The Newton Schulz iteration for matrix inversion
The Newton Schulz method is well-known, and the proof of convergence is widely available on the internet. Yet the derivation of the method itself is more obscure. Here it is:
We seek the zero of \( f: \mathbb{R}^2 \rightarrow \mathbb{R}^2\), defined as follows: \( f(X) = X^{-1} – A\).
The derivative of \(f\) at \(X\), applied to matrix \(B\), operates on \(B\) as follows: \[ \begin{array}{l} f'(X)[B] = -X^{-1} B X^{-1}. \end{array} \]
We can then prove that \( f'^{-1} \) at \( X \), applied to matrix \( B \), operates on \( B \) as follows: \( \begin{array}{l} f'^{-1}(X)[B] = -X B X \end{array} \).
To see this, notice that \[ \begin{array}{l} B \\ = f'^{-1}(X)\Big[f'(X)[B]\Big] \\ = -X \Big[-X^{-1} B X^{-1}\Big] X \\ = B. \end{array} \]
The Newton method for root finding has at each iterate: \[ \begin{array}{l} X_{t+1} \\ = X_t – f'^{-1}(X_t)\Big[f(X_t)\Big]\\ = X_t – f'^{-1}(X_t)\Big[X^{-1} – A\Big] \\ = X_t – X_t[X^{-1}_t-A] X_t \\ = X_t – [-X_t + X_t A X_t]\\ = 2 X_t – X_t A X_t \end{array} \].
This was originally published here: https://calvinmccarter.wordpress.com/2021/11/18/the-newton-schulz-iteration-for-matrix-inversion/