Why Intervals Why Interval Matrices Why Products of . . . The Problem of . . . Interval Computations Computing the Cube of an Computing the . . . Why Power of a Matrix Interval Matrix Is NP-Hard Feasible Algorithm for . . . Interval Matrix . . . Computing the Cube . . . Olga Kosheleva, Vladik Kreinovich Acknowledgments NASA Pan-American Center for Proof Earth and Environmental Studies Proof (cont-d) University of Texas at El Paso Proof (final part) El Paso, TX 79968, USA Title Page vladik@cs.utep.edu ◭◭ ◮◮ G¨ unter Mayer ◭ ◮ Dept. of Math., University of Rostock, Germany Page 1 of 15 Hung T. Nguyen Go Back Dept. of Math. Sciences, New Mexico State University Las Cruces, NM 88003, USA, hunguyen@nmsu.edu Full Screen March 9, 2005 Close Quit
Why Intervals 1. Why Intervals Why Interval Matrices Why Products of . . . • In many real-life situations, we do not know the exact value of a physical The Problem of . . . quantity x . Interval Computations Computing the . . . • We only know the interval x of possible values of x . Why Power of a Matrix • This happens, e.g.: Feasible Algorithm for . . . Interval Matrix . . . – if our information about x comes from measurement, and Computing the Cube . . . – the only information that we have about the possible error of the mea- Acknowledgments suring instrument is that this error is ≤ a certain bound ∆. Proof Proof (cont-d) • In this case, let the measurement result is � x . Proof (final part) • We know that that | � x − x |≤ ∆, where x is the (unknown) actual value of the Title Page measured quantity. ◭◭ ◮◮ • We can conclude that x belongs to the interval ◭ ◮ def x = [ � x − ∆ , � x + ∆] . Page 2 of 15 Go Back Full Screen Close Quit
Why Intervals 2. Why Interval Matrices Why Interval Matrices Why Products of . . . • In some physical situations, quantities form a matrix The Problem of . . . Interval Computations a 11 . . . a 1 j . . . a 1 n Computing the . . . . . . . . . . . . . . . . . . Why Power of a Matrix A = a i 1 . . . a ij . . . a in . Feasible Algorithm for . . . . . . . . . . . . . . . . . . Interval Matrix . . . a m 1 . . . a mj . . . a mn Computing the Cube . . . Acknowledgments • Example: system’s dynamics Proof s i ( t + 1) = f i ( s 1 ( t ) , . . . , s n ( t )) . Proof (cont-d) Proof (final part) def = s i ( t ) − s (0) • Often, we are interested in small deviations ∆ s i ( t ) from the Title Page i stable state s (0) . ◭◭ ◮◮ n � • Linearization leads to ∆ s i ( t + 1) = a ij · ∆ s i ( t ) or ∆ s ( t + 1) = A ∆ s ( t ). ◭ ◮ j =1 • Often, for each i and j , we only know the interval a ij of possible values of Page 3 of 15 a ij – an interval matrix . Go Back Full Screen Close Quit
Why Intervals 3. Why Products of Interval Matrices Why Interval Matrices Why Products of . . . • Why product: The Problem of . . . Interval Computations – if transition t → t + 1 is described by a matrix A , Computing the . . . – transition t + 1 → t + 2 is described by B , Why Power of a Matrix – then transition t → t + 2 is described by the product C = BA , with Feasible Algorithm for . . . entries Interval Matrix . . . n � Computing the Cube . . . c ij = a ik · b kj . Acknowledgments k =1 Proof • In case of interval uncertainty, we know A and B , and we want to know Proof (cont-d) Proof (final part) def ( AB ) ij = { ( AB ) ij ; A ∈ A , B ∈ B } . Title Page • Similar, for a transition t → t + 3, we must know: ◭◭ ◮◮ def ( ABC ) ij = { ( ABC ) ij ; A ∈ A , B ∈ B , C ∈ C } . ◭ ◮ • Problem: How can we compute these products? Page 4 of 15 Go Back Full Screen Close Quit
Why Intervals 4. The Problem of Multiplying Interval Matrices is a Why Interval Matrices Why Products of . . . Particular Case of a General Problem The Problem of . . . Interval Computations • General problem: Computing the . . . – we have a function f ( x 1 , . . . , x n ) of n variables, Why Power of a Matrix Feasible Algorithm for . . . – we know the interval x i of possible values of each of these variables, and Interval Matrix . . . – we must find the range Computing the Cube . . . def Acknowledgments f ( x 1 , . . . , x n ) = { f ( x 1 , . . . , x n ); x 1 ∈ x 1 , . . . , x n ∈ x n } Proof of this function when x i ∈ x i . Proof (cont-d) Proof (final part) • This general problem is called the problem of interval computations . Title Page • Known: in general, NP-hard. ◭◭ ◮◮ ◭ ◮ Page 5 of 15 Go Back Full Screen Close Quit
Why Intervals 5. Interval Computations Why Interval Matrices Why Products of . . . • Interval arithmetic: explicit formulas when f = +, − , · , etc.: The Problem of . . . Interval Computations [ x 1 , x 1 ] + [ x 2 , x 2 ] = [ x 1 + x 2 , x 1 + x 2 ]; Computing the . . . Why Power of a Matrix [ x 1 , x 1 ] · [ x 2 , x 2 ] = [min( x 1 · x 2 , x 1 · x 2 , x 1 · x 2 , x 1 · x 2 ) , Feasible Algorithm for . . . max( x 1 · x 2 , x 1 · x 2 , x 1 · x 2 , x 1 · x 2 )] . Interval Matrix . . . Computing the Cube . . . • Straightforward interval computations: Acknowledgments – replace each operation forming the algorithm f Proof Proof (cont-d) – with the corresponding operation from interval arithmetic. Proof (final part) • Case of single-use expressions (SUE): exact result. Title Page • Conclusion: we get the exact product of two interval matrices: ◭◭ ◮◮ n � c ij = a ik · b kj . ◭ ◮ k =1 Page 6 of 15 Go Back Full Screen Close Quit
Why Intervals 6. Computing the Product of Three Interval Matrices Why Interval Matrices Why Products of . . . is NP-Hard The Problem of . . . Interval Computations • Problem: computing the product D = ABC of three interval matrices. Computing the . . . n n � � Why Power of a Matrix • Situation: the expression d ij = a ik · b kl · c lj is not SUE. Feasible Algorithm for . . . k =1 l =1 Interval Matrix . . . • Conclusion: we can only guarantee that the straightforward interval compu- Computing the Cube . . . tation leads to an enclosure – i.e., the result may not be always exact. Acknowledgments Proof • Our first (simple) result: The problem of computing the exact product of Proof (cont-d) three interval matrices is NP-hard. Proof (final part) • Idea of the proof: it is NP-hard, given a square matrix B = ( b ij ) i,j , to Title Page compute the range of x T By , where x i = y j = [ − 1 , 1]. ◭◭ ◮◮ ◭ ◮ Page 7 of 15 Go Back Full Screen Close Quit
Why Intervals 7. Why Power of a Matrix Why Interval Matrices Why Products of . . . • Situation: in many practical situations, we know that the system is station- The Problem of . . . ary . Interval Computations Computing the . . . • This means that the transition from each moment of time to the next is Why Power of a Matrix described by the same matrix A . Feasible Algorithm for . . . • Then: Interval Matrix . . . Computing the Cube . . . – transition t → t + 2 is described by A 2 , Acknowledgments – transition t → t + 3 is described by A 3 , etc. Proof Proof (cont-d) • In case of interval uncertainty, we only know that A ∈ A for a given interval Proof (final part) matrix A . Title Page • Problem: compute, for every i and j , the set (interval) of possible values of A 2 and/or A 3 : ◭◭ ◮◮ def def ( A 2 ) ij = { ( A 2 ) ij ; A ∈ A } ; ( A 3 ) ij = { ( A 3 ) ij ; A ∈ A } . ◭ ◮ Page 8 of 15 Go Back Full Screen Close Quit
Why Intervals 8. Feasible Algorithm for Computing the Square of an Why Interval Matrices Why Products of . . . Interval Matrix The Problem of . . . Interval Computations • Situation: for B = A 2 , the expression Computing the . . . n � Why Power of a Matrix b ij = a ik · a kj Feasible Algorithm for . . . k =1 Interval Matrix . . . is not SUE. Computing the Cube . . . Acknowledgments • Example: for i � = j , we have two occurrences of a ij : a ij · a jj (when k = j ) Proof and a ii · a ij (when k = j ). Proof (cont-d) • Idea: reformulate into SUE: Proof (final part) � Title Page b ij = a ik · a kj + a ij · ( a ii + a jj ) ( i � = j ); k : k � = i,k � = j ◭◭ ◮◮ � a ik · a ki + a 2 b ii = ii . ◭ ◮ k : k � = i • Solution: a feasible algorithm for computing A 2 : Page 9 of 15 � b ij = a ik · a kj + a ij · ( a ii + a jj ) ( i � = j ); Go Back k : k � = i,k � = j Full Screen � a ik · a ki + a 2 b ii = ii . Close k : k � = i Quit
Why Intervals 9. Interval Matrix Product Is Not Associative: Why Interval Matrices Why Products of . . . Example The Problem of . . . Interval Computations � 1 � � [1 , 2] � Computing the . . . [0 , 1] [ − 1 , 1] A = ; then A ∗ s A = ; Why Power of a Matrix 1 − 1 0 [1 , 2] Feasible Algorithm for . . . � [1 , 2] � [ − 1 , 3] Interval Matrix . . . A ∗ s ( A ∗ s A ) = ; [1 , 2] [ − 3 , 0] Computing the Cube . . . � [0 , 3] � Acknowledgments [ − 1 , 3] ( A ∗ s A ) ∗ s A = � = A ∗ s ( A ∗ s A ) . Proof [1 , 2] [ − 2 , − 1] � 1 � � 1 + a 12 � Proof (cont-d) a 12 0 ; so A 2 = Proof (final part) Here, A = ; 1 − 1 0 1 + a 12 Title Page � 1 + a 12 � a 12 + a 2 A 3 = 12 ; hence 1 + a 12 − (1 + a 12 ) ◭◭ ◮◮ � [1 , 2] � � [1 , 2] � 0 [0 , 2] A 2 = A 3 = ◭ ◮ ; . 0 [1 , 2] [1 , 2] [ − 2 , − 1] Page 10 of 15 Go Back Full Screen Close Quit
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