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New inclusion sets for singular values.

Jun He¹, Yan-Min Liu¹, Jun-Kang Tian¹

¹School of Mathematics, Zunyi Normal College, Zunyi, Guizhou 563006 P.R. China.

Journal of Inequalities and Applications

|April 8, 2017

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New inclusion sets for matrix singular values were established, offering tighter bounds than existing Geršgorin-type and Brauer-type sets. Numerical experiments confirm the improved efficiency of these novel results in matrix analysis.

Area of Science:

Numerical analysis
Linear algebra
Matrix theory

Background:

Singular values are fundamental properties of matrices with broad applications.
Existing methods like Geršgorin and Brauer sets provide bounds for singular values.
There is a continuous need for tighter and more efficient inclusion sets.

Purpose of the Study:

To establish novel inclusion sets for the singular values of a given matrix.
To demonstrate that these new sets are tighter than existing Geršgorin-type and Brauer-type inclusion sets.
To validate the practical efficiency of the proposed inclusion sets through numerical experiments.

Main Methods:

Derivation of new inclusion sets based on matrix properties and parameters.
Theoretical comparison of the newly derived sets with established Geršgorin-type and Brauer-type sets.

Keywords:

inclusion sets matrix singular value

Related Experiment Videos

Implementation of a numerical experiment to compare the tightness and efficiency of the sets.

Main Results:

New inclusion sets for singular values of matrices are successfully established.
The new inclusion sets are proven to be strictly tighter than the Geršgorin-type sets by Qi and Brauer-type sets by Li.
Numerical experiments confirm the superior efficiency and tightness of the new inclusion sets.

Conclusions:

The newly developed inclusion sets provide a significant improvement in bounding matrix singular values.
These tighter bounds have implications for numerical stability and computational efficiency in matrix computations.
The findings offer a valuable enhancement to the existing toolkit for matrix analysis.