Optimization of extrapolated Cayley transform with non-Hermitian positive definite matrix

Abstract

For the extrapolated Cayley transform, we give necessary and sufficient conditions for guaranteeing its convergence and contraction (in the Euclidean norm). We derive upper bounds for the convergence and the contraction factors, and compute the optimal parameters minimizing these upper bounds and the corresponding optimal values of these upper bounds. Numerical computations show that these upper bounds are reasonably sharp compared with the exact convergence and the exact contraction factors of the extrapolated Cayley transform, respectively.

A new generalization of the Hermitian and skew-Hermitian splitting method for solving the continuous Sylvester equation

“…Aghazadeh et al (2016) have presented the following lemma which plays an essential role in proving the convergence property of the TGHSS method. Similar versions of the next lemma have been presented by Johnson and Krukier (2009) and Bai and Hadjidimos (2014).…”

A class of modified DPSS preconditioners for generalized saddle-point linear systems

“…In the following theorems, using Lemma 2, we determine the quasi-optimal iteration parameter α, which minimizes some upper bounds of the spectral radius of the PMDPSS-I iteration method. This idea is similar to and inspired by the optimization idea for the HSS method in Bai et al (2003), see also the optimization results for extrapolated Cayley transform in Bai and Hadjidimos (2014).…”

Generalized Shift-Splitting Preconditioner for Saddle Point Problems with Block Three-by-Three Structure

“…It is known that the GSS iteration method (5) enjoys the unconditional convergence if the spectral radius of the iteration matrix GSS Γ is less than 1. The following results are adapted from [22] and [23] i.e., the GSS iteration method converges to the unique solution of (1) unconditionally.…”