Computing inverse of a matrix when a good starting guess is available? I need to compute inverse of matrix $A$ where inverse of a nearby matrix $B$ is known, is there a standard solution? I searched for "adaptive algorithm for computing inverse" with no results.
 A: One may use the Schulz iteration
$$ X_{k+1} = X_k (2I - AX_k) = (2I - X_kA) X_k$$
where $X_k$ converges to $A^{-1}$ (or more generally to the pseudo-inverse if $A$ is rectangular or defective). As $X_0$ one can take the initial approximation of the inverse $A^{-1}$. 
The Schulz iteration converge if $X_0$ is sufficiently close to $A^{-1}$ and is numerically stable. It is also known, that this method converges to $A^{-1}$ if $X_0 = \alpha A^T$ and $0 < \alpha < 2/\|A\|_2^2$. 
Note however, that numerical cost of one Schulz iteration is higher, than the cost of matrix inversion through LU decomposition for dence matrices (but can be much lower for sparse matrices).
A: The problem is that we need very well-conditioned systems to be able to use nearby matrices.
For instance, I copy/paste this famous example invented by R.S.Wilson

Suppose we know how to inverse $B=\begin{pmatrix} 10 & 7 & 8 & 7 \\ 7 & 5 & 6 & 5\\ 8 & 6 & 10 & 9 \\ 7 & 5 & 9 & 10\end{pmatrix}\ B^{-1}=\begin{pmatrix} 25 & -41 & 10 & -6 \\ -41 & 68 & -17 & 10\\ 10 & -17 & 5 & -3 \\ -6 & 10 & -3 & 2\end{pmatrix}$
This matrix is nice, it is symetric, has determinant $1$ and it belongs to $\mathbb Z^{4\times 4}$.
The system $BX=(32,23,33,31)^T$ also has super nice solution $X=(1,1,1,1)^T$.

Now let's consider the nearby matrix $A=\begin{pmatrix} 10 & 7 & 8.1 & 7.2 \\ 7.08 & 5.04 & 6 & 5\\ 8 & 5.98 & 9.89 & 9 \\ 6.99 & 4.99 & 9 & 9.98\end{pmatrix}$
Solving the same system $AY=(32,23,33,31)^T$ give a solution $Y=(-81,137,-34,22)^T$quite far away from $X$.
I do not write the matrix $A^{-1}$ but it is not nice, it has terms going from $-2063.26...$ to $3279.08...$ and resembles nothing near $B^{-1}$.
So initiating an algorithm with a nearby matrix may not be much better than a random choice if the system is badly conditioned. And if the system is well conditioned then any choice of initial point will most surely converge efficiently anyway. That's probably why you found no paper on adaptative algorithms. 
Research on system solving focused on improving the condition-number by solving derived systems (via preconditioner), rather than trying to improve the initiation of the algorithm.
A: You can apply the Newton method for the inverse. More directly, if $R=I-AB$ or $AB=I-R$, then by the geometric or Neumann series
$$
I=AB(I+R+R^2+R^3+…)
$$
so that $B(I+R)=B(2I-AB)$ is a better approximation of the inverse with error $O(R^2)$.
This quantifies "a good starting guess" to the condition that $\|R\|\ll 1$ for some operator norm.
