Reduced row echelon form: Difference between revisions
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=={{header|Python}}== |
=={{header|Python}}== |
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This closely follows the pseudocode given |
This closely follows the pseudocode given. |
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<lang python>def ToReducedRowEchelonForm( M): |
<lang python>def ToReducedRowEchelonForm( M): |
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if not M: return |
if not M: return |
Revision as of 06:49, 1 September 2009
This page uses content from Wikipedia. The original article was at Rref#Pseudocode. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance) |
You are encouraged to solve this task according to the task description, using any language you may know.
Show how to compute the reduced row echelon form (a.k.a. row canonical form) of a matrix. The matrix can be stored in any datatype that is convenient (for most languages, this will probably be a two-dimensional array). Built-in functions or this pseudocode (from Wikipedia) may be used:
function ToReducedRowEchelonForm(Matrix M) is 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 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
ALGOL 68
<lang algol>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, $l$))</lang> 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))
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.
<lang cpp>
- 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 // dfault 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_typ 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;
} </lang> Output:
1 0 0 -8 -0 1 0 1 -0 -0 1 -2
D
Be warned that this is definitely not the most efficient implementation, as it was coded from a thought process. <lang d> import std.stdio; real[][]example = [
[1.0,2,-1,-4], [2,3,-1,-11], [-2,0,-3,22]
];
real getFirstNZ(real[]row) {
int i = getOrder(row); if (i == row.length) return 0.0; return row[i];
}
int getOrder(real[]row) {
foreach(i,ele;row) { if (ele != 0) return i; } return row.length;
}
real[][]rref(real[][]input) {
real[][]ret; // copy the input matrix foreach(row;input) { real[]currow = row.dup; // fix leading digit to be positive to make things easier, // by inverting the signs of all elements of the row if neccesary if (getFirstNZ(currow) < 0) { currow[] *= -1; } // normalize to the first element currow[] /= getFirstNZ(currow); ret ~= currow; } ret.sort.reverse; // get the matrix into reduced row echelon form for(int i = 0;i<ret.length;i++) { int order = getOrder(ret[i]); if (order == ret[i].length) { // this row has no non-zero digits in it, so we're done, // since any subsequent rows will also have all 0s, because of the sorting done earlier break; } foreach(j,ref row;ret) { // prevent us from erasing the current row if (i == j) { continue; } // see if we need to modify the current row if (row[order] != 0.0) { // cancel rows with awesome vector ops! real[]tmp = ret[i].dup; tmp[]*= -row[order]; row[]+=tmp[]; // ensure row is renormalized int corder = getOrder(row); if (corder < row.length && row[corder] != 1) { row[]/=row[corder]; } } } } return ret;
}
void main() {
auto result = rref(example); writefln(result);
} </lang>
Fortran
<lang 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</lang>
<lang fortran>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</lang>
Haskell
This program was produced by translating from the Python and gradually refactoring the result into a more functional style.
<lang haskell>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</lang>
Java
<lang java>public class RREF {
public static void toRREF(double[][] M) { int rowCount = M.length; if (rowCount == 0) return;
int columnCount = M[0].length;
int lead = 0; for (int r = 0; r < rowCount; r++) { if (lead >= columnCount) break; { int i = r; while (M[i][lead] == 0) { i++; if (i == rowCount) { i = r; lead++; if (lead == columnCount) return; } } double[] temp = M[r]; M[r] = M[i]; M[i] = temp; }
{ double lv = M[r][lead]; for (int j = 0; j < columnCount; j++) M[r][j] /= lv; }
for (int i = 0; i < rowCount; i++) { if (i != r) { double lv = M[i][lead]; for (int j = 0; j < columnCount; j++) M[i][j] -= lv * M[r][j]; } } lead++; } }
public static void main(String[] args) { double[][] mtx = { { 1, 2, -1, -4}, { 2, 3, -1,-11}, {-2, 0, -3, 22}};
toRREF(mtx);
for (double [] row : mtx) { for (double x : row) System.out.print(x + ", "); System.out.println(); } }
}</lang>
Mathematica
<lang Mathematica>
RowReduce[{{1, 2, -1, -4}, {2, 3, -1, -11}, {-2, 0, -3, 22}}]
</lang> gives back: <lang Mathematica>
{{1, 0, 0, -8}, {0, 1, 0, 1}, {0, 0, 1, -2}}
</lang>
OCaml
<lang 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</lang>
Octave
<lang octave>A = [ 1, 2, -1, -4; 2, 3, -1, -11; -2, 0, -3, 22]; refA = rref(A); disp(refA);</lang>
Perl
Note that the function defined here takes an array reference, not an array. <lang perl>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;}}</lang>
Python
This closely follows the pseudocode given. <lang 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 / 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) )
</lang>
R
<lang 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))</lang>
Ruby
<lang ruby>require 'rational'
- 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_rational(ary) # 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
def convert_to_rational(ary)
new = [] ary.each_index do |row| new << ary[row].collect {|elem| Rational(elem)} end new
end
- type should be one of :to_s, :to_i, :to_f, :to_r
def convert_to(ary, type)
new = [] ary.each_index do |row| new << ary[row].collect {|elem| elem.send(type)} end new
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].to_s}; puts ""}
end
mtx = [
[ 1, 2, -1, -4], [ 2, 3, -1,-11], [-2, 0, -3, 22]
] print_matrix convert_to(reduced_row_echelon_form(mtx), :to_s) puts ""
mtx = [
[ 1, 2, 3, 7], [-4, 7,-2, 7], [ 3, 3, 0, 7]
] reduced = reduced_row_echelon_form(mtx) print_matrix convert_to(reduced, :to_r) print_matrix convert_to(reduced, :to_f) </lang>
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 0.666666666666667 0.0 1.0 0.0 1.66666666666667 0.0 0.0 1.0 1.0
Tcl
Using utility procs defined at Matrix Transpose#Tcl <lang 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]</lang> outputs
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-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.
<lang Ursala>#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.>></lang>
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. <lang Ursala>
- import lin
rref = @ySzSX msolve; ^plrNCTS\~& ~&iiDlSzyCK9+ :/1.+ 0.!*t</lang>