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

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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

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

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

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

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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, …

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Special cases United solution set Tolerable solution set Controllable solution set

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

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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]

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

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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 )

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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…

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

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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

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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

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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

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

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

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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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Chapter 8 Objectives Understanding matrix notation.

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