AI 中文总结
本文研究有限精度下对称 Krylov 方法的数值分歧，通过精确舍入示例和块 Paige 恒等式，提出变块 Lanczos 模型，并给出精度界和收敛性分析。
AI 中文摘要
在精确算术中等价的短递推 Krylov 方法在浮点算术中常常出现分歧。为说明这一点，我们为具有递归更新残差的最速下降法提供了一个精确可表示的二周期。一个互补的收敛定理给出了存储残差几何衰减的充分条件。然后我们证明，第二个正定族对于 Hestenes--Stiefel 共轭梯度（CG）实现和直接 Lanczos--Galerkin 方法产生不同的结果。CG 在四次更新后找到精确解，而两步投影系统变得不一致。在给定的扰动和转移假设下，一个共同的谱包含给出了可比较的收敛界。对于稠密、从左到右的求值顺序，我们证明了对于 n×n 矩阵，在 n 次更新内达到规定后向误差的充分精度界，以及一个可计算的停止检验和指定的指数范围假设。我们使用多项式估计和数值实验来研究工作精度与达到规定后向误差所需的 CG 迭代次数之间的权衡。我们还与 Paige 和 Greenbaum 的框架比较了不精确矩阵-向量乘积和预处理的结果。对于块 Lanczos，我们分析了当块大小变化时的 Householder 正交化和奇异值截断。在分量wise和范数wise误差界下，计算出的系数满足受控的局部递推，并且对于更大空间中的邻近对称问题满足精确的块 Lanczos 关系。Paige 恒等式的块形式限定了与 Ritz 向量沿其残差坐标方向的重叠。一个块间递推描述了重叠的演变，一个 Gram 矩阵论证给出了额外邻近 Ritz 值的计数。
英文摘要
Short-recurrence Krylov methods that are equivalent in exact arithmetic often diverge in floating-point arithmetic. To illustrate this, we provide an exactly representable two-cycle for steepest descent with a recursively updated residual. A complementary convergence theorem gives a sufficient condition under which the stored residual decreases geometrically. We then show that a second positive definite family yields different outcomes for a Hestenes--Stiefel Conjugate Gradient (CG) implementation and a direct Lanczos--Galerkin approach. CG finds the exact solution after four updates, whereas the two-step projected system becomes inconsistent. Under stated perturbation and transfer hypotheses, a common spectral enclosure gives comparable convergence bounds. For a dense, left-to-right evaluation order, we prove a sufficient precision bound for a prescribed backward error within $n$ updates for an $n\times n$ matrix, together with a computable stopping test and specified exponent-range assumptions. We use polynomial estimates and numerical experiments to examine the trade-off between working precision and the number of CG iterations needed to achieve a prescribed backward error. We also compare results on inexact matrix-vector products and preconditioning with the frameworks of Paige and Greenbaum. For block Lanczos, we analyze Householder orthogonalization and singular-value truncation as the block size changes. Under componentwise and normwise error bounds, the computed coefficients satisfy a controlled local recurrence and an exact block Lanczos relation for a nearby symmetric problem in a larger space. A block form of Paige's identity bounds the overlap with a Ritz vector along its residual coordinate direction. An inter-block recurrence describes the evolution of overlap, and a Gram-matrix argument gives a count of additional nearby Ritz values.
