Interpolating functions of matrices on zeros of quasi-kernel polynomials

The computation of functions of matrices is a classical topic in numerical linear algebra. In the recent years research in this area has received new impulse due to the introduction of Krylov subspace techniques for the treatment of functions of large and sparse matrices, in particular in the context of the solution of di erential problems. Such techniques are projective in nature, since they resort to computing functions of matrices in lower dimensions, but, at the same time, they can also be viewed as polynomial interpolation methods. More precisely, for the computation of y=f(A)v, where f is a function analytic in a domain containing the spectrum of the square matrix A, and v is a given vector, these methods produce approximations of the type ym=pf;m−1(A)v, where the polynomial pf;m−1, of degree m− 1, interpolates f (in the Hermite sense) in suitably chosen points. Therefore a great attention must be payed to the choice of such points, with the aim of re ecting the eigenvalue distribution of the argument matrix and maintaining at the same time a limited cost for the arising algorithm. The Ritz values associated to Krylov subspace methods (see References [1–4]) are of course natural candidates to this task. If some information on the spectrum is available, it turns out also to be e ective the use of zeros of Chebychev or Faber polynomials (see Reference [5]), or of