The Gram–Schmidt process uses orthogonal projection to construct the A = QR factorization of a matrix. When Q has linearly independent columns, the operator P = I − Q(Q^TQ)⁻¹Q^T defines an orthogonal projection onto Q^⊥. In finite precision, Q loses orthogonality as the factorization progresses. A family of approximate projections is derived with the form P = I − QTQ^T, with correction matrix T. When T = (Q^TQ)⁻¹, and T is triangular, it is postulated that the best achievable orthogonality is $𝒪(\epsilon )\kappa (A)$. We present new variants of modified (MGS) and classical Gram–Schmidt algorithms that require one global reduction step. An interesting form of the projector leads to a compact WY representation for MGS. In particular, the inverse compact WY MGS algorithm is equivalent to a lower triangular solve. Our main contribution is to introduce a backward normalization lag into the compact WY representation, resulting in a $𝒪(\epsilon )\kappa ([r_0,A{V}_m])$ stable Generalized Minimal Residual Method (GMRES) algorithm that requires only one global reduce per iteration. Further improvements in performance are achieved by accelerating GMRES on GPUs.

中文翻译：

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低同步Gram–Schmidt和广义最小残差算法

Gram–Schmidt过程使用正交投影来构造矩阵的A  =  QR分解。当Q具有线性独立的列中，操作者P  = 我 -  Q（Q ^Ť Q）^-1 Q ^Ť限定正交投影Q ^⊥。以有限的精度，随着分解的进行，Q失去正交性。近似突起的家庭是衍生自具有形式P  = 我 -  QTQ ^Ť，用校正矩阵Ť。什么时候T  =（Q ^T Q）^-1，并且T为三角形，假设最佳可实现正交性为$𝒪（\epsilon ）\kappa （\mathrm{一个}）$。我们提出了改进的（MGS）和经典的Gram–Schmidt算法的新变体，它们需要一个全局归约步骤。投影机的一种有趣形式导致了MGS的紧凑WY表示。特别地，逆紧凑WY MGS算法等效于下三角解。我们的主要贡献是在压缩WY表示中引入了反向标准化滞后，$𝒪（\epsilon ）\kappa （[{\mathrm{[R}}_0，\mathrm{一个}{V}_{米}]）$稳定的通用最小残差方法（GMRES）算法，每次迭代仅需要全局减少一次。通过加速GPU上的GMRES，可以进一步提高性能。