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function [L U P] = LU_pivot (A)
[n _] = size (A);
[L P U] = deal (eye (n), eye (n), A);
for k = 1:n
[pivot m] = max (abs (U(k:n, k)));
m = m + k - 1;
if (m ~= k)
U([m k], :) = U([k m], :); % interchange rows m and k in U
P([m k], :) = P([k m], :); % interchange rows m and k in P
if k >= 2; % very important point
% interchange rows m and k in columns 1:k-1 of L
L([m k], 1:k-1) = L([k m], 1:k-1);
end
end
for i = k + 1 : n
L(i, k) = U(i, k) / U(k, k);
U(i, :) -= L(i, k)*U(k, :);
end
end
end
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