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Continuation Theory

Continuation Theory. Troy Thomas 6/28/13. What is C ontinuation Theory?. Computing the curve or set of solutions to a nonlinear equation Takes the form F( x,a ) = 0, where a is some parameter that is varying. Overview. Predictor/Corrector Method Alternate Corrector Steps Validation Step

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Continuation Theory

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  1. Continuation Theory Troy Thomas 6/28/13

  2. What is Continuation Theory? • Computing the curve or set of solutions to a nonlinear equation • Takes the form F(x,a) = 0, where a is some parameter that is varying

  3. Overview • Predictor/Corrector Method • Alternate Corrector Steps • Validation Step • Future Work

  4. Predictor/Corrector

  5. Example Problem F(x,a) = a – (x * x) = 0

  6. Pseudo-arclength Corrector

  7. Moore Penrose Corrector

  8. Validation • Banach’s Fixed Point Theorem Let (X, d) be a non-empty complete metric space. Let f : X → X be a contraction mapping on X, i.e.: there is a nonnegative real number K < 1 such that d(f(a),f(b)) ≤ K d(a,b) for all x, y in X. Then the map fadmits one and only one fixed point x* in X

  9. Future Work • Alternate Corrector Methods • Interval Arithmetic • Validation Information

  10. Acknowledgements • Advisors Dr. Lamar and Day • Charles Center Monroe Scholar Freshman Research Program

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