Respectively scaled splitting iteration method for a class of block 4-by-4 linear systems from eddy current electromagnetic problems

On the solution of the distributed optimal control problem with time‐periodic parabolic equations

“…Recently, Salkuyeh [13] presented a stationary iteration method which was called Alternating Symmetric positive definite and Scaled symmetric positive semidefinite Splitting (ASSS) for solving the system (1). He used the idea of Zeng [14] and so rewrote the system (1) as the 4-by-4 block real system…”

“…Recently, Salkuyeh [13] presented a stationary iteration method which was called Alternating Symmetric positive definite and Scaled symmetric positive semidefinite Splitting (ASSS) for solving the system (). He used the idea of Zeng [14] and so rewrote the system () as the 4‐by‐4 block real system $$$$ \mathcal{A}x\equiv \left(\begin{array}{cccc}M& 0& \sqrt{\nu }K& \omega \sqrt{\nu }M\\ {}0& M& -\omega \sqrt{\nu }M& \sqrt{\nu }K\\ {}\sqrt{\nu }K& -\omega \sqrt{\nu }M& -M& 0\\ {}\omega \sqrt{\nu }M& \sqrt{\nu }K& 0& -M\end{array}\right)\left(\begin{array}{c}\mathfrak{\Re}\left(\overline{y}\right)\\ {}\mathfrak{\Im}\left(\overline{y}\right)\\ {}\mathfrak{\Re}\left(\overline{q}\right)\\ {}\mathfrak{\Im}\left(\overline{q}\right)\end{array}\right)=\left(\begin{array}{c}\mathfrak{\Re}\left( at{y}_d\right)\\ {}\mathfrak{\Im}\left( at{y}_d\right)\\ {}0\\ {}0\end{array}\right)\equiv \hat{\mathbf{b}}. $$$$ The ASSS iteration method can be stated as (see Salkuyeh [13]) …”