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I have two systems $$ A \vec{x} = f(\vec{x}, \vec{y}), \:\:\: A \vec{y} = g(\vec{x}, \vec{y}) $$ Both have the same constant, square, invertible matrix $A$. I implemented an iterative algorithm with an initial value of $\vec{x_{0}}, \vec{y_{0}}$. Then $$ \vec{x_{n+1}} = A^{-1} f(\vec{x_{n}},\vec{y_{n}}), \:\:\: \vec{y_{n+1}} = A^{-1} g(\vec{x_{n}},\vec{y_{n}}) $$ And it turns out that they are always convergent when I compute in Matlab, for a particular matrix $A$.

Is there a theorem that specifies the necessary and sufficient conditions for the convergence?

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    $\begingroup$ This website is for research mathematics only, your post will likely be closed soon. I recommend asking your question on our companion website, math.overflow. And for the record, the keyword you're looking for is "Picard–Lindelöf fixed point theorem" $\endgroup$
    – leo monsaingeon
    Commented 3 hours ago
  • $\begingroup$ The right place to ask is math.stackexchange $\endgroup$
    – Moishe Kohan
    Commented 50 mins ago

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