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

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

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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 diamond painting strand en zee 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 here region of the complex plane.Finally, two numerical examples are presented to verify the theoretical results.