Skip to content
DraconiticResearch
Search Sources API
197,788 studies31 sourcesUpdated 28 min ago
Peer-reviewedCC BY 4.0Axioms

On a PSOR-like Method for Generalized Saddle Point Problems

  • Rongxia Shao1
  • Xuanle Guo1
  • Shuo Li1
  • Jinfeng Zhang2
  • Guodong Li3,4,5
5 affiliations
  1. 1School of Statistics and Data Science, Xinjiang University of Finance and Economics, Urumqi 830012, China
  2. 2School of Marxism, Changji University, Changji 831100, China
  3. 3Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China
  4. 4Center for Applied Mathematics of Guangxi (GUET), Guilin 541004, China
  5. 5Guangxi Artificial Intelligence College, Guilin University of Electronic Technology, Guilin 541004, China
Open at sourceDOICite

Abstract

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.

Related

  1. The transposition method in saddle-point problems
    Dietmar Gallistl, Zhen Liu
    arXivPreprintCC BY-NC-ND 4.0Full textNumerical analysis & scientific computingarXiv
  2. A parameterized block LU preconditioning for double saddle point problems
    Hamed Aslani, Mehdi Najafi Kalyani, Davod Khojasteh Salkuyeh
    arXivPreprintCC0Full textNumerical analysis & scientific computingarXiv
  3. Fluid Problems Posed on Closed Hypersurfaces: Circumventing the Saddle-Point Structure
    Ricardo H. Nochetto, Mansur Shakipov
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  4. Projection-Free Algorithms for Nonsmooth Stochastic Convex-Concave Saddle-Point Problems
    Khanh-Hung Giang-Tran, Soroosh Shafiee
    arXivPreprintCC BY 4.0Full textOptimisation & controlarXiv
  5. Tikhonov-Regularized Dynamics and Inexact Algorithms for Nonlinear Saddle-Point Problems
    Hong-lu Li, Yi-bin Xiao
    arXivPreprintarXiv licenceFull textOptimisation & controlarXiv

Cite

BibTeXRIS
BibTeX
@article{shao2026psorlike,
  title = {{On a PSOR-like Method for Generalized Saddle Point Problems}},
  author = {Rongxia Shao and Xuanle Guo and Shuo Li and Jinfeng Zhang and Guodong Li},
  journal = {Axioms},
  year = {2026},
  month = nov,
  doi = {10.3390/axioms15110822},
  url = {https://www.mdpi.com/2075-1680/15/11/822},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - On a PSOR-like Method for Generalized Saddle Point Problems
AU  - Rongxia Shao
AU  - Xuanle Guo
AU  - Shuo Li
AU  - Jinfeng Zhang
AU  - Guodong Li
PY  - 2026
DA  - 2026/11/04
JO  - Axioms
DO  - 10.3390/axioms15110822
UR  - https://www.mdpi.com/2075-1680/15/11/822
AB  - 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.
SN  - 2075-1680
ER  -