1. Interval Newton Method Projektseminar1 18.01.12 Johannes Kananen Interval Newton Method Projektseminar zur Optimierung

2. Agenda Introduction Scud accident Interval arithmetic Newton method Interval Newton method Properties of the interval Newton method Matlab Implementation Numerical examples2 18.01.12 Johannes Kananen Interval Newton Method Projektseminar zur Optimierung 3. Patriot vs. ScudMIM-104 Patriot SS-1c Scud Surface-to-air missile Short-range ballistic system missile Originally designed to Missile speed: 1700 m/s engage against aircraft(~5 mach) Missile speed airborne: ~5 Range: 300 km mach Photos: Wikipedia Commons3 18.01.12 Johannes Kananen Interval Newton Method Projektseminar zur Optimierung 4. United States General Accounting Office 1992Outcome: No InterceptionScud hits an American barracks => 28 dead4 18.01.12Johannes Kananen Interval Newton Method Projektseminar zur Optimierung 5. Causes of the accident5 18.01.12 Johannes Kananen Interval Newton Method Projektseminar zur Optimierung 6. Interval Arithmetic/AnalysisEvolved in the 1950-1960s on the advent ofcomputational mathematics (pioneering work inthe University of Karlsruhe)Idea: Bounding the rounding errorsHow: Counting with intervals instead of realnumbersDefinines mathematical operations ((+),(-),(*),(/))for intervals [-5,5]+[2,4]=[-3,9][1,2]*[-4,7]=[-8,14]6 18.01.12 Johannes Kananen Interval Newton Method Projektseminar zur Optimierung 7. SpecialitiesDependency problem In interval arithmetic, each variable is handled independentlyThis causes widening of intervals in computations anddecreases the accuracy of the results in calculations.Extended interval arithmetic A rule set for division by intervals containing 07 18.01.12 Johannes Kananen Interval Newton Method Projektseminar zur Optimierung 8. Newton Method8 18.01.12 Johannes Kananen Interval Newton Method Projektseminar zur Optimierung 9. Newton method http://math.fullerton.edu/mathews/a2001/animations/rootfinding/NewtonMethod/NewtonMethod.html9 18.01.12 Johannes Kananen Interval Newton Method Projektseminar zur Optimierung 10. However, sometimes it does not converge! http://math.fullerton.edu/mathews/a2001/animations/rootfinding/NewtonMethod/NewtonMethod.html10 18.01.12 Johannes Kananen Interval Newton Method Projektseminar zur Optimierung 11. Interval Newton method Moore et al., Introduction to interval analysis, 200911 18.01.12Johannes Kananen Interval Newton Method Projektseminar zur Optimierung 12. Example Graph: www.woflramalpha.com12 18.01.12 Johannes Kananen Interval Newton Method Projektseminar zur Optimierung 13. 13 18.01.12 Johannes Kananen Interval Newton Method Projektseminar zur Optimierung 14. 14 18.01.12 Johannes Kananen Interval Newton Method Projektseminar zur Optimierung 15. Properties of the interval Newton method15 18.01.12 Johannes Kananen Interval Newton Method Projektseminar zur Optimierung 16. Properties of the interval Newton method16 18.01.12 Johannes Kananen Interval Newton Method Projektseminar zur Optimierung 17. Properties of the interval Newton method17 18.01.12 Johannes Kananen Interval Newton Method Projektseminar zur Optimierung 18. Matlab implementation The non-extended interval Newton method implemented in Matlab using the INTLAB toolbox18 18.01.12 Johannes Kananen Interval Newton Method Projektseminar zur Optimierung 19. Numerical examples19 18.01.12 Johannes Kananen Interval Newton Method Projektseminar zur Optimierung 20. Thank for your attention!20 18.01.12Johannes Kananen Interval Newton Method Projektseminar zur Optimierung