# Derivatives of Approximate Matrix inverses

I have a question concerning the derivatives of approximate matrix inverses. I have a system, $$Ax = b$$ which I solve approximately (and with an iterative method) with: $$\Delta x = x - \tilde{A}^{-1}b$$ I would like to take the derivative of this process, i.e. find $$\frac{d\Delta x}{dD} = \frac{dx}{dD} - \frac{d\tilde{A}^{-1}}{dD}b - \tilde{A}^{-1}\frac{db}{dD}$$ I know that if I had an ideal inverse, my expression would be without tildes as I'd have the exact result: $$\frac{d\Delta x}{dD} = \frac{dx}{dD} - {A}^{-1}\frac{dA}{dD}{A}^{-1}b - \tilde{A}^{-1}\frac{db}{dD}$$ But for this, I cannot start with the typical assumption that $$AA^{-1} = I$$ and instead have $$A\tilde{A}^{-1} = C \neq I$$ Does anyone have any ideas for resources or techniques for such problems? Thank you.

• What is $D$?$\;\;\;$ – Algebraic Pavel Oct 16 '18 at 8:52
• The variable that I'm taking the derivative with respect to. The state variables depend on it through the nonlinear solve. – EMP Oct 16 '18 at 13:04
• what iterative method did you use? – Shogun Oct 16 '18 at 22:41
• I used Point Jacobi as my iterative solver – EMP Oct 17 '18 at 23:02