I have the expression for the inversion of a matrix with the Sherman-Morrison formula as follows (given that $A^{-1}$ is known).
$B^{-1} = A^{-1} + \frac{A^{-1}\mathbf{uv^*}A^{-1}}{1-\mathbf{v^*}A^{-1}\mathbf{u}}$
Now I'm trying to figure out how many FLOPS this would require to calculate. This is what I get from looking at it:
- one matrix-vector multiplication: $2m^2-m$
- one vector-matrix multiplication: $2m^2-m$
- one outer product: $m^2$
- one Sesquilinear form multiplication ($\mathbf{v^*}A^{-1}u$): $2m^2 +m-1$
- one subtraction: $1$ and
- one matrix addition: $m^2$
Totaling $8m^2-m$
But everywhere I look it says that the Sherman-Morrison formula is supposed to be able to be calculated in $\sim3m^2$ FLOPS.
What am I missing here?