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