead {
return
}
}
}
m[i], m[r] = m[r], m[i]
f := 1 / m[r][lead]
for j, _ := range m[r] {
m[r][j] *= f
}
for i = 0; i < rowCount; i++ {
if i != r {
f = m[i][lead]
for j, e := range m[r] {
m[i][j] -= e * f
}
}
}
lead++
}
}
- Output:
(not so pretty, sorry)
[1 2 -1 -4] [2 3 -1 -11] [-2 0 -3 22] [1 0 0 -8] [-0 1 0 1] [-0 -0 1 -2]
Flat representation
package main
import "fmt"
type matrix struct {
ele []float64
stride int
}
func matrixFromRows(rows [][]float64) *matrix {
if len(rows) == 0 {
return &matrix{nil, 0}
}
m := &matrix{make([]float64, len(rows)*len(rows[0])), len(rows[0])}
for rx, row := range rows {
copy(m.ele[rx*m.stride:(rx+1)*m.stride], row)
}
return m
}
func (m *matrix) print(heading string) {
if heading > "" {
fmt.Print("\n", heading, "\n")
}
for e := 0; e < len(m.ele); e += m.stride {
fmt.Printf("%6.2f ", m.ele[e:e+m.stride])
fmt.Println()
}
}
func (m *matrix) rref() {
lead := 0
for rxc0 := 0; rxc0 < len(m.ele); rxc0 += m.stride {
if lead >= m.stride {
return
}
ixc0 := rxc0
for m.ele[ixc0+lead] == 0 {
ixc0 += m.stride
if ixc0 == len(m.ele) {
ixc0 = rxc0
lead++
if lead == m.stride {
return
}
}
}
for c, ix, rx := 0, ixc0, rxc0; c < m.stride; c++ {
m.ele[ix], m.ele[rx] = m.ele[rx], m.ele[ix]
ix++
rx++
}
if d := m.ele[rxc0+lead]; d != 0 {
d := 1 / d
for c, rx := 0, rxc0; c < m.stride; c++ {
m.ele[rx] *= d
rx++
}
}
for ixc0 = 0; ixc0 < len(m.ele); ixc0 += m.stride {
if ixc0 != rxc0 {
f := m.ele[ixc0+lead]
for c, ix, rx := 0, ixc0, rxc0; c < m.stride; c++ {
m.ele[ix] -= m.ele[rx] * f
ix++
rx++
}
}
}
lead++
}
}
func main() {
m := matrixFromRows([][]float64{
{1, 2, -1, -4},
{2, 3, -1, -11},
{-2, 0, -3, 22},
})
m.print("Input:")
m.rref()
m.print("Reduced:")
}
- Output:
Input: [ 1.00 2.00 -1.00 -4.00] [ 2.00 3.00 -1.00 -11.00] [ -2.00 0.00 -3.00 22.00] Reduced: [ 1.00 0.00 0.00 -8.00] [ -0.00 1.00 0.00 1.00] [ -0.00 -0.00 1.00 -2.00]
Groovy
This solution implements the transformation to reduced row echelon form with optional pivoting. Options are provided for both partial pivoting and scaled partial pivoting. The default option is no pivoting at all.
enum Pivoting {
NONE({ i, it -> 1 }),
PARTIAL({ i, it -> - (it[i].abs()) }),
SCALED({ i, it -> - it[i].abs()/(it.inject(0) { sum, elt -> sum + elt.abs() } ) });
public final Closure comparer
private Pivoting(Closure c) {
comparer = c
}
}
def isReducibleMatrix = { matrix ->
def m = matrix.size()
m > 1 && matrix[0].size() > m && matrix[1..<m].every { row -> row.size() == matrix[0].size() }
}
def reducedRowEchelonForm = { matrix, Pivoting pivoting = Pivoting.NONE ->
assert isReducibleMatrix(matrix)
def m = matrix.size()
def n = matrix[0].size()
(0..<m).each { i ->
matrix[i..<m].sort(pivoting.comparer.curry(i))
matrix[i][i..<n] = matrix[i][i..<n].collect { it/matrix[i][i] }
((0..<i) + ((i+1)..<m)).each { k ->
(i..<n).reverse().each { j ->
matrix[k][j] -= matrix[i][j]*matrix[k][i]
}
}
}
matrix
}
This test first demonstrates the test case provided, and then demonstrates another test case designed to show the dangers of not using pivoting on an otherwise solvable matrix. Both test cases exercise all three pivoting options.
def matrixCopy = { matrix -> matrix.collect { row -> row.collect { it } } }
println "Tests for matrix A:"
def a = [
[1, 2, -1, -4],
[2, 3, -1, -11],
[-2, 0, -3, 22]
]
a.each { println it }
println()
println "pivoting == Pivoting.NONE"
reducedRowEchelonForm(matrixCopy(a)).each { println it }
println()
println "pivoting == Pivoting.PARTIAL"
reducedRowEchelonForm(matrixCopy(a), Pivoting.PARTIAL).each { println it }
println()
println "pivoting == Pivoting.SCALED"
reducedRowEchelonForm(matrixCopy(a), Pivoting.SCALED).each { println it }
println()
println "Tests for matrix B (divides by 0 without pivoting):"
def b = [
[1, 2, -1, -4],
[2, 4, -1, -11],
[-2, 0, -6, 24]
]
b.each { println it }
println()
println "pivoting == Pivoting.NONE"
try {
reducedRowEchelonForm(matrixCopy(b)).each { println it }
println()
} catch (e) {
println "KABOOM! ${e.message}"
println()
}
println "pivoting == Pivoting.PARTIAL"
reducedRowEchelonForm(matrixCopy(b), Pivoting.PARTIAL).each { println it }
println()
println "pivoting == Pivoting.SCALED"
reducedRowEchelonForm(matrixCopy(b), Pivoting.SCALED).each { println it }
println()
- Output:
Tests for matrix A: [1, 2, -1, -4] [2, 3, -1, -11] [-2, 0, -3, 22] pivoting == Pivoting.NONE [1, 0, 0, -8] [0, 1, 0, 1] [0, 0, 1, -2] pivoting == Pivoting.PARTIAL [1, 0.0, 0E-11, -7.9999999997000000000150] [0, 1, 0E-10, 0.999999999700000000010] [0, 0.0, 1, -2.00000000030] pivoting == Pivoting.SCALED [1, 0, 0, -8] [0, 1, 0, 1] [0, 0, 1, -2] Tests for matrix B (divides by 0 without pivoting): [1, 2, -1, -4] [2, 4, -1, -11] [-2, 0, -6, 24] pivoting == Pivoting.NONE KABOOM! Division undefined pivoting == Pivoting.PARTIAL [1, 0, 0.00, -3.00] [0, 1, 0.00, -2.00] [0, 0, 1, -3] pivoting == Pivoting.SCALED [1, 0, 0, -3] [0, 1, 0, -2] [0, 0, 1, -3]
Haskell
This program was produced by translating from the Python and gradually refactoring the result into a more functional style.
import Data.List (find)
rref :: Fractional a => [[a]] -> [[a]]
rref m = f m 0 [0 .. rows - 1]
where rows = length m
cols = length $ head m
f m _ [] = m
f m lead (r : rs)
| indices == Nothing = m
| otherwise = f m' (lead' + 1) rs
where indices = find p l
p (col, row) = m !! row !! col /= 0
l = [(col, row) |
col <- [lead .. cols - 1],
row <- [r .. rows - 1]]
Just (lead', i) = indices
newRow = map (/ m !! i !! lead') $ m !! i
m' = zipWith g [0..] $
replace r newRow $
replace i (m !! r) m
g n row
| n == r = row
| otherwise = zipWith h newRow row
where h = subtract . (* row !! lead')
replace :: Int -> a -> [a] -> [a]
{- Replaces the element at the given index. -}
replace n e l = a ++ e : b
where (a, _ : b) = splitAt n l
Icon and Unicon
Works in both languages:
procedure main(A)
tM := [[ 1, 2, -1, -4],
[ 2, 3, -1,-11],
[ -2, 0, -3, 22]]
showMat(rref(tM))
end
procedure rref(M)
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
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
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");
}
}
Without external libraries
import java.util.List;
public final class ReducedRowEchelonForm {
public static void main(String[] args) {
double[][] first = new double[][] { { 1.0, 2.0, -1.0, -4.0 },
{ 2.0, 3.0, -1.0, -11.0 },
{ -2.0, 0.0, -3.0, 22.0 } };
double[][] second = new double[][] { { 1.0, 2.0, 3.0, 4.0, 3.0, 1.0 },
{ 2.0, 4.0, 6.0, 2.0, 6.0, 2.0 },
{ 3.0, 6.0, 18.0, 9.0, 9.0, -6.0 },
{ 4.0, 8.0, 12.0, 10.0, 12.0, 4.0 },
{ 5.0, 10.0, 24.0, 11.0, 15.0, -4.0 } };
List.of( first, second ).forEach( m -> {
display("Original matrix:", m);
display("Reduced row echelon form:", reducedRowEchelonForm(m));
} );
}
private static double[][] reducedRowEchelonForm(double[][] matrix) {
final int rowCount = matrix.length;
final int colCount = matrix[0].length;
int lead = 0;
for ( int row = 0; row < rowCount; row++ ) {
if ( colCount <= lead ) {
return matrix;
}
int ro = row;
while ( matrix[ro][lead] == 0.0 ) {
ro += 1;
if ( rowCount == ro ) {
ro = row;
lead += 1;
if ( colCount == lead ) {
return matrix;
}
}
}
final double[] temp = matrix[row];
matrix[row] = matrix[ro];
matrix[ro] = temp;
if ( matrix[row][lead] != 0.0 ) {
final double divisor = matrix[row][lead];
for ( int co = 0; co < matrix[0].length; co++ ) {
matrix[row][co] /= divisor;
}
}
for ( int j = 0; j < rowCount; j++ ) {
if ( j != row ) {
final double multiple = matrix[j][lead];
for ( int co = 0; co < matrix[0].length; co++ ) {
matrix[j][co] -= matrix[row][co] * multiple;
}
}
}
lead += 1;
}
return matrix;
}
private static void display(String title, double[][] matrix) {
System.out.println(title);
final int rowCount = matrix.length;
final int colCount = matrix[0].length;
for ( int row = 0; row < rowCount; row++ ) {
for ( int col = 0; col < colCount; col++ ) {
if ( matrix[row][col] == -0.0 ) {
matrix[row][col] = 0.0;
}
System.out.print(String.format("%6.1f", matrix[row][col]));
}
System.out.println();
}
System.out.println();
}
}
- Output:
Original 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 row echelon form: 1.0 0.0 0.0 -8.0 0.0 1.0 0.0 1.0 0.0 0.0 1.0 -2.0 Original matrix: 1.0 2.0 3.0 4.0 3.0 1.0 2.0 4.0 6.0 2.0 6.0 2.0 3.0 6.0 18.0 9.0 9.0 -6.0 4.0 8.0 12.0 10.0 12.0 4.0 5.0 10.0 24.0 11.0 15.0 -4.0 Reduced row echelon form: 1.0 2.0 0.0 0.0 3.0 4.0 0.0 0.0 1.0 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 0.0 0.0 0.0 0.0
JavaScript
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
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 == $j then .
else .[$i] as $t
| .[$i] = .[$j]
| .[$j] = $t
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(. * $mult)))
else .
end )
| .lead += 1
else .
end )
| .a;
[ [ 1, 2, -1, -4],
[ 2, 3, -1, -11],
[-2, 0, -3, 22] ],
[ [1, 2, -1, -4],
[2, 4, -1, -11],
[-2, 0, -6, 24] ]
| "Original:", matrix_print, "",
"RREF:", (toReducedRowEchelonForm|matrix_print), "\n"- Output:
Invocation: jq -nrc -f reduced-row-echelon-form.jq
Original: 1 2 -1 -4 2 3 -1 -11 -2 0 -3 22 RREF: 1 0 0 -8 0 1 0 1 0 0 1 -2 Original: 1 2 -1 -4 2 4 -1 -11 -2 0 -6 24 RREF: 1 0 0 -3 0 1 0 -2 0 0 1 -3
Julia
RowEchelon.jl offers the function rref to compute the reduced-row echelon form:
julia> matrix = [1 2 -1 -4 ; 2 3 -1 -11 ; -2 0 -3 22]
3x4 Int32 Array:
1 2 -1 -4
2 3 -1 -11
-2 0 -3 22
julia> rref(matrix)
3x4 Array{Float64,2}:
1.0 0.0 0.0 -8.0
0.0 1.0 0.0 1.0
0.0 0.0 1.0 -2.0
Kotlin
// version 1.1.51
typealias Matrix = Array<DoubleArray>
/* changes the matrix to RREF 'in place' */
fun Matrix.toReducedRowEchelonForm() {
var lead = 0
val rowCount = this.size
val colCount = this[0].size
for (r in 0 until rowCount) {
if (colCount <= lead) return
var i = r
while (this[i][lead] == 0.0) {
i++
if (rowCount == i) {
i = r
lead++
if (colCount == lead) return
}
}
val temp = this[i]
this[i] = this[r]
this[r] = temp
if (this[r][lead] != 0.0) {
val div = this[r][lead]
for (j in 0 until colCount) this[r][j] /= div
}
for (k in 0 until rowCount) {
if (k != r) {
val mult = this[k][lead]
for (j in 0 until colCount) this[k][j] -= this[r][j] * mult
}
}
lead++
}
}
fun Matrix.printf(title: String) {
println(title)
val rowCount = this.size
val colCount = this[0].size
for (r in 0 until rowCount) {
for (c in 0 until colCount) {
if (this[r][c] == -0.0) this[r][c] = 0.0 // get rid of negative zeros
print("${"% 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
function ToReducedRowEchelonForm ( M )
local lead = 1
local n_rows, n_cols = #M, #M[1]
for r = 1, n_rows do
if n_cols <= lead then break end
local i = r
while M[i][lead] == 0 do
i = i + 1
if n_rows == i then
i = r
lead = lead + 1
if n_cols == lead then break end
end
end
M[i], M[r] = M[r], M[i]
local m = M[r][lead]
for k = 1, n_cols do
M[r][k] = M[r][k] / m
end
for i = 1, n_rows do
if i ~= r then
local m = M[i][lead]
for k = 1, n_cols do
M[i][k] = M[i][k] - m * M[r][k]
end
end
end
lead = lead + 1
end
end
M = { { 1, 2, -1, -4 },
{ 2, 3, -1, -11 },
{ -2, 0, -3, 22 } }
res = ToReducedRowEchelonForm( M )
for i = 1, #M do
for j = 1, #M[1] do
io.write( M[i][j], " " )
end
io.write( "\n" )
end
- Output:
1 0 0 -8 0 1 0 1 0 0 1 -2
M2000 Interpreter
low bound 1 for array
Module Base1 {
dim base 1, A(3, 4)
A(1, 1)= 1, 2, -1, -4, 2 , 3, -1, -11, -2 , 0 , -3, 22
lead=1
rowcount=3
columncount=4
gosub disp()
for r=1 to rowcount {
if columncount<lead then exit
i=r
while A(i,lead)=0 {
i++
if rowcount=i then i=r : lead++ : if columncount<lead then exit
}
for c =1 to columncount {
swap A(i, c), A(r, c)
}
if A(r, lead)<>0 then {
div1=A(r,lead)
For c =1 to columncount {
A( r, c)/=div1
}
}
for i=1 to rowcount {
if i<>r then {
mult=A(i,lead)
for j=1 to columncount {
A(i,j)-=A(r,j)*mult
}
}
}
lead=lead+1
}
disp()
sub disp()
local i, j
for i=1 to rowcount
for j=1 to columncount
Print A(i, j),
Next j
if pos>0 then print
Next i
End sub
}
Base1Low 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
}
base0Maple
with(LinearAlgebra):
ReducedRowEchelonForm(<<1,2,-2>|<2,3,0>|<-1,-1,-3>|<-4,-11,22>>);- Output:
[1 0 0 -8]
[ ]
[0 1 0 1]
[ ]
[0 0 1 -2]
Mathematica /Wolfram Language
RowReduce[{{1, 2, -1, -4}, {2, 3, -1, -11}, {-2, 0, -3, 22}}]
- Output:
{{1, 0, 0, -8}, {0, 1, 0, 1}, {0, 0, 1, -2}}
MATLAB
rref([1, 2, -1, -4; 2, 3, -1, -11; -2, 0, -3, 22])
Maxima
rref(a):=block([p,q,k],[p,q]:matrix_size(a),a:echelon(a),
k:min(p,q),
for i thru min(p,q) do (if a[i,i]=0 then (k:i-1,return())),
for i:k thru 2 step -1 do (for j from i-1 thru 1 step -1 do a:rowop(a,j,i,a[j,i])),
a)$
a: matrix([12,-27,36,44,59],
[26,41,-54,24,23],
[33,70,59,15,-68],
[43,16,29,-52,-61],
[-43,20,71,88,11])$
rref(a);
matrix([1,0,0,0,1/2],[0,1,0,0,-1],[0,0,1,0,-1/2],[0,0,0,1,1],[0,0,0,0,0])
Nim
Using rationals
To avoid rounding issues, we can use rationals and convert to floats only at the end.
import rationals, strutils
type Fraction = Rational[int]
const Zero: Fraction = 0 // 1
type Matrix[M, N: static Positive] = array[M, array[N, Fraction]]
func toMatrix[M, N: static Positive](a: array[M, array[N, int]]): Matrix[M, N] =
## Convert a matrix of integers to a matrix of integer fractions.
for i in 0..<M:
for j in 0..<N:
result[i][j] = a[i][j] // 1
func transformToRref(mat: var Matrix) =
## Transform the given matrix to reduced row echelon form.
var lead = 0
for r in 0..<mat.M:
if lead >= mat.N: return
var i = r
while mat[i][lead] == Zero:
inc i
if i == mat.M:
i = r
inc lead
if lead == mat.N: return
swap mat[i], mat[r]
if (let d = mat[r][lead]; d) != Zero:
for item in mat[r].mitems:
item /= d
for i in 0..<mat.M:
if i != r:
let m = mat[i][lead]
for c in 0..<mat.N:
mat[i][c] -= mat[r][c] * m
inc lead
proc `$`(mat: Matrix): string =
## Display a matrix.
for row in mat:
var line = ""
for val in row:
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
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
class RowEchelon {
function : Main(args : String[]) ~ Nil {
matrix := [
[1, 2, -1, -4 ]
[2, 3, -1, -11 ]
[-2, 0, -3, 22]
];
matrix := Rref(matrix);
sizes := matrix->Size();
for(i := 0; i < sizes[0]; i += 1;) {
for(j := 0; j < sizes[1]; j += 1;) {
IO.Console->Print(matrix[i,j])->Print(",");
};
IO.Console->PrintLine();
};
}
function : native : Rref(matrix : Int[,]) ~ Int[,] {
lead := 0;
sizes := matrix->Size();
rowCount := sizes[0];
columnCount := sizes[1];
for(r := 0; r < rowCount; r+=1;) {
if (columnCount <= lead) {
break;
};
i := r;
while(matrix[i, lead] = 0) {
i+=1;
if (i = rowCount) {
i := r;
lead += 1;
if (columnCount = lead) {
lead-=1;
break;
};
};
};
for (j := 0; j < columnCount; j+=1;) {
temp := matrix[r, j];
matrix[r, j] := matrix[i, j];
matrix[i, j] := temp;
};
div := matrix[r, lead];
for(j := 0; j < columnCount; j+=1;) {
matrix[r, j] /= div;
};
for(j := 0; j < rowCount; j+=1;) {
if (j <> r) {
sub := matrix[j, lead];
for (k := 0; k < columnCount; k+=1;) {
matrix[j, k] -= sub * matrix[r, k];
};
};
};
lead+=1;
};
return matrix;
}
}OCaml
let swap_rows m i j =
let tmp = m.(i) in
m.(i) <- m.(j);
m.(j) <- tmp;
;;
let rref m =
try
let lead = ref 0
and rows = Array.length m
and cols = Array.length m.(0) in
for r = 0 to pred rows do
if cols <= !lead then
raise Exit;
let i = ref r in
while m.(!i).(!lead) = 0 do
incr i;
if rows = !i then begin
i := r;
incr lead;
if cols = !lead then
raise Exit;
end
done;
swap_rows m !i r;
let lv = m.(r).(!lead) in
m.(r) <- Array.map (fun v -> v / lv) m.(r);
for i = 0 to pred rows do
if i <> r then
let lv = m.(i).(!lead) in
m.(i) <- Array.mapi (fun i iv -> iv - lv * m.(r).(i)) m.(i);
done;
incr lead;
done
with Exit -> ()
;;
let () =
let m =
[| [| 1; 2; -1; -4 |];
[| 2; 3; -1; -11 |];
[| -2; 0; -3; 22 |]; |]
in
rref m;
Array.iter (fun row ->
Array.iter (fun v ->
Printf.printf " %d" v
) row;
print_newline()
) m
Another implementation:
let rref m =
let nr, nc = Array.length m, Array.length m.(0) in
let add r s k =
for i = 0 to nc-1 do m.(r).(i) <- m.(r).(i) +. m.(s).(i)*.k done in
for c = 0 to min (nc-1) (nr-1) do
for r = c+1 to nr-1 do
if abs_float m.(c).(c) < abs_float m.(r).(c) then
let v = m.(r) in (m.(r) <- m.(c); m.(c) <- v)
done;
let t = m.(c).(c) in
if t <> 0.0 then
begin
for r = 0 to nr-1 do if r <> c then add r c (-.m.(r).(c)/.t) done;
for i = 0 to nc-1 do m.(c).(i) <- m.(c).(i)/.t done
end
done;;
let mat = [|
[| 1.0; 2.0; -.1.0; -.4.0;|];
[| 2.0; 3.0; -.1.0; -.11.0;|];
[|-.2.0; 0.0; -.3.0; 22.0;|]
|] in
let pr v = Array.iter (Printf.printf " %9.4f") v; print_newline() in
let show = Array.iter pr in
show mat;
print_newline();
rref mat;
show mat
Octave
A = [ 1, 2, -1, -4; 2, 3, -1, -11; -2, 0, -3, 22];
refA = rref(A);
disp(refA);
PARI/GP
PARI has a built-in matrix type, but no commands for row-echelon form. This is a basic one implementing Gauss-Jordan reduction.
matrref(M)=
{
my(s=matsize(M),t=s[1]);
for(i=1,s[2],
if(M[t,i]==0, next);
M[t,] /= M[t,i];
for(j=1,t-1,
M[j,] -= M[j,i]*M[t,]
);
for(j=t+1,s[1],
M[j,] -= M[j,i]*M[t,]
);
if(t--<1,break)
);
M;
}
addhelp(matrref, "matrref(M): Returns the reduced row-echelon form of the matrix M.");A faster, dimension-limited one can be constructed from the built-in matsolve command:
rref(M)={
my(d=matsize(M));
if(d[1]+1 != d[2], error("Bad size in rref"), d=d[1]);
concat(matid(d), matsolve(matrix(d,d,x,y,M[x,y]), M[,d+1]))
};Example:
rref([1,2,-1,-4;2,3,-1,-11;-2,0,-3,22])- Output:
%1 = [1 0 0 -8] [0 1 0 1] [0 0 1 -2]
Perl
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
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
<?php
function rref($matrix)
{
$lead = 0;
$rowCount = count($matrix);
if ($rowCount == 0)
return $matrix;
$columnCount = 0;
if (isset($matrix[0])) {
$columnCount = count($matrix[0]);
}
for ($r = 0; $r < $rowCount; $r++) {
if ($lead >= $columnCount)
break; {
$i = $r;
while ($matrix[$i][$lead] == 0) {
$i++;
if ($i == $rowCount) {
$i = $r;
$lead++;
if ($lead == $columnCount)
return $matrix;
}
}
$temp = $matrix[$r];
$matrix[$r] = $matrix[$i];
$matrix[$i] = $temp;
} {
$lv = $matrix[$r][$lead];
for ($j = 0; $j < $columnCount; $j++) {
$matrix[$r][$j] = $matrix[$r][$j] / $lv;
}
}
for ($i = 0; $i < $rowCount; $i++) {
if ($i != $r) {
$lv = $matrix[$i][$lead];
for ($j = 0; $j < $columnCount; $j++) {
$matrix[$i][$j] -= $lv * $matrix[$r][$j];
}
}
}
$lead++;
}
return $matrix;
}
?>
PicoLisp
(de reducedRowEchelonForm (Mat)
(let (Lead 1 Cols (length (car Mat)))
(for (X Mat X (cdr X))
(NIL
(loop
(T (seek '((R) (n0 (get R 1 Lead))) X)
@ )
(T (> (inc 'Lead) Cols)) ) )
(xchg @ X)
(let D (get X 1 Lead)
(map
'((R) (set R (/ (car R) D)))
(car X) ) )
(for Y Mat
(unless (== Y (car X))
(let N (- (get Y Lead))
(map
'((Dst Src)
(inc Dst (* N (car Src))) )
Y
(car X) ) ) ) )
(T (> (inc 'Lead) Cols)) ) )
Mat )- Output:
(reducedRowEchelonForm '(( 1 2 -1 -4) ( 2 3 -1 -11) (-2 0 -3 22)) ) -> ((1 0 0 -8) (0 1 0 1) (0 0 1 -2))
Python
def ToReducedRowEchelonForm( M):
if not M: return
lead = 0
rowCount = len(M)
columnCount = len(M[0])
for r in range(rowCount):
if lead >= columnCount:
return
i = r
while M[i][lead] == 0:
i += 1
if i == rowCount:
i = r
lead += 1
if columnCount == lead:
return
M[i],M[r] = M[r],M[i]
lv = M[r][lead]
M[r] = [ mrx / float(lv) for mrx in M[r]]
for i in range(rowCount):
if i != r:
lv = M[i][lead]
M[i] = [ iv - lv*rv for rv,iv in zip(M[r],M[i])]
lead += 1
mtx = [
[ 1, 2, -1, -4],
[ 2, 3, -1, -11],
[-2, 0, -3, 22],]
ToReducedRowEchelonForm( mtx )
for rw in mtx:
print ', '.join( (str(rv) for rv in rw) )
R
rref <- function(m) {
pivot <- 1
norow <- nrow(m)
nocolumn <- ncol(m)
for(r in 1:norow) {
if ( nocolumn <= pivot ) break;
i <- r
while( m[i,pivot] == 0 ) {
i <- i + 1
if ( norow == i ) {
i <- r
pivot <- pivot + 1
if ( nocolumn == pivot ) return(m)
}
}
trow <- m[i, ]
m[i, ] <- m[r, ]
m[r, ] <- trow
m[r, ] <- m[r, ] / m[r, pivot]
for(i in 1:norow) {
if ( i != r )
m[i, ] <- m[i, ] - m[r, ] * m[i, pivot]
}
pivot <- pivot + 1
}
return(m)
}
m <- matrix(c(1, 2, -1, -4,
2, 3, -1, -11,
-2, 0, -3, 22), 3, 4, byrow=TRUE)
print(m)
print(rref(m))
Racket
#lang racket
(require math)
(define (reduced-echelon M)
(matrix-row-echelon M #t #t))
(reduced-echelon
(matrix [[1 2 -1 -4]
[2 3 -1 -11]
[-2 0 -3 22]]))
- Output:
(mutable-array
#[#[1 0 0 -8]
#[0 1 0 1]
#[0 0 1 -2]])
Raku
(formerly Perl 6)
Following pseudocode
sub rref (@m) {
my ($lead, $rows, $cols) = 0, @m, @m[0];
for ^$rows -> $r {
return @m unless $lead < $cols;
my $i = $r;
until @m[$i;$lead] {
next unless ++$i == $rows;
$i = $r;
return @m if ++$lead == $cols;
}
@m[$i, $r] = @m[$r, $i] if $r != $i;
@m[$r] »/=» $ = @m[$r;$lead];
for ^$rows -> $n {
next if $n == $r;
@m[$n] »-=» @m[$r] »×» (@m[$n;$lead] // 0);
}
++$lead;
}
@m
}
sub rat-or-int ($num) {
return $num unless $num ~~ Rat;
return $num.narrow if $num.narrow ~~ Int;
$num.nude.join: '/';
}
sub say_it ($message, @array) {
say "\n$message";
$_».&rat-or-int.fmt(" %5s").say for @array;
}
my @M = (
[ # base test case
[ 1, 2, -1, -4 ],
[ 2, 3, -1, -11 ],
[ -2, 0, -3, 22 ],
],
[ # mix of number styles
[ 3, 0, -3, 1 ],
[ .5, 3/2, -3, -2 ],
[ .2, 4/5, -1.6, .3 ],
],
[ # degenerate case
[ 1, 2, 3, 4, 3, 1],
[ 2, 4, 6, 2, 6, 2],
[ 3, 6, 18, 9, 9, -6],
[ 4, 8, 12, 10, 12, 4],
[ 5, 10, 24, 11, 15, -4],
],
[ # larger matrix
[1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0],
[0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0],
[0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0],
[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1],
[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0],
]
);
for @M -> @matrix {
say_it( 'Original Matrix', @matrix );
say_it( 'Reduced Row Echelon Form Matrix', rref(@matrix) );
say "\n";
}
Raku handles rational numbers internally as a ratio of two integers to maintain precision. For some situations it is useful to return the ratio rather than the floating point result.
- Output:
Original Matrix
1 2 -1 -4
2 3 -1 -11
-2 0 -3 22
Reduced Row Echelon Form Matrix
1 0 0 -8
0 1 0 1
0 0 1 -2
Original Matrix
3 0 -3 1
1/2 3/2 -3 -2
1/5 4/5 -8/5 3/10
Reduced Row Echelon Form Matrix
1 0 0 -41/2
0 1 0 -217/6
0 0 1 -125/6
Original Matrix
1 2 3 4 3 1
2 4 6 2 6 2
3 6 18 9 9 -6
4 8 12 10 12 4
5 10 24 11 15 -4
Reduced Row Echelon Form Matrix
1 2 0 0 3 4
0 0 1 0 0 -1
0 0 0 1 0 0
0 0 0 0 0 0
0 0 0 0 0 0
Original Matrix
1 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0
1 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0
1 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0
0 1 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0
0 1 0 0 0 0 0 0 1 0 0 -1 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 0
0 0 1 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0
0 0 1 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0
0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 -1 0 0
0 0 0 1 0 0 0 0 0 1 0 0 -1 0 0 0 0 0
0 0 0 0 1 0 0 1 0 0 0 0 0 -1 0 0 0 0
0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 -1 0
0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 -1 0 0
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1
0 0 0 0 0 1 0 0 0 0 1 0 0 0 -1 0 0 0
Reduced Row Echelon Form Matrix
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17/39
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4/13
0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20/39
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 28/39
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 19/39
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 8/39
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 11/39
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1/3
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 20/39
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 25/39
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 28/39
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 10/13
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 20/39
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 32/39
Row operations, procedural code
Re-implemented as elementary matrix row operations. Follow links for background on row operations and reduced row echelon form
sub scale-row ( @M, \scale, \r ) { @M[r] = @M[r] »×» scale }
sub shear-row ( @M, \scale, \r1, \r2 ) { @M[r1] = @M[r1] »+» ( @M[r2] »×» scale ) }
sub reduce-row ( @M, \r, \c ) { scale-row @M, 1/@M[r;c], r }
sub clear-column ( @M, \r, \c ) { shear-row @M, -@M[$_;c], $_, r for @M.keys.grep: * != r }
my @M = (
[< 1 2 -1 -4 >],
[< 2 3 -1 -11 >],
[< -2 0 -3 22 >],
);
my $column-count = @M[0];
my $col = 0;
for @M.keys -> $row {
reduce-row( @M, $row, $col );
clear-column( @M, $row, $col );
last if ++$col == $column-count;
}
say @$_».fmt(' %4g') for @M;
- Output:
[ 1 0 0 -8] [ 0 1 0 1] [ 0 0 1 -2]
Row operations, object-oriented code
The same code as previous section, recast into OO. Also, scale and shear are recast as unscale and unshear, which fit the problem better.
class Matrix is Array {
method unscale-row ( @M: \scale, \row ) { @M[row] = @M[row] »/» scale }
method unshear-row ( @M: \scale, \r1, \r2 ) { @M[r1] = @M[r1] »-» @M[r2] »×» scale }
method reduce-row ( @M: \row, \col ) { @M.unscale-row( @M[row;col], row ) }
method clear-column ( @M: \row, \col ) { @M.unshear-row( @M[$_;col], $_, row ) for @M.keys.grep: * != row }
method reduced-row-echelon-form ( @M: ) {
my $column-count = @M[0];
my $col = 0;
for @M.keys -> $row {
@M.reduce-row( $row, $col );
@M.clear-column( $row, $col );
return if ++$col == $column-count;
}
}
}
my $M = Matrix.new(
[< 1 2 -1 -4 >],
[< 2 3 -1 -11 >],
[< -2 0 -3 22 >],
);
$M.reduced-row-echelon-form;
say @$_».fmt(' %4g') for @$M;
- Output:
[ 1 0 0 -8] [ 0 1 0 1] [ 0 0 1 -2]
REXX
Reduced Row Echelon Form (a.k.a. row canonical form) of a matrix, with optimization added.
/*REXX pgm performs Reduced Row Echelon Form (RREF), AKA row canonical form on a matrix)*/
cols= 0; w= 0; @. =0 /*max cols in a row; max width; matrix.*/
mat.=; mat.1= ' 1 2 -1 -4 '
mat.2= ' 2 3 -1 -11 '
mat.3= ' -2 0 -3 22 '
do r=1 until mat.r==''; _=mat.r /*build @.row.col from (matrix) mat.X*/
do c=1 until _=''; parse var _ @.r.c _
w= max(w, length(@.r.c) + 1) /*find the maximum width of an element.*/
end /*c*/
cols= max(cols, c) /*save the maximum number of columns. */
end /*r*/
rows= r-1 /*adjust the row count (from DO loop). */
call showMat 'original matrix' /*display the original matrix──►screen.*/
!= 1 /*set the working column pointer to 1.*/
/* ┌──────────────────────◄────────────────◄──── Reduced Row Echelon Form on matrix.*/
do r=1 for rows while cols>! /*begin to perform the heavy lifting. */
j= r /*use a subsitute index for the DO loop*/
do while @.j.!==0; j= j + 1
if j==rows then do; j= r; != ! + 1; if cols==! then leave r; end
end /*while*/
/* [↓] swap rows J,R (but not if same)*/
do _=1 for cols while j\==r; parse value @.r._ @.j._ with @.j._ @._._
end /*_*/
?= @.r.!
do d=1 for cols while ?\=1; @.r.d= @.r.d / ?
end /*d*/ /* [↑] divide row J by @.r.p ──unless≡1*/
do k=1 for rows; ?= @.k.! /*subtract (row K) @.r.s from row K.*/
if k==r | ?=0 then iterate /*skip if row K is the same as row R.*/
do s=1 for cols; @.k.s= @.k.s - ? * @.r.s
end /*s*/
end /*k*/ /* [↑] for the rest of numbers in row.*/
!= !+1 /*bump the working column pointer. */
end /*r*/
call showMat 'matrix RREF' /*display the reduced row echelon form.*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
showMat: parse arg title; say; say center(title, 3 + (cols+1) * w, '─'); say
do r=1 for rows; _=
do c=1 for cols
if @.r.c=='' then do; say "***error*** matrix element isn't defined:"
say 'row' r", column" c'.'; exit 13
end
_= _ right(@.r.c, w)
end /*c*/
say _ /*display a matrix row to the terminal.*/
end /*r*/; return
- output when using the default (internal) input:
────original matrix────
1 2 -1 -4
2 3 -1 -11
-2 0 -3 22
──────matrix RREF──────
1 0 0 -8
0 1 0 1
0 0 1 -2
Ring
# Project : Reduced row echelon form
matrix = [[1, 2, -1, -4],
[2, 3, -1, -11],
[ -2, 0, -3, 22]]
ref(matrix)
for row = 1 to 3
for col = 1 to 4
if matrix[row][col] = -0
see "0 "
else
see "" + matrix[row][col] + " "
ok
next
see nl
next
func ref(m)
nrows = 3
ncols = 4
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
nextOutput:
1 0 0 -8 0 1 0 1 0 0 1 -2
RPL
The RREF built-in intruction is available for HP-48G and newer models.
[[1 2 -1 -4] [2 3 -1 -11] [-2 0 -3 22]] RREF
- Output:
1: [[ 1 0 0 -8 ]
[ 0 1 0 1 ]
[ 0 0 1 -2 ]]
Ruby
# 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
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
sage: m = matrix(ZZ, [[1,2,-1,-4],[2,3,-1,-11],[-2,0,-3,22]])
sage: m.rref()
[ 1 0 0 -8]
[ 0 1 0 1]
[ 0 0 1 -2]
Scheme
Works with: Scheme version 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
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
func rref (M) {
var (j, rows, cols) = (0, M.len, M[0].len)
for r in (^rows) {
j < cols || return M
var i = r
while (!M[i][j]) {
++i == rows || next
i = r
++j == cols && return M
}
M[i, r] = M[r, i] if (r != i)
M[r] = (M[r] »/» M[r][j])
for n in (^rows) {
next if (n == r)
M[n] = (M[n] »-« (M[r] »*» M[n][j]))
}
++j
}
return M
}
func say_it (message, array) {
say "\n#{message}";
array.each { |row|
say row.map { |n| " %5s" % n.as_rat }.join
}
}
var M = [
[ # base test case
[ 1, 2, -1, -4 ],
[ 2, 3, -1, -11 ],
[ -2, 0, -3, 22 ],
],
[ # mix of number styles
[ 3, 0, -3, 1 ],
[ .5, 3/2, -3, -2 ],
[ .2, 4/5, -1.6, .3 ],
],
[ # degenerate case
[ 1, 2, 3, 4, 3, 1],
[ 2, 4, 6, 2, 6, 2],
[ 3, 6, 18, 9, 9, -6],
[ 4, 8, 12, 10, 12, 4],
[ 5, 10, 24, 11, 15, -4],
],
];
M.each { |matrix|
say_it('Original Matrix', matrix);
say_it('Reduced Row Echelon Form Matrix', rref(matrix));
say '';
}
- Output:
Original Matrix
1 2 -1 -4
2 3 -1 -11
-2 0 -3 22
Reduced Row Echelon Form Matrix
1 0 0 -8
0 1 0 1
0 0 1 -2
Original Matrix
3 0 -3 1
1/2 3/2 -3 -2
1/5 4/5 -8/5 3/10
Reduced Row Echelon Form Matrix
1 0 0 -41/2
0 1 0 -217/6
0 0 1 -125/6
Original Matrix
1 2 3 4 3 1
2 4 6 2 6 2
3 6 18 9 9 -6
4 8 12 10 12 4
5 10 24 11 15 -4
Reduced Row Echelon Form Matrix
1 2 0 0 3 4
0 0 1 0 0 -1
0 0 0 1 0 0
0 0 0 0 0 0
0 0 0 0 0 0
Swift
var lead = 0
for r in 0..<rows {
if (cols <= lead) { break }
var i = r
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
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
Builtin function: rref()
rref([[1,2,-1,-4][2,3,-1,-11][-2,0,-3,22]])- Output:
[[1 0 0 -8]
[0 1 0 1]
[0 0 1 -2]]
TI-89 BASIC
rref([1,2,–1,–4; 2,3,–1,–11; –2,0,–3,22])Output (in prettyprint mode):
Matrices can also be stored in variables, and entered interactively using the Data/Matrix Editor.
Ursala
The most convenient representation for a matrix in Ursala is as a list of lists. Several auxiliary functions are defined to make this task more manageable. The pivot function reorders the rows to position the first column entry with maximum magnitude in the first row. The descending function is a second order function abstracting the pattern of recursion down the major diagonal of a matrix. The reflect function allows the code for the first phase in the reduction to be reused during the upward traversal by appropriately permuting the rows and columns. The row_reduce function adds a multiple of the top row to each subsequent row so as to cancel the first column. These are all combined in the main rref function.
#import std
#import flo
pivot = -<x fleq+ abs~~bh
descending = ~&a^&+ ^|ahPathS2fattS2RpC/~&
reflect = ~&lxPrTSx+ *iiD ~&l-~brS+ zipp0
row_reduce = ^C/vid*hhiD *htD minus^*p/~&r times^*D/vid@bh ~&l
rref = reflect+ (descending row_reduce)+ reflect+ descending row_reduce+ pivot
#show+
test =
printf/*=*'%8.4f' rref <
<1.,2.,-1.,-4.>,
<2.,3.,-1.,-11.>,
<-2.,0.,-3.,22.>>- Output:
1.0000 0.0000 0.0000 -8.0000 0.0000 1.0000 0.0000 1.0000 0.0000 0.0000 1.0000 -2.0000
An alternative and more efficient solution is to use the msolve library function as shown, which interfaces with the lapack library if available. This solution is applicable only if the input is a non-singular augmented square matrix.
#import lin
rref = @ySzSX msolve; ^plrNCTS\~& ~&iiDlSzyCK9+ :/1.+ 0.!*tVBA
Private Function ToReducedRowEchelonForm(M As Variant) As Variant
Dim lead As Integer: lead = 0
Dim rowCount As Integer: rowCount = UBound(M)
Dim columnCount As Integer: columnCount = UBound(M(0))
Dim i As Integer
For r = 0 To rowCount
If lead >= columnCount Then
Exit For
End If
i = r
Do While M(i)(lead) = 0
i = i + 1
If i = rowCount Then
i = r
lead = lead + 1
If lead = columnCount Then
Exit For
End If
End If
Loop
Dim tmp As Variant
tmp = M(r)
M(r) = M(i)
M(i) = tmp
If M(r)(lead) <> 0 Then
div = M(r)(lead)
For t = LBound(M(r)) To UBound(M(r))
M(r)(t) = M(r)(t) / div
Next t
End If
For j = 0 To rowCount
If j <> r Then
subt = M(j)(lead)
For t = LBound(M(j)) To UBound(M(j))
M(j)(t) = M(j)(t) - subt * M(r)(t)
Next t
End If
Next j
lead = lead + 1
Next r
ToReducedRowEchelonForm = M
End Function
Public Sub main()
r = ToReducedRowEchelonForm(Array( _
Array(1, 2, -1, -4), _
Array(2, 3, -1, -11), _
Array(-2, 0, -3, 22)))
For i = LBound(r) To UBound(r)
Debug.Print Join(r(i), vbTab)
Next i
End Sub
- Output:
1 0 0 -8 0 1 0 1 0 0 1 -2
Visual FoxPro
Translation of Fortran.
CLOSE DATABASES ALL
LOCAL lnRows As Integer, lnCols As Integer, lcSafety As String
LOCAL ARRAY matrix[1]
lcSafety = SET("Safety")
SET SAFETY OFF
CLEAR
CREATE CURSOR results (c1 B(6), c2 B(6), c3 B(6), c4 B(6))
CREATE CURSOR curs1(c1 I, c2 I, c3 I, c4 I)
INSERT INTO curs1 VALUES (1,2,-1,-4)
INSERT INTO curs1 VALUES (2,3,-1,-11)
INSERT INTO curs1 VALUES (-2,0,-3,22)
lnRows = RECCOUNT() && 3
lnCols = FCOUNT() && 4
SELECT * FROM curs1 INTO ARRAY matrix
IF RREF(@matrix, lnRows, lnCols)
SELECT results
APPEND FROM ARRAY matrix
BROWSE NORMAL IN SCREEN
ENDIF
SET SAFETY &lcSafety
FUNCTION RREF(mat, tnRows As Integer, tnCols As Integer) As Boolean
LOCAL lnPivot As Integer, i As Integer, r As Integer, j As Integer, ;
p As Double. llResult As Boolean, llExit As Boolean
llResult = .T.
llExit = .F.
lnPivot = 1
FOR r = 1 TO tnRows
IF lnPivot > tnCols
EXIT
ENDIF
i = r
DO WHILE mat[i,lnPivot] = 0
i = i + 1
IF i = tnRows
i = r
lnPivot = lnPivot + 1
IF lnPivot > tnCols
llExit = .T.
EXIT
ENDIF
ENDIF
ENDDO
IF llExit
EXIT
ENDIF
ASwapRows(@mat, i, r)
p = mat[r,lnPivot]
IF p # 0
FOR j = 1 TO tnCols
mat[r,j] = mat[r,j]/p
ENDFOR
ELSE
? "Divison by zero."
llResult = .F.
EXIT
ENDIF
FOR i = 1 TO tnRows
IF i # r
p = mat[i,lnPivot]
FOR j = 1 TO tnCols
mat[i,j] = mat[i,j] - mat[r,j]*p
ENDFOR
ENDIF
ENDFOR
lnPivot = lnPivot + 1
ENDFOR
RETURN llResult
ENDFUNC
PROCEDURE ASwapRows(arr, tnRow1 As Integer, tnRow2 As Integer)
*!* Interchange rows tnRow1 and tnRow2 of array arr.
LOCAL n As Integer
n = ALEN(arr,2)
LOCAL ARRAY tmp[1,n]
STORE 0 TO tmp
ACPY2(@arr, @tmp, tnRow1, 1)
ACPY2(@arr, @arr, tnRow2, tnRow1)
ACPY2(@tmp, @arr, 1, tnRow2)
ENDPROC
PROCEDURE ACPY2(m1, m2, tnSrcRow As Integer, tnDestRow As Integer)
*!* Copy m1[tnSrcRow,*] to m2[tnDestRow,*]
*!* m1 and m2 must have the same number of columns.
LOCAL n As Integer, e1 As Integer, e2 As Integer
n = ALEN(m1,2)
e1 = AELEMENT(m1,tnSrcRow,1)
e2 = AELEMENT(m2,tnDestRow,1)
ACOPY(m1, m2, e1, n, e2)
ENDPROC
- Output:
C1 C2 C3 C4 1.000000 0.000000 0.000000 -8.000000 0.000000 1.000000 0.000000 1.000000 0.000000 0.000000 1.000000 -2.000000
Wren
The above module has a method for this built in as it's needed to implement matrix inversion using the Gauss-Jordan method. However, as in the example here, it's not just restricted to square matrices.
import "./matrix" for Matrix
import "./fmt" for Fmt
var m = Matrix.new([
[ 1, 2, -1, -4],
[ 2, 3, -1, -11],
[-2, 0, -3, 22]
])
System.print("Original:\n")
Fmt.mprint(m, 3, 0)
System.print("\nRREF:\n")
m.toReducedRowEchelonForm
Fmt.mprint(m, 3, 0)
- Output:
Original: | 1 2 -1 -4| | 2 3 -1 -11| | -2 0 -3 22| RREF: | 1 0 0 -8| | 0 1 0 1| | 0 0 1 -2|
XPL0
proc ReducedRowEchelonForm(M, Rows, Cols);
\Replace M with its reduced row echelon form
real M; int Rows, Cols;
int Lead, R, C, I;
real RLead, ILead, T;
[Lead:= 0;
for R:= 0 to Rows-1 do
[if Lead >= Cols then return;
I:= R;
while M(I, Lead) = 0. do
[I:= I+1;
if I = Rows-1 then
[I:= R;
Lead:= Lead+1;
if Lead = Cols-1 then return;
];
];
\Swap rows I and R
T:= M(I); M(I):= M(R); M(R):= T;
if M(R, Lead) # 0. then
\Divide row R by M[R, Lead]
[RLead:= M(R, Lead);
for C:= 0 to Cols-1 do
M(R, C):= M(R, C) / RLead;
];
for I:= 0 to Rows-1 do
[if I # R then
\Subtract M[I, Lead] multiplied by row R from row I
[ILead:= M(I, Lead);
for C:= 0 to Cols-1 do
M(I, C):= M(I, C) - ILead * M(R, C);
];
];
Lead:= Lead+1;
];
];
real M;
int R, C;
[M:= [ [ 1., 2., -1., -4.],
[ 2., 3., -1.,-11.],
[-2., 0., -3., 22.] ];
ReducedRowEchelonForm(M, 3, 4);
Format(4,1);
for R:= 0 to 3-1 do
[for C:= 0 to 4-1 do
RlOut(0, M(R,C));
CrLf(0);
];
]- Output:
1.0 0.0 0.0 -8.0 0.0 1.0 0.0 1.0 0.0 0.0 1.0 -2.0
Yabasic
// Rosetta Code problem: https://rosettacode.org/wiki/Reduced_row_echelon_form
// by Jjuanhdez, 06/2022
dim matrix (3, 4)
matrix(1, 1) = 1 : matrix(1, 2) = 2 : matrix(1, 3) = -1 : matrix(1, 4) = -4
matrix(2, 1) = 2 : matrix(2, 2) = 3 : matrix(2, 3) = -1 : matrix(2, 4) = -11
matrix(3, 1) = -2 : matrix(3, 2) = 0 : matrix(3, 3) = -3 : matrix(3, 4) = 22
RREF (matrix())
for row = 1 to 3
for col = 1 to 4
if matrix(row, col) = 0 then
print "0", chr$(9);
else
print matrix(row, col), chr$(9);
end if
next
print
next
end
sub RREF(x())
local nrows, ncols, lead, r, i, j, n
nrows = arraysize(matrix(), 1) //3
ncols = arraysize(matrix(), 2) //4
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 subzkl
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
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