The use of the Preconditioned Conjugate Gradient (PCG) method for computing the Generalized Least Squares (GLS) estimator of the General Linear Model (GLM) is considered. The GLS estimator is expressed in terms of the solution of an augmented system. That system is solved by means of the PCG method using an indefinite preconditioner. The resulting method iterates a sequence Ordinary Least Squares (OLS) estimations that converges, in exact precision, to the GLS estimator within a finite number of steps. The numerical and statistical properties of the estimator computed at an intermediate step are analytically and numerically studied. This approach allows to combine direct methods, used in the OLS step, with those of iterative methods. This advantage is exploited to design PCG methods for the estimation of Constrained GLMs and of some structured multivariate GLMs. The structure of the matrices involved are exploited as much as possible, in the OLS step. The iterative method then solves for the unexploited structure. Numerical experiments shows that the proposed methods can achieve, for these structured problems, the same precision of state of the art direct methods, but in a fraction of the time.

翻译：本文探讨了使用预处理共轭梯度（PCG）方法计算广义线性模型（GLM）的广义最小二乘（GLS）估计量。GLS估计量通过增广系统的解来表示。该系统采用不定预处理子的PCG方法求解。所得方法通过迭代一系列普通最小二乘（OLS）估计，在精确计算条件下于有限步内收敛至GLS估计量。本文从解析和数值角度研究了中间步骤所得估计量的数值与统计特性。该方法能够将OLS步骤中使用的直接法与迭代法相结合。基于此优势，我们设计了用于估计约束GLM及特定结构化多元GLM的PCG方法。在OLS步骤中，我们尽可能利用所涉及矩阵的结构特性，而迭代法则用于处理未被利用的结构部分。数值实验表明，针对此类结构化问题，所提方法能够达到与最先进直接法相同的精度，但仅需其部分计算时间。