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

• Main Authors: Quan Shi, Qin-Qin Shen, Lin-Quan Yao
• Format: Article
• Language: English
• Published: SpringerOpen 2016-10-01
• Series: Journal of Inequalities and Applications
• Subjects: singular nonsymmetric saddle-point matrix; shift-splitting preconditioner; eigenvalue
• Online Access: http://link.springer.com/article/10.1186/s13660-016-1193-y

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.