Let A ∈ R p×p be a diagonalizable matrix and f a smooth function. We are interested in the problem of approximating the action of f (A) on a vector b ∈ R p , i.e., f (A)b, without explicitly computing the matrix f (A). In the present work, we derive families of one-term, two-term, and three-term inexpensive approximations to the quantity f (A)b via an extrapolation procedure. For a given diagonalizable matrix A, the proposed families of vector estimates allow us to approximate the form W T f (A)U , for any matrices W, U ∈ R p×m , 1 ≤ m p, not necessarily biorthogonal. We present several numerical examples to illustrate the effectiveness of our method for several functions f for both the quantity f (A)b and the form W T f (A)U .
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