LU factorization with decreasing elements on the main diagonal of U

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Sara 2023-9-25

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评论： Bruno Luong 2023-9-25

Does

select P so that

is decreasing?

If not, how can I request it?

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Bruno Luong 2023-9-25

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https://ww2.mathworks.cn/matlabcentral/answers/2025312-lu-factorization-with-decreasing-elements-on-the-main-diagonal-of-u#answer_1317392

编辑：Bruno Luong 2023-9-25

在 MATLAB Online 中打开

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