Why TMW does not improve INV?
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If TMW thinks the legacy algorithm behind INV applied on full square matrix (base on LU and LDT) is not accurate enough (*), why don't they add user-option to use better algorithm if user needs, for example the same algorithm used in
B = A \ eye(size(A))
B = eye(size(A)) / A
(*) NOTE: I personally INV precision is good enough and usage of INV should't be discourage in many practical cases.