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

Quan Shi,Lin-Quan Yao,Qin-Qin Shen

doi:10.1186/s13660-016-1193-y

Quan Shi, Lin-Quan Yao + Show 1 more

Open Access

https://doi.org/10.1186/s13660-016-1193-y

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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.

Highlights

Consider the large and sparse nonsymmetric saddle point problems Au ≡ A BT x = f ≡ b, ( . )–B y g where A = AT ∈ Rn×n is positive definite, B ∈ Rm×n is a rectangular matrix with m ≤ n, and u = [xT, yT ]T and b = [f T, gT ]T with x, f ∈ Rn and y, g ∈ Rm are the unknown and given right-hand side vectors, respectively
We show that all eigenvalues having nonzero imaginary parts are located in an intersection of two circles and all nonzero real eigenvalues are located in a positive interval
We study the extreme eigenvalues of the symmetric positive definite matrix

Summary

Introduction

In the particular case α = , the eigenvalue distribution of the shift-splitting preconditioned nonsingular saddle-point matrix PS–S A was studied in [ ]. Since the spectral distribution of the preconditioned matrix is closely related to the convergence rate of Krylov subspace iteration methods [ ], we hope that the resulting preconditioned saddle-point matrices have desired eigenvalue distributions, that is, tightly clustered spectra or positive real spectra, and so on; see, for example, [ , – ]. ), in this paper, we provide eigenvalue bounds for the nonzero eigenvalues of the shift-splitting preconditioned singular nonsymmetric saddle-point matrices PS–S A that depend only on the extremal eigenvalues of A and nonzero extremal singular values of B. We first present the eigenvalue distribution of the shift-splitting preconditioned matrix PS–S A and give a bound for its nonzero eigenvalues To this end, we first give a useful lemma.

B T – αIn – V T AV β Ir

B T – AβIr –B

B T – A

Conclusion