A forward-only Lanczos gradient approximation for Hermitian matrix function bilinear forms whose error scales with the same residual norm as the forward approximation and appears stable without reorthogonalization.
Weinberger, David Bindel, and Andrew Gordon Wilson
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Fast and Stable Gradient Approximation for Bilinear Forms of Hermitian Matrix Functions
A forward-only Lanczos gradient approximation for Hermitian matrix function bilinear forms whose error scales with the same residual norm as the forward approximation and appears stable without reorthogonalization.