The global extended-rational Arnoldi method for matrix function approximation
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The numerical computation of matrix functions such as f(A)V, where A is an n× n large and sparse square matrix, V is an n × p block with p≪ n and f is a nonlinear matrix function, arises in various applications such as network analysis (f(t)=exp(t) or f(t)=t^3), machine learning (f(t)=log(t)), theory of quantum chromodynamics (f(t)=t^1/2), electronic structure computation, and others. In this work, we propose the use of global extended-rational Arnoldi method for computing approximations of such expressions. The derived method projects the initial problem onto an global extended-rational Krylov subspace RK^e_m(A,V)=span({∏_i=1^m(A-s_iI_n)^-1V,...,(A-s_1I_n)^-1V,V,AV, ...,A^m-1V}) of a low dimension. An adaptive procedure for the selection of shift parameters {s_1,...,s_m} is given. The proposed method is also applied to solve parameter dependent systems. Numerical examples are presented to show the performance of the global extended-rational Arnoldi for these problems.
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