Suppose I factored real symmetric quasi-definite $ A_0= L_0 \cdot D_0 \cdot L_0^T$ and the factorization exists, with $D$ diagonal and $L$ unit lower-triangular; and suppose $L$ and $D$ are badly scaled: e.g., $L$ has large off-diagonal elements and the diagonal elements of $D$ differ vastly in scale.
Question: How do I alter the factorization at "low cost" into $A_1 = L_1 \cdot D_1 \cdot L_1^T$ such that $\|A_0-A_1\|$ remains "small"?
Remark: The question is deliberately vague as to cost and measure because different norms or minimization/optimality/criteria/heuristics could be suitable. I care about practical solutions ;)
