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Speeding up Krylov subspace methods for computing f(A)b via randomization

Alice CortinovisDaniel KressnerYuji Nakatsukasa
Dec 2022
摘要
This work is concerned with the computation of the action of a matrixfunction f(A), such as the matrix exponential or the matrix square root, on avector b. For a general matrix A, this can be done by computing the compressionof A onto a suitable Krylov subspace. Such compression is usually computed byforming an orthonormal basis of the Krylov subspace using the Arnoldi method.In this work, we propose to compute (non-orthonormal) bases in a faster way andto use a fast randomized algorithm for least-squares problems to compute thecompression of A onto the Krylov subspace. We present some numerical exampleswhich show that our algorithms can be faster than the standard Arnoldi methodwhile achieving comparable accuracy.
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