Consider the generalized linear least squares (GLS) problem $\min\|Lx\|_2 \ \mathrm{s.t.} \ \|M(Ax-b)\|_2=\min$. The weighted pseudoinverse $A_{ML}^{\dag}$ is the matrix that maps $b$ to the minimum 2-norm solution of this GLS problem. By introducing a linear operator induced by $\{A, M, L\}$ between two finite-dimensional Hilbert spaces, we show that the minimum 2-norm solution of the GLS problem is equivalent to the minimum norm solution of a linear least squares problem involving this linear operator, and $A_{ML}^{\dag}$ can be expressed as the composition of the Moore-Penrose pseudoinverse of this linear operator and an orthogonal projector. With this new interpretation, we establish the generalized Moore-Penrose equations that completely characterize the weighted pseudoinverse, give a closed-form expression of the weighted pseudoinverse using the generalized singular value decomposition (GSVD), and propose a generalized LSQR (gLSQR) algorithm for iteratively solving the GLS problem. We construct several numerical examples to test the proposed iterative algorithm for solving GLS problems. Our results highlight the close connections between GLS, weighted pseudoinverse, GSVD and gLSQR, providing new tools for both analysis and computations.

翻译：考虑广义线性最小二乘（GLS）问题 $\min\|Lx\|_2 \ \mathrm{s.t.} \ \|M(Ax-b)\|_2=\min$。加权伪逆 $A_{ML}^{\dag}$ 是将 $b$ 映射至该 GLS 问题最小 2-范数解的矩阵。通过引入由 $\{A, M, L\}$ 在两个有限维希尔伯特空间之间诱导的线性算子，我们证明了 GLS 问题的最小 2-范数解等价于涉及该线性算子的线性最小二乘问题的最小范数解，且 $A_{ML}^{\dag}$ 可表示为该线性算子的 Moore-Penrose 伪逆与一个正交投影算子的复合。基于这一新解释，我们建立了完全刻画加权伪逆的广义 Moore-Penrose 方程组，利用广义奇异值分解（GSVD）给出了加权伪逆的闭式表达式，并提出了一种用于迭代求解 GLS 问题的广义 LSQR（gLSQR）算法。我们构造了若干数值算例来测试所提迭代算法求解 GLS 问题的性能。研究结果揭示了 GLS、加权伪逆、GSVD 与 gLSQR 之间的紧密联系，为理论分析与数值计算提供了新工具。