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The Dynamical Functional Particle Method

We present a new algorithm which is named the Dynamical Functional Particle Method, DFPM. It is based on the idea of formulating a finite dimensional damped dynamical system whose stationary points are the solution to the original equations. The resulting Hamiltonian dynamical system makes it possible to apply efficient symplectic integrators. Other attractive properties of DFPM are that it has an exponential convergence rate, automatically includes a sparse formulation and in many cases can solve nonlinear problems without any special treatment. We study the convergence and convergence rate of DFPM. It is shown that for the discretized symmetric eigenvalue problems the computational complexity is given by $\mathcal{O}(N^{(d+1)/{d}})$, where \emph{d} is the dimension of the problem and \emph{N} is the vector size. An illustrative example of this is made for the 2-dimensional Schrödinger equation. Comparisons are made with the standard numerical libraries ARPACK and LAPACK. The conjugated gradient method and shifted power method are tested as well. It is concluded that DFPM is both versatile and efficient.