basics and applications of interval mathematics
TRANSCRIPT
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S O H A M
U N D E R T H E G U I D A N C E O F
P R O F . D R . A N D R E A S K A M P M A N N
BASICS & APPLICATIONS OF
INTERVAL MATHEMATICS
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Basics
Interval Notation in Mathematics:The interval of numbers between a and b, including a and b, is denoted by [a,b]
Interval Arithmetic: a general numerical computing technique thatautomatically provides guaranteed enclosures for arbitrary formulas,
in the presence of uncertainties, mathematical approximations,and arithmetic round-off.
When i/p has Specific Ranges or Uncertainties instead of definiteknown values.
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General Syntax of Interval Computations
Interval Operation where
We can get interval for the result c,
WhereExamples
Simulation\AreaPerimeterAndDiagonalOfARectangleWithUncertainSides.cdf
c a b [ , ]; [ , ]a a a b b b
{ | [ , ], [ , ] &
[min( , , , ), max( , , , )]
c a b a a a b b b
c a b a b a b a b a b a b a b a b
&a a a a a a
http://examples%20simulation/AreaPerimeterAndDiagonalOfARectangleWithUncertainSides.cdfhttp://examples%20simulation/AreaPerimeterAndDiagonalOfARectangleWithUncertainSides.cdfhttp://examples%20simulation/AreaPerimeterAndDiagonalOfARectangleWithUncertainSides.cdfhttp://examples%20simulation/AreaPerimeterAndDiagonalOfARectangleWithUncertainSides.cdfhttp://examples%20simulation/AreaPerimeterAndDiagonalOfARectangleWithUncertainSides.cdfhttp://examples%20simulation/AreaPerimeterAndDiagonalOfARectangleWithUncertainSides.cdfhttp://examples%20simulation/AreaPerimeterAndDiagonalOfARectangleWithUncertainSides.cdfhttp://examples%20simulation/AreaPerimeterAndDiagonalOfARectangleWithUncertainSides.cdf -
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Challenges Faced
Integral Arithmetic generally overestimates the actual bounds of therange, to overcome this we use the extension of Integral Arithmetic
Conversion between IA and Affine Arithmetic.
An affine form is created from an interval as follows:
x0 = (xH + xL)/2;
x1 = (xH - xL)/2;
xi = 0; i > 1
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Pattern recognition & computational Geometry
(a) Surface intersection using AA. (b) IA (top) versus AA (bottom).
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Applications
Space-Applications: In Spacecraft design for taking photos ofdistant planets.
constraints between weight & cost.
several possible solutions.
Select a possible range of most applicable solutions andFormulate for satisfying given constraints.
Numerical Computing: General methods yield approximate solutions. E.g. Solution of
Optimization problems.
Have Iterative Methods => The more the iterations, the moreaccurate solution. But never exact.
Repeatedly compute estimates for the same quantities. Usepreviously achieved result x(k) , to compute the next estimate x(k+1)
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Applications
By Interval Operations, we can compute intervals for thesetwo successive steps. Hence for the next step x(k+2) , we takesmaller of the two intervals for the first two step.
i.e. Interval of
Hence Converge faster & Avoid Overestimates.
( ) ( 2)k kx x
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Applications
Truncation Errors:Taylor series of the exponential function:
ex = 1 +x+ (x2 /2!)et
where t [0,x]. For x < 0, ex 1+x+ (x2 /2!) [0,1].
In particular, with,
x =-0.531, we get e(-0:531) 1-0.531+ ((-0:531)2/2!) [0,1]
= 0:469+[0:140,0:141][0,1]= [0:469;0:610].
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Advantages & Drawbacks
Advantages: Very useful when the working data is subjected to
measurement errors or uncertainties.
An alternative error estimation approach; i.e. we get errors
estimations simultaneous to the iterations. Whereas in normalmethods we get errors only after iteration process. Hence,Savings in Computation time.
A very powerful technique for controlling errors in
computations. Any contiguous set of real numbers (a continuum) can be
represented by containing interval.
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Drawbacks: Interval arithmetic can be slow, and often gives overly
pessimistic results for real-world computations
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References
Weisstein, Eric W. "Interval Arithmetic." FromMathWorld--AWolfram Web Resource.http://mathworld.wolfram.com/IntervalArithmetic.html
http://www.cs.utep.edu/interval-comp/
Mainstream Contributions of Interval Computations in Engineeringand Scientific Computing -R. Baker Kearfott,Department ofMathematics,University of Louisiana at Lafayette
Introduction to Numerical Analysis by J.Stoer and R.Bulirsch Applications of interval computations by R. Baker Kearfott, Vladik
Kreinovich
http://mathworld.wolfram.com/about/author.htmlhttp://mathworld.wolfram.com/http://mathworld.wolfram.com/IntervalArithmetic.htmlhttp://www.cs.utep.edu/interval-comp/http://www.cs.utep.edu/interval-comp/http://www.cs.utep.edu/interval-comp/http://www.cs.utep.edu/interval-comp/http://mathworld.wolfram.com/IntervalArithmetic.htmlhttp://mathworld.wolfram.com/http://mathworld.wolfram.com/about/author.html -
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Thank You!