basics and applications of interval mathematics

7/31/2019 Basics and Applications of Interval Mathematics

1/12

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


2/12

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.


3/12

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


4/12

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


5/12

Pattern recognition & computational Geometry

(a) Surface intersection using AA. (b) IA (top) versus AA (bottom).


6/12

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)


7/12

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


8/12

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].


9/12

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.


10/12

Drawbacks: Interval arithmetic can be slow, and often gives overly

pessimistic results for real-world computations


11/12

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


12/12

Thank You!

basics and applications of interval mathematics

Documents