Eigenvalue bounds of the shift-splitting preconditioned singular nonsymmetric saddle-point matrices

Abstract

For singular nonsymmetric saddle-point problems, a shift-splitting preconditioner was studied in (Appl. Math. Comput. 269:947-955, 2015). To further show the efficiency of the shift-splitting preconditioner, we provide eigenvalue bounds for the nonzero eigenvalues of the shift-splitting preconditioned singular nonsymmetric saddle-point matrices. For real parts of the eigenvalues, the bound is provided by valid inequalities. For eigenvalues having nonzero imaginary parts, the bound is a combination of two inequalities proving their clustering in a confined region of the complex plane. Finally, two numerical examples are presented to verify the theoretical results. MSC: 65F10

A modification of the generalized shift-splitting method for singular saddle point problems

“…In the section of the numerical experiments, we choose the matrix Q as Q = αI + βBB T where β > 0. In this case, when α, β → 0 we deduce that n eigenvalues of the preconditioned matrix are approximately equal to 1 and from (23) it follows that the others tends to 0 or 1.…”

“…In [18], Ren et al investigated the eigenvalue distribution of the shift-splitting preconditioned saddle point matrix and showed that all eigenvalues having nonzero imaginary parts are located in an intersection of two circles and all real eigenvalues are located in a positive interval. Shi et al in [23] provided eigenvalue bounds for the nonzero eigenvalues of the shift-splitting preconditioned singular nonsymmetric saddle point matrices.…”

An extension of the positive-definite and skew-Hermitian splitting method for preconditioning of generalized saddle point problems

“…(b) When A is PD, the P GSS preconditioner was studied in [16,17,34,36,38] and P ESS preconditioner was studied in [35].…”

A Local Positive (Semi)Definite Shift-Splitting Preconditioner for Saddle Point Problems with Applications to Time-Harmonic Eddy Current Models

“…Theoretical analysis shows that the stationary iteration method (1.5) for saddle point problems is unconditionally convergent. For saddle point problems with a Hermitian positive definite (1, 1) block matrix, the spectral properties of the shift-splitting preconditioned matrix are discussed in [37,44]. A generalised shift-splitting preconditioner with an additional parameter β has been studied in [23,24,[41][42][43].…”

“…A generalised shift-splitting preconditioner with an additional parameter β has been studied in [23,24,[41][42][43]. Numerical results suggest that the shift-splitting preconditioner SS and its generalisations are efficient in saddle point problems of computational fluid dynamics [22][23][24]44] and in meshfree discretisations of elasticity problems [37]. However, for time-harmonic current eddy models there are implementation difficulties associated with special block structure of the non-Hermitian saddle point matrix in (1.3).…”