论文标题
Schur分解的迭代改进
Iterative Refinement of Schur decompositions
论文作者
论文摘要
方形矩阵$ a $的SCHUR分解是用于解决特征值问题，矩阵函数和矩阵方程的最新数值算法的重要中间步骤。这项工作涉及以下任务：从给定的近似Schur分解中计算$ a $的（更多）精确的Schur分解。例如，在参数依赖的特征值问题和混合精度计算的背景下，此任务出现。我们已经开发了一种类似牛顿的算法，该算法需要在每种迭代中的三角矩阵方程和近似正交步骤的解决方案。我们证明了具有相互不同特征值的矩阵的局部二次收敛，并在实践中观察到快速收敛。在混合的低调环境中，我们的算法本质上仅减少了四个高精度矩阵矩阵乘积。当完善双重到四倍的精度时，通常只需要3-4次迭代，这将计算四倍精度Schur分解的时间减少到10-20倍。
The Schur decomposition of a square matrix $A$ is an important intermediate step of state-of-the-art numerical algorithms for addressing eigenvalue problems, matrix functions, and matrix equations. This work is concerned with the following task: Compute a (more) accurate Schur decomposition of $A$ from a given approximate Schur decomposition. This task arises, for example, in the context of parameter-dependent eigenvalue problems and mixed precision computations. We have developed a Newton-like algorithm that requires the solution of a triangular matrix equation and an approximate orthogonalization step in every iteration. We prove local quadratic convergence for matrices with mutually distinct eigenvalues and observe fast convergence in practice. In a mixed low-high precision environment, our algorithm essentially reduces to only four high-precision matrix-matrix multiplications per iteration. When refining double to quadruple precision, it often needs only 3-4 iterations, which reduces the time of computing a quadruple precision Schur decomposition by up to a factor of 10-20.
