I was wondering how the following approximate bound can be derived?
1 Like
The derivation is usually done for each term separately, assuming the other terms are zero, as the complete derivation may be too complicated.
The b-bhat and r terms are easily proved.
The A-Ahat term is fairly complex.
I will give a general view: First show x-xhat = -invA (A-Ahat)xhat
Then take norms and also use xhat = x + xhat - x
Then arrive to
rel error <= kappa(A)/(1-||invA|| ||A-Ahat||) ||A-Ahat||/||A||.
Then, for 1/(1-||invA|| ||A-Ahat||), use some Taylor series approximation
to get the final result.
Needless to say, this is not what I would ask in a final exam,
but those interested in deriving it, could try to follow the steps above.