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Estimation of AE-solution sets of square linear interval systems

Estimation of AE-solution sets of square linear interval systems. Alexandre Goldsztejn University of Nice-Sophia Antipolis. Outline. United solution sets and Gauss-Seidel AE-solution sets Generalized intervals Generalized Gauss-Seidel Jacoby algorithm for AE-solution sets

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Estimation of AE-solution sets of square linear interval systems

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  1. Estimation of AE-solution setsof 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 forstrongly 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: Problem Resolution Solution

  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-SeidelOuter estimation (1/2) • Gauss Seidel for united solution set • Generalized Gauss Seidel •  for parameter Aij : Aij→ dual(Aij) •  for parameter bi : bi→ dual(bi)

  12. Generalized Gauss-SeidelOuter 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 Aij 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 estimates Skew Boxes estimates

  18. Example(2/2) • Neumaier example United solution set Tolerable 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?

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