A class of new extended shift-splitting preconditioners for saddle point problems

A parameterized extended shift‐splitting preconditioner for nonsymmetric saddle point problems

“…In this section, using the idea of References 18 and 59 for nonsymmetric saddle point problems, the following new splitting of $$$ \mathfrak{A} $$$ is given as where $$$ \hat{\alpha}\ge 0 $$$, $$$ \hat{\beta}>0 $$$, $$$ l\in {\mathbb{R}}^{+} $$$, and $$$ P\in {\mathbb{R}}^{m\times m} $$$, $$$ Q\in {\mathbb{R}}^{n\times n} $$$ are symmetric positive definite matrices. Therefore, applying (2) emerges the following new method.…”

“…Huang et al 18 recently utilized constant $$$ l $$$ within the matrix $$$ \mathfrak{A} $$$ to introduce the following parameterized $$$ GSS $$$ preconditioner $$$ (PGSS) $$$ such that $$$ \hat{\alpha}\ge 0 $$$, $$$ \hat{\beta}>0 $$$. Wang et al 59 presented a class of new extended shift‐splitting ($$$ NESS $$$) preconditioners for solving symmetric positive definite saddle point and they presented the $$$ NESS $$$ preconditioner as …”

“…such that α ≥ 0, β > 0. Wang et al 59 presented a class of new extended shift-splitting (NESS) preconditioners for solving symmetric positive definite saddle point and they presented the NESS preconditioner as…”

“…There is no discussion on the NESS 59 and PGSS 54 iteration methods for the nonsymmetric saddle point problem with nonsymmetric positive definite A. In this article, inspired by the  NESS and  PGSS , we propose a parameterized extended shift-splitting (PESS) preconditioner for nonsymmetric and nonsingular saddle point problems with nonsymmetric (1,1)-block.…”