Inverse of matrix with QR method

Question

What is the complexity of finding the inverse of matrix by QR decomposition? A is a $n×n$ with full rank.

Answer 1

As far as I recall, the QR decomposition stage requires $O\left(n^3\right)$ operations.

If I assume that you are not finding the inverse, but solving the linear system $Ax=b$, then once you have found the QR decomposition of $A$, the remaining operations are all $O\left(n^2\right)$.

If you are explicitly finding the inverse, then this will require an additional $O\left(n^3\right)$ operations once you have found the QR decomposition.

Answer 2

Calculating the $QR$ decomposition has complexity $O(n^3)$, after which $A = QR$ can be inverted into $R^{-1} Q^T$ also by $O(n^3)$ operations. In many of these algorithms, it is not the complexity that is important, but rather the numerical stability. It is well-known that first decomposing into $QR$ before inverting provides greater stability.