Sherman–Morrison rank-one update formula 

Sherman–Morrison-Woodbury formula

can be used when A^{-1} is known

Neumann Series

we have to be careful because (I− A)−1 need not always exist. However, we are safe when the entries in A are sufficiently small.

Approximation of (I-A)^{-1}

(A+B)&{-1}

the Neumann series allows us to say something when B has small entries relative to A, or vice versa.

and a first-order approximation is

The change in A^{-1} is 

Measure the error

Sensitivity and Conditioning

from later topic

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The term on the left is the relative change in the inverse

“magnification factor” -- the condition number 

if κ is small relative to 1 (i.e., if A is well conditioned), then a small relative change (or error) in A cannot produce a large relative change (or error) in the inverse,

Perturbation in the linear system

Since the right-hand side of (3.8.8) k is only an estimate of the relative error in the solution, the exact value of κ is not as important as its order of magnitude.

Floating point and accuracy

if t -digit floating-point arithmetic is used, then, assuming no other source of error exists, it can be argued that when κ is of order 10p, the computed solution is expected to be accurate to at least t − p significant digits, more or less.

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