On a PSOR-like Method for Generalized Saddle Point Problems

We study a parameterized SOR-like (PSOR-like) representation of the parameterized inexact Uzawa (PIU) iteration for generalized saddle point systems. An explicit parameter change establishes equivalence to PIU with an exact first-block solve, and to GSOR when stabilization is zero. A quadratic matrix polynomial characterizes the complete iteration spectrum, including unit relaxation. We give a self-contained necessary and sufficient matrix-inequality convergence criterion for this specialization, considered alongside existing generalized PIU analyses without claiming a new iteration or priority for Uzawa convergence theory. The derivation avoids Rayleigh-quotient denominator assumptions and yields computable bounds for definite preconditioners and an exact relation to the preconditioned spectrum. Sparse experiments on two synthetic families compare the stationary iteration, an explicitly defined CGPIU${}^{*}$ predictor–corrector comparator and GMRES. Archived tests use a common true residual tolerance and report setup and solve times separately. New manufactured-solution tests use a common forward-error threshold. At the common forward-error threshold, PSOR/PIU and CGPIU${}^{*}$ require identical stationary counts; P-GMRES uses fewer inner steps on both synthetic families.