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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 Troy Thomas 6/28/13
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
Overview • Predictor/Corrector Method • Alternate Corrector Steps • Validation Step • Future Work
Example Problem F(x,a) = a – (x * x) = 0
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
Future Work • Alternate Corrector Methods • Interval Arithmetic • Validation Information
Acknowledgements • Advisors Dr. Lamar and Day • Charles Center Monroe Scholar Freshman Research Program