LU factorization with decreasing elements on the main diagonal of U

Does select P so that is decreasing?

If not, how can I request it?

Answer 1

[Bruno Luong]

Obviously not

A=[1 10 9;
    5 1 9;
    2 8 1]

A = 3×3

1 10 9 5 1 9 2 8 1

[L,U,P]=lu(A)

L = 3×3

1.0000 0 0 0.2000 1.0000 0 0.4000 0.7755 1.0000

U = 3×3

5.0000 1.0000 9.0000 0 9.8000 7.2000 0 0 -8.1837

P = 3×3

0 1 0 1 0 0 0 0 1

But you can fix the progression of abs(diag(U)) in any arbitray decrasing sequance you want, just scale appropiately L.

Here I select the sequene of U(1,1).*2.^(-(1:n-1))

% Fix it, assuming A is not singular
Ukk = abs(U(1,1));
for k=2:size(U,1)
    s = Ukk/(2*abs(U(k,k)));
    U(k,:) = s*U(k,:);
    L(:,k) = L(:,k)/s;
    Ukk = abs(U(k,k));
end
L

L = 3×3

1.0000 0 0 0.2000 3.9200 0 0.4000 3.0400 6.5469

U

U = 3×3

5.0000 1.0000 9.0000 0 2.5000 1.8367 0 0 -1.2500

P'*L*U % close to A

ans = 3×3

1 10 9 5 1 9 2 8 1

In some sense the question of scaling U alone is trivial and sort of useless if you don't specify what L should be.

Note that MATLAB returns L such that diag(L) are 1.

[Sara]

Thank you.

I would like to leave the diagonal elements of L equal to 1 and I would like to order the diagonal elements of U in such a way that their modules are decreasing.

[Bruno Luong]

Then it is not possible in general.

If you need similar property, use qr with permutation.

[Bruno Luong]

I'll give you a simple counter example where no permutation can achieve what you ask for. The example is 2 x 2, there is 2 pemutations of 1:2, both them don't provide U with absolute values of the diagonal decreasing.

In both case the diagonal od U is [1,2]; which is non decreasing

Note that once you impose P, and diag(L)=1, the LU factorization is unique.

A = [1 0; 0 2]

A = 2×2

1 0 0 2

p = perms(1:2);
for k=1:size(p,1)
    P = eye(2);
    P = P(:,p(k,:));
    [L,U] = lu(P*A)
end

L = 2×2

0 1 1 0

U = 2×2

1 0 0 2

L = 2×2

1 0 0 1

U = 2×2

1 0 0 2

% and of course
[L,U,P] = lu(A)

L = 2×2

1 0 0 1

U = 2×2

1 0 0 2

P = 2×2

1 0 0 1

[Sara]

Thank you, you are right.

Last question, is there any option in matlab to generate a PLU factorization such that is decreasing without having to modify a lu factorization previously calculated by matlab?

[Bruno Luong]

No