Mixed Precision Symmetric Eigensolvers: Proof of Concept

代数特征值问题的数值解是支撑计算科学和工程许多领域的关键技术，包括声学、航空、控制理论、流体力学、人口建模、量子物理、机器人和结构工程。在所有这些领域，需要快速和数值可靠的解决方案的特征值问题。现代硬件（多核处理器和加速器，如图形处理单元（GPU））越来越多地支持半精度浮点运算。这些低精度为大大加速线性代数计算提供了新的机会。这项“小额资助”提案中的研究是下一代高效且数值稳定的特征值求解器的概念证明，这些特征值求解器利用现代硬件的不同算术精度，同时保持数值稳定性。我们的研究主要集中在对称特征值问题上，对于对称特征值问题，特征值是真实的，具有一个完整的正交特征向量集，但任何进展将直接影响到未来的算法，非对称特征值问题，广义特征值问题，奇异值分解。将分析它们的数值稳定性以及它们在算术成本和通信成本方面的效率，以确定哪些应该在最先进的数值线性代数库中完全实现，例如免费提供的GPU和多核架构上的矩阵代数（MAGMA）库。

The numerical solution of algebraic eigenvalue problems is a key technology underpinning many areas of computational science and engineering, including acoustics, aeronautics, control theory, fluid mechanics, population modelling, quantum physics, robotics, and structural engineering. In all these areas, the need for fast and numerically reliable solution of eigenvalue problems arises. The problems can be large so that time to solution can be unacceptably long.Modern hardware (multicore processors and accelerators such as graphics processing units (GPUs)) increasingly supports half precision floating-point arithmetics. These low precisions provide new opportunities to considerably accelerate linear algebra computations.The research in this "small grant" proposal is a proof of concept for the next generation of efficient and numerically stable eigensolvers that exploit the different arithmetic precisions of modern hardware while maintaining numerical stability. Our investigation concentrates on the symmetric eigenvalue problem, for which eigenvalues are real with a full set of orthonormal eigenvectors, but any advances will have direct impact on future algorithms for nonsymmetric eigenproblems, generalized eigenproblems, and the singular value decomposition.The algorithms will be developed as prototypes in MATLAB, using simulated half precision. Their numerical stability will be analyzed as well as their efficiency in terms of arithmetic costs and communications costs so as to determine which one(s) should be fully implemented in state of the art numerical linear algebra libraries such as the freely available matrix algebra on GPU and multicore architectures (MAGMA) library.