Question

In Matlab create a rref() function from scratch given the matrix A

Answer 1

`Hey,

Note: If you have any queries related the answer please do comment. I would be very happy to resolve all your queries.

clc%clears screen
clear all%clears history
close all%closes all files
format long
clear all
clc
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MAKES RANDOM MATRIX A
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
m = 3;
A=randi([1,10],3,3);
BB=A;
[m,n] = size(A);
type='f';
tol=1e-5;
do_full = ~issparse(A) || strcmp(type,'f');
if do_full
% Loop over the entire matrix.
i = 1;
j = 1;
jb = [];
% t1 = clock;
while (i <= m) && (j <= n)
% Find value and index of largest element in the remainder of column j.
[p,k] = max(abs(A(i:m,j))); k = k+i-1;
if (p <= tol)
% The column is negligible, zero it out.
A(i:m,j) = 0; %(faster for sparse) %zeros(m-i+1,1);
j = j + 1;
else
% Remember column index
jb = [jb j];
% Swap i-th and k-th rows.
A([i k],j:n) = A([k i],j:n);
% Divide the pivot row by the pivot element.
Ai = A(i,j:n)/A(i,j);
% Subtract multiples of the pivot row from all the other rows.
A(:,j:n) = A(:,j:n) - A(:,j)*Ai;
A(i,j:n) = Ai;
i = i + 1;
j = j + 1;
end
end
else
% Non-pivoted Q-less QR decomposition computed by Matlab actually
% produces the right structure (similar to rref) to identify independent
% columns.
R = qr(A);
% i_dep = pivot columns = dependent variables
% = left-most non-zero column (if any) in each row
% indep_rows (binary vector) = non-zero rows of R
[indep_rows, i_dep] = max(R ~= 0, [], 2);
indep_rows = full(indep_rows); % probably more efficient
i_dep = i_dep(indep_rows);
i_indep = setdiff(1:n, i_dep);
% solve R(indep_rows, i_dep) x = R(indep_rows, i_indep)
% to eliminate all the i_dep columns
% (i.e. we want F(indep_rows, i_dep) = Identity)
F = sparse([],[],[], m, n);
F(indep_rows, i_indep) = R(indep_rows, i_dep) \ R(indep_rows, i_indep);
F(indep_rows, i_dep) = speye(length(i_dep));
% result
A = F;
jb = i_dep;
end
disp('RREF is');
disp(A);

Kindly revert for any queries

Thanks.