1. Estimation of AE-solution sets of square linear interval systems Alexandre Goldsztejn University of Nice-Sophia Antipolis

2. Outline United solution sets and Gauss-Seidel AE-solution sets Generalized intervals Generalized Gauss-Seidel Jacoby algorithm for AE-solution sets Construction of skew boxes

3. United solution set Consider an interval linear system: United solution set:

4. Gauss-Seidel iteration (1/2) Gauss-Seidel Algorithm: Fixed point algorithm for outer estimation. Component wise expression:

5. Gauss-Seidel iteration (2/2) Preconditioned system: Relationship original/preconditioned systems Gauss-Seidel applied to preconditioned system: OK for strongly regular matrices

6. AE-solution sets [Shary2004] Consider an interval linear system: Consider a quantifier for each parameter. A stands for universally quantified entries of A, A stands for existentially quantified entries of A, …

7. Special cases United solution set Tolerable solution set Controllable solution set

8. Generalized intervals (1/3) [Kaucher1973] Main tool for AE-solution sets Analogy with complex numbers: ProblemResolutionSolution

9. Generalized intervals (1/3) Set of generalized interval denoted Generalized intervals: classical (proper) intervals, [-1,1] Improper intervals, [1,-1] Proper improper intervals: dual[-1,1]=[1,-1] ; dual[1,-1]=[-1,1]

10. Generalized intervals (1/3) Interval arithmetic replaced by Kaucher arithmetic Addition and subtraction idem classical case. Multiplication and division idem classical for proper intervals.

11. Generalized Gauss-Seidel Outer estimation (1/2) Gauss Seidel for united solution set Generalized Gauss Seidel for parameter A ij : A ij dual(A ij ) for parameter b i : b i dual(b i )

12. Generalized Gauss-Seidel Outer estimation (2/2) x contains the AE-solution set Compatible with preconditioning Applied to the generalized interval matrix Works for strongly regular matrices Convention proper inversed for A ij inside Shary[2004] Inversion proposed here is motivated by a simpler presentation…

13. Jacoby iteration(1/2) Inner estimation Jacoby iteration Component wise expression:

14. Jacoby iteration(2/2) Inner estimation x inside the AE-solution set if proper Not compatible with preconditioning !!! Works only for strictly diagonally dominant matrix Not useful in practical situations

15. Postconditioning (1/2) Consider the postconditioned system Estimation of the postconditioned system using previously introduced iterations outer/inner boxes in an other base Apply so as to build a skew box in the original base

16. Postconditioning (2/2) Advantages: skew boxes more adapted to the shape of the solution set Compatible with both Gauss-Seidel and Jacoby Jacoby will now converge for any strongly regular matrices

17. Example(1/2) Hansen example, united solution set: Boxes estimatesSkew Boxes estimates

18. Example(2/2) Neumaier example United solution setTolerable solution set

19. Conclusion Skew boxes enhance both precision and convergence Can they be used in practical applications ? Strong limitation: Linear systems. Modal intervals for non-linear systems?