Tridiagonalizing Complex Symmetric Matrices in Waveguide Simulations

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Abstract

We discuss a method for solving complex symmetric (non-Hermitian) eigenproblems Ax = λBx arising in an application from optoelectronics, where reduced accuracy requirements provide an opportunity for trading accuracy for performance. In this case, the objective is to exploit the structural symmetry. Consequently, our focus is on a non-Hermitian tridiagonalization process. For solving the resulting complex symmetric tridiagonal problem, a variant of the Lanczos algorithm is used. Based on Fortran implementations of these algorithms, we provide extensive experimental evaluations. Runtimes and numerical accuracy are compared to the standard routine for non-Hermitian eigenproblems, LAPACK/zgeev. Although the performance results reveal that more work is needed in terms of increasing the fraction of Level 3 Blas in our tridiagonalization routine, the numerical accuracy achieved with the non-Hermitian tridiagonalization process is very encouraging and indicates important research directions for this class of eigenproblems.