We present a high-performance solver for dense skew-symmetric matrix eigenvalue problems. Our work is motivated by applications in computational quantum physics, where one solution approach to solve the Bethe-Salpeter equation involves the solution of a large, dense, skew-symmetric eigenvalue problem. The computed eigenpairs can be used to compute the optical absorption spectrum of molecules and crystalline systems. One state-of-the art high-performance solver package for symmetric matrices is the ELPA (Eigenvalue SoLvers for Petascale Applications) library. We exploit a link between tridiagonal skew-symmetric and symmetric matrices in order to extend the methods available in ELPA to skew-symmetric matrices. This way, the presented solution method can benefit from the optimizations available in ELPA that make it a well-established, efficient and scalable library. The solution strategy is to reduce a matrix to tridiagonal form, solve the tridiagonal eigenvalue problem and perform a back-transformation for eigenvectors of interest. ELPA employs a one-step or a two-step approach for the tridiagonalization of symmetric matrices. We adapt these to suit the skew-symmetric case. The two-step approach is generally faster as memory locality is exploited better. If all eigenvectors are required, the performance improvement is counteracted by the additional back transformation step. We exploit the symmetry in the spectrum of skew-symmetric matrices, such that only half of the eigenpairs need to be computed, making the two-step approach the favorable method. We compare performance and scalability of our method to the only available high-performance approach for skew-symmetric matrices, an indirect route involving complex arithmetic. In total, we achieve a performance that is up to 3.67 times higher than the reference method using Intel’s ScaLAPACK implementation. Our method is freely available in the current release of the ELPA library.

我们为密集的偏对称矩阵特征值问题提供了一种高性能求解器。我们的工作是由计算量子物理学中的应用推动的，其中一种解决Bethe-Salpeter方程的解决方法涉及解决一个大的，密集的，偏对称特征值问题。计算出的本征对可用于计算分子和晶体系统的光吸收谱。ELPA（用于Petascale应用的特征值求解器）库是一种用于对称矩阵的最新高性能求解器程序包。我们利用三对角斜对称矩阵与对称矩阵之间的联系，以将ELPA中可用的方法扩展到斜对称矩阵。这样，所提出的解决方案方法可以受益于ELPA中可用的优化，这些优化使其成为公认的，高效且可扩展的库。解决方案是将矩阵简化为三对角形式，解决三对角特征值问题，并对感兴趣的特征向量执行逆变换。ELPA采用一步或两步方法对对称矩阵进行三对角线化。我们对它们进行了调整，以适应倾斜对称情况。由于可以更好地利用内存局部性，因此两步方法通常更快。如果需要所有特征向量，则可以通过附加的反变换步骤来抵消性能的提高。我们利用倾斜对称矩阵频谱中的对称性，因此只需要计算一半特征对，这使两步法成为一种有利的方法。我们将方法的性能和可伸缩性与偏对称矩阵唯一可用的高性能方法进行了比较，涉及复杂算术的间接路线。总体而言，我们实现的性能比使用英特尔ScaLAPACK实现的参考方法高出3.67倍。当前版本的ELPA库免费提供我们的方法。