Given a Hermitian matrix $A$ and arbitrary vector $v$, the Lanczos decomposition yields $(Q, T)$. With this data, the "Lanczos-FA" method approximates $f(A) v \approx \Vert v \Vert Q f(T) e_1$ for arbitrary function $f$. Lanczos-FA is near-optimal within the Krylov subspace, as was shown in Amsel et al., arXiv:2303.03358. Remarkably, the Lanczos-FA method is backward-stable -- one can use the bare Lanczos recurrence directly, in finite precision, without orthogonalization. That is, $f(A) v \approx \Vert v \Vert Q f(T) e_1$ remains a good approximation even when the columns of $Q$ have lost their orthonormality due to numerical roundoff. Proofs are discussed in Sec. 6.2 of the recent review by Tyler Chen, arXiv:2410.11090. To avoid memory costs of storing $Q$, one has the option to employ a two-pass algorithm: The first run of Lanczos generates $T$. Then, once the coefficients $c = f(T) e_1$ are known, a second run of Lanczos forms the linear combination $Q c$ by accumulating columns of $Q$ as they become available.
Is this Lanczos-FA method possibly in scope for KrylovKit?
Given a Hermitian matrix$A$ and arbitrary vector $v$ , the Lanczos decomposition yields $(Q, T)$ . With this data, the "Lanczos-FA" method approximates $f(A) v \approx \Vert v \Vert Q f(T) e_1$ for arbitrary function $f$ . Lanczos-FA is near-optimal within the Krylov subspace, as was shown in Amsel et al., arXiv:2303.03358. Remarkably, the Lanczos-FA method is backward-stable -- one can use the bare Lanczos recurrence directly, in finite precision, without orthogonalization. That is, $f(A) v \approx \Vert v \Vert Q f(T) e_1$ remains a good approximation even when the columns of $Q$ have lost their orthonormality due to numerical roundoff. Proofs are discussed in Sec. 6.2 of the recent review by Tyler Chen, arXiv:2410.11090. To avoid memory costs of storing $Q$ , one has the option to employ a two-pass algorithm: The first run of Lanczos generates $T$ . Then, once the coefficients $c = f(T) e_1$ are known, a second run of Lanczos forms the linear combination $Q c$ by accumulating columns of $Q$ as they become available.
Is this Lanczos-FA method possibly in scope for KrylovKit?