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

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.