The local convergence condition is proposed and the optimal parameter is given. Numerical experiments are used to show the efficiency of the NPHSS-KPIK iteration method for solving the Sylvester equations arising from the time-periodic fractional diffusion equations. | Turk J Math (2016) 40: 1325 – 1339 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article On the NPHSS-KPIK iteration method for low-rank complex Sylvester equations arising from time-periodic fractional diffusion equations 1 Min-Li ZENG1,2 , Guo-Feng ZHANG2,∗ School of Mathematics and Statistics, Lanzhou University, Lanzhou, . China 2 School of Mathematics, Putian University, Putian, . China Received: • Accepted/Published Online: • Final Version: Abstract: Based on the Hermitian and skew-Hermitian (HS) splitting for non-Hermitian matrices, a nonalternating preconditioned Hermitian and skew-Hermitian splitting-Krylov plus inverted Krylov subspace (NPHSS-KPIK) iteration method for solving a class of large and low-rank complex Sylvester equations arising from the two-dimensional timeperiodic fractional diffusion problem is established. The local convergence condition is proposed and the optimal parameter is given. Numerical experiments are used to show the efficiency of the NPHSS-KPIK iteration method for solving the Sylvester equations arising from the time-periodic fractional diffusion equations. Key words: Sylvester equation, Krylov-plus-inverted-Krylov subspace method, time-periodic fractional diffusion equation, NPHSS method, low-rank 1. Introduction Consider the following Sylvester equation AU + U B = C, () where A ∈ Cm×m , B ∈ Cn×n , and C ∈ Cm×n are given complex matrices. Assume (A1 ) A , B , and C are large matrices; (A2 ) A = W1 + iT1 , B = W2 + iT2 , with W1 and W2 being both symmetric positive matrices, at least one of T1 and T2 being a nonzero matrix; (A3 ) C = F GT has much lower rank than the problem size, . F ∈ Cm×s , G ∈ Cn×s , where s ≪ m and s ≪ n . The Sylvester equation of the form () arises in numerous applications, such as control and system theory, image restoration, and the discretized approximation of fractional
