标题： 混合精度正交化自由投影方法求解特征值和奇异值问题 显示英文标题

标题： Mixed Precision Orthogonalization-Free Projection Methods for Eigenvalue and Singular Value Problems

摘要： 混合精度算术为大规模矩阵计算任务提供了显著的计算优势，但在特征值问题和奇异值分解（SVD）中保持准确性和稳定性仍然具有挑战性。 本文介绍了一种方法，该方法消除了传统瑞利-里兹投影方法中的正交化要求。 所提出的方法采用在降精度下计算的非正交基，从而得到无需内积计算的基。 主要关注点在于保持基向量的线性独立性。 通过广泛评估合成测试用例和实际应用，我们证明了所提出的方法在充分利用混合精度算术的同时实现了所需的准确性。 显示英文摘要

摘要： Mixed-precision arithmetic offers significant computational advantages for large-scale matrix computation tasks, yet preserving accuracy and stability in eigenvalue problems and the singular value decomposition (SVD) remains challenging. This paper introduces an approach that eliminates orthogonalization requirements in traditional Rayleigh-Ritz projection methods. The proposed method employs non-orthogonal bases computed at reduced precision, resulting in bases computed without inner-products. A primary focus is on maintaining the linear independence of the basis vectors. Through extensive evaluation with both synthetic test cases and real-world applications, we demonstrate that the proposed approach achieves the desired accuracy while fully taking advantage of mixed-precision arithmetic.