We present two open-source implementations of the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) algorithm to find a few eigenvalues and eigenvectors of large, possibly sparse matrices. We then test LOBPCG for various quantum chemistry problems, encompassing medium to large, dense to sparse, wellbehaved to ill-conditioned ones, where the standard method typically used is Davidson's diagonalization. Numerical tests show that, while Davidson's method remains the best choice for most applications in quantum chemistry, LOBPCG represents a competitive alternative, especially when memory is an issue, and can even outperform Davidson for ill-conditioned, non diagonally dominant problems.

翻译：我们给出了局部最优块预处理共轭梯度（LOBPCG）算法的两个开源实现，用于求解大型、可能稀疏的矩阵的少量特征值和特征向量。随后，我们在多种量子化学问题中测试了LOBPCG算法，这些问题涵盖中等至大型、稠密至稀疏、良态至病态的不同类型，而标准方法通常采用戴维森对角化。数值测试表明，尽管戴维森方法仍是量子化学中大多数应用的最佳选择，但LOBPCG是一种有竞争力的替代方案，尤其在内存受限的情况下，并且在病态、非对角占优问题上甚至可能优于戴维森方法。