How to update cholesky decomposition of the normalized SPD matrix?

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Vlad Gorohov
Vlad Gorohov 2015-6-24
回答： SAI SRUJAN 2024-9-25
I try cholrank1 update ( wikipedia ) of the symmetric positive definite matrix:
function [L] = cholupdate(L,x)
p = length(x);
for k=1:p
r = sqrt(L(k,k)^2 + x(k)^2);
c = r / L(k, k);
s = x(k) / L(k, k);
L(k, k) = r;
L(k+1:p,k) = (L(k+1:p,k) + s*x(k+1:p)) / c;
x(k+1:p) = c*x(k+1:p) - s*L(k+1:p,k);
end
end
Get SPD matrix and factorize it:
H = J'*J
L = chol(H)';
L_updated = cholupdate(L, new_J_row');
It works well. But how can I modify algorithm when I need to do normalization of the SPD matrix?
% normalization
n = 1 ./ sqrt(diag(H));
Hn = diag(n) * H * diag(n);
Ln = chol(Hn)';
Ln_updated = ???

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回答（1 个）

SAI SRUJAN
SAI SRUJAN 2024-9-25
Hi Vlad,
Updating the Cholesky decomposition of a normalized symmetric positive definite (SPD) matrix involves a couple of additional steps compared to the standard update. When you normalize an SPD matrix, you are essentially scaling it, which affects both the matrix and its Cholesky factor.
Please go through the following code sample to proceed further,
% normalization
n = 1 ./ sqrt(diag(H));
Hn = diag(n) * H * diag(n);
Ln = chol(Hn)';
% To update with a new row new_J_row, first normalize the new row:
new_J_row_normalized = new_J_row .* n';
Ln_updated = cholupdate(Ln, new_J_row_normalized');
For a comprehensive understanding of the "chol" function in MATLAB, please refer to the following documentation.
https://www.mathworks.com/help/matlab/ref/chol.html
I hope this helps!

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