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I need it to invert a matrix. Wikipedia explains that there is a generalization of the Newton Method for matrices. However, there is nothing mentioned about the error bounds.

Suppose we have, as input, a matrix $M \in \mathbb{R}$ or $M \in \mathbb{C}$. However, this input is not exact, and the correct values would be $M'$. So the input error can be stated as $||M-M'||$.

Which error do we have after $n \in \mathbb{N}$ iterations of Newton? For which starting seed does the error bound hold?

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I think you missed reading the well-known literatures such as "T. Soderstrom, G.W. Stewart, On the numerical properties of an iterative method for computing the Moore-Penrose generalized inverse, SIAM J. Numer. Anal. 11 (1974), 61-74." in this field.

According to the above-mentiond paper and under a typical seed with machine precision, basically ALMOST $2Log_2[K(A)]$ is required for the Newton's method to converge to the $A^{-1}$, where $Log_2[.]$ stands for the logarithm in base 2 and $K(A)$ is the condition number of the matrix $A$.

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