The document discusses the interval Newton method, which is an extension of the Newton method to interval arithmetic. It aims to guarantee convergence of the root-finding process for functions where the standard Newton method may fail. The interval Newton method works by computing the image of an interval under iteration, rather than a single point. The document outlines interval arithmetic, the standard Newton method, properties of the interval Newton method, a Matlab implementation, and provides numerical examples.

1. Interval Newton Method
Projektseminar
1 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 examples
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3. Patriot vs. Scud
MIM-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 Commons
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4. United States General Accounting Office 1992
Outcome: No Interception
Scud hits an American barracks => 28 dead
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5. Causes of the accident
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6. Interval Arithmetic/Analysis
Evolved in the 1950-1960s on the advent of
computational mathematics (pioneering work in
the University of Karlsruhe)
Idea: Bounding the rounding errors
How: Counting with intervals instead of real
numbers
Definines mathematical operations ((+),(-),(*),(/))
for intervals
[-5,5]+[2,4]=[-3,9] [1,2]*[-4,7]=[-8,14]
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7. Specialities
Dependency problem
In interval arithmetic, each variable is handled
independently
This causes widening of intervals in computations and
decreases the accuracy of the results in calculations.
Extended interval arithmetic
A rule set for division by intervals containing 0
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8. Newton Method
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9. Newton method
http://math.fullerton.edu/mathews/a2001/animations/rootfinding/NewtonMethod/NewtonMethod.html
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10. However, sometimes it does not converge!
http://math.fullerton.edu/mathews/a2001/animations/rootfinding/NewtonMethod/NewtonMethod.html
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11. Interval Newton method
Moore et al., Introduction to interval analysis, 2009
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12. Example
Graph: www.woflramalpha.com
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15. Properties of the interval Newton method
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16. Properties of the interval Newton method
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17. Properties of the interval Newton method
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18. Matlab implementation
The non-extended interval Newton method
implemented in Matlab using the INTLAB toolbox
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19. Numerical examples
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20. Thank for your attention!
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