2,234 research outputs found
Computing a matrix function for exponential integrators
AbstractAn efficient numerical method is developed for evaluating ϕ(A), where A is a symmetric matrix and ϕ is the function defined by ϕ(x)=(ex−1)/x=1+x/2+x2/6+⋯. This matrix function is useful in the so-called exponential integrators for differential equations. In particular, it is related to the exact solution of the ODE system dy/dt=Ay+b, where A and b are t-independent. Our method avoids the eigenvalue decomposition of the matrix A and it requires about 10n3/3 operations for a general symmetric n×n matrix. When the matrix is tridiagonal, the required number of operations is only O(n2) and it can be further reduced to O(n) if only a column of the matrix function is needed. These efficient schemes for tridiagonal matrices are particularly useful when the Lanczos method is used to calculate the product of this matrix function (for a large symmetric matrix) with a given vector
Computing diffraction anomalies as nonlinear eigenvalue problems
When a plane electromagnetic wave impinges upon a diffraction grating or
other periodic structures, reflected and transmitted waves propagate away from
the structure in different radiation channels. A diffraction anomaly occurs
when the outgoing waves in one or more radiation channels vanish. Zero
reflection, zero transmission and perfect absorption are important examples of
diffraction anomalies, and they are useful for manipulating electromagnetic
waves and light. Since diffraction anomalies appear only at specific
frequencies and/or wavevectors, and may require the tuning of structural or
material parameters, they are relatively difficult to find by standard
numerical methods. Iterative methods may be used, but good initial guesses are
required. To determine all diffraction anomalies in a given frequency interval,
it is necessary to repeatedly solve the diffraction problem for many
frequencies. In this paper, an efficient numerical method is developed for
computing diffraction anomalies. The method relies on nonlinear eigenvalue
formulations for scattering anomalies and solves the nonlinear eigenvalue
problems by a contour-integral method. Numerical examples involving periodic
arrays of cylinders are presented to illustrate the new method
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