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It seems to me that a useful concept to define when studying Krylov subspace methods is the idea of a vector $v$ that belongs to a Krylov subspace $\mathcal{K}_{n+1}(A,b)$ but not to the preceding one $\mathcal{K}_{n}(A,b)$, i.e., $v$ can be written as a polynomial $v = p(A)b$ of degree exactly $n$, with a nonzero leading coefficient.

Is there a standard name or notation for vectors that belong to $\mathcal{K}_{n+1}(A,b) \setminus \mathcal{K}_{n}(A,b)$? Or, how can I describe or denote them? Something like "$v$ is a proper vector in $\mathcal{K}_{n+1}(A,b)$" sounds good, but I don't want to invent a new term when there already exists a word for it.

"Continuation vector" is something close, but it is too vague and specific to Arnoldi, when the concept appears in many other contexts.

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    $\begingroup$ I have never seen a word for this. I need it when proving that Arnoldi's method works. I tend to shorten $K_j(A,b)$ to $K_j$ and just write $x \in K_{j+1}\setminus K_j$. $\endgroup$ Commented Dec 12, 2023 at 17:38
  • $\begingroup$ It is possible that we need to cast a wider net. I am thinking of an infinite union of increasing sets as an onion. We are looking for the elements in the (j+1)$st layer. This is not a phrase that have seen used, but it occured to me that there might be a good established turn of phrase outside of our very specialized context. $\endgroup$ Commented Dec 13, 2023 at 23:24

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