Using interval unions to solve linear systems of equations with uncertainties.

Area of Science:

• Numerical analysis
• Computational mathematics
• Interval arithmetic

Background:

• Solving linear systems with uncertain parameters is crucial in many scientific and engineering fields.
• Traditional methods struggle with the rigorous enclosure of solution sets when uncertainties are represented by intervals or interval unions.

Purpose of the Study:

• To introduce a novel interval union Gauss-Seidel procedure for robustly enclosing solutions of linear systems with interval uncertainties.
• To develop and evaluate interval union midpoint and Gauss-Jordan preconditioners for enhanced performance.

Main Methods:

• The core method is the interval union Gauss-Seidel procedure, designed for interval arithmetic.
• Interval union midpoint and Gauss-Jordan preconditioners are developed and integrated.
• A mixed strategy employing the Gauss-Jordan preconditioner is utilized to optimize the algorithm's quality and efficiency.

Main Results:

• The interval union Gauss-Seidel procedure effectively provides rigorous enclosures for the solution sets of interval linear systems.
• The developed preconditioners contribute to improved accuracy and computational speed.
• Numerical experiments demonstrate the practical capabilities of the proposed approach.

Conclusions:

• The interval union Gauss-Seidel procedure offers a powerful tool for handling linear systems with interval uncertainties.
• The integration of preconditioners enhances the efficiency and reliability of the solution enclosure process.
• This work advances the field of interval computation for uncertainty quantification.