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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

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Why Intervals Why Interval Matrices Why Products of . . . The Problem of . . . Interval Computations Computing the . . . Why Power of a Matrix Feasible Algorithm for . . . Interval Matrix . . . Computing the Cube . . . Acknowledgments Proof Proof (cont-d) Proof (final part) Title Page ◭◭ ◮◮ ◭ ◮ Page 1 of 15 Go Back Full Screen Close Quit

Computing the Cube of an Interval Matrix Is NP-Hard

Olga Kosheleva, Vladik Kreinovich

NASA Pan-American Center for Earth and Environmental Studies University of Texas at El Paso El Paso, TX 79968, USA vladik@cs.utep.edu

G¨ unter Mayer

  • Dept. of Math., University of Rostock, Germany

Hung T. Nguyen

  • Dept. of Math. Sciences, New Mexico State University

Las Cruces, NM 88003, USA, hunguyen@nmsu.edu March 9, 2005

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Why Intervals Why Interval Matrices Why Products of . . . The Problem of . . . Interval Computations Computing the . . . Why Power of a Matrix Feasible Algorithm for . . . Interval Matrix . . . Computing the Cube . . . Acknowledgments Proof Proof (cont-d) Proof (final part) Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 15 Go Back Full Screen Close Quit

1. Why Intervals

  • In many real-life situations, we do not know the exact value of a physical

quantity x.

  • We only know the interval x of possible values of x.
  • This happens, e.g.:

– if our information about x comes from measurement, and – the only information that we have about the possible error of the mea- suring instrument is that this error is ≤ a certain bound ∆.

  • In this case, let the measurement result is

x.

  • We know that that |

x−x|≤ ∆, where x is the (unknown) actual value of the measured quantity.

  • We can conclude that x belongs to the interval

x

def

= [ x − ∆, x + ∆].

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Why Intervals Why Interval Matrices Why Products of . . . The Problem of . . . Interval Computations Computing the . . . Why Power of a Matrix Feasible Algorithm for . . . Interval Matrix . . . Computing the Cube . . . Acknowledgments Proof Proof (cont-d) Proof (final part) Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 15 Go Back Full Screen Close Quit

2. Why Interval Matrices

  • In some physical situations, quantities form a matrix

A =       a11 . . . a1j . . . a1n . . . . . . . . . . . . . . . ai1 . . . aij . . . ain . . . . . . . . . . . . . . . am1 . . . amj . . . amn       .

  • Example: system’s dynamics

si(t + 1) = fi(s1(t), . . . , sn(t)).

  • Often, we are interested in small deviations ∆si(t)

def

= si(t) − s(0)

i

from the stable state s(0).

  • Linearization leads to ∆si(t + 1) =

n

  • j=1

aij · ∆si(t) or ∆s(t + 1) = A ∆s(t).

  • Often, for each i and j, we only know the interval aij of possible values of

aij – an interval matrix.

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Why Intervals Why Interval Matrices Why Products of . . . The Problem of . . . Interval Computations Computing the . . . Why Power of a Matrix Feasible Algorithm for . . . Interval Matrix . . . Computing the Cube . . . Acknowledgments Proof Proof (cont-d) Proof (final part) Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 15 Go Back Full Screen Close Quit

3. Why Products of Interval Matrices

  • Why product:

– if transition t → t + 1 is described by a matrix A, – transition t + 1 → t + 2 is described by B, – then transition t → t + 2 is described by the product C = BA, with entries cij =

n

  • k=1

aik · bkj.

  • In case of interval uncertainty, we know A and B, and we want to know

(AB)ij

def

= {(AB)ij; A ∈ A, B ∈ B}.

  • Similar, for a transition t → t + 3, we must know:

(ABC)ij

def

= {(ABC)ij; A ∈ A, B ∈ B, C ∈ C}.

  • Problem: How can we compute these products?
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Why Intervals Why Interval Matrices Why Products of . . . The Problem of . . . Interval Computations Computing the . . . Why Power of a Matrix Feasible Algorithm for . . . Interval Matrix . . . Computing the Cube . . . Acknowledgments Proof Proof (cont-d) Proof (final part) Title Page ◭◭ ◮◮ ◭ ◮ Page 5 of 15 Go Back Full Screen Close Quit

4. The Problem of Multiplying Interval Matrices is a Particular Case of a General Problem

  • General problem:

– we have a function f(x1, . . . , xn) of n variables, – we know the interval xi of possible values of each of these variables, and – we must find the range f(x1, . . . , xn)

def

= {f(x1, . . . , xn); x1 ∈ x1, . . . , xn ∈ xn}

  • f this function when xi ∈ xi.
  • This general problem is called the problem of interval computations.
  • Known: in general, NP-hard.
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Why Intervals Why Interval Matrices Why Products of . . . The Problem of . . . Interval Computations Computing the . . . Why Power of a Matrix Feasible Algorithm for . . . Interval Matrix . . . Computing the Cube . . . Acknowledgments Proof Proof (cont-d) Proof (final part) Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 15 Go Back Full Screen Close Quit

5. Interval Computations

  • Interval arithmetic: explicit formulas when f = +, −, ·, etc.:

[x1, x1] + [x2, x2] = [x1 + x2, x1 + x2]; [x1, x1] · [x2, x2] = [min(x1 · x2, x1 · x2, x1 · x2, x1 · x2), max(x1 · x2, x1 · x2, x1 · x2, x1 · x2)].

  • Straightforward interval computations:

– replace each operation forming the algorithm f – with the corresponding operation from interval arithmetic.

  • Case of single-use expressions (SUE): exact result.
  • Conclusion: we get the exact product of two interval matrices:

cij =

n

  • k=1

aik · bkj.

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Why Intervals Why Interval Matrices Why Products of . . . The Problem of . . . Interval Computations Computing the . . . Why Power of a Matrix Feasible Algorithm for . . . Interval Matrix . . . Computing the Cube . . . Acknowledgments Proof Proof (cont-d) Proof (final part) Title Page ◭◭ ◮◮ ◭ ◮ Page 7 of 15 Go Back Full Screen Close Quit

6. Computing the Product of Three Interval Matrices is NP-Hard

  • Problem: computing the product D = ABC of three interval matrices.
  • Situation: the expression dij =

n

  • k=1

n

  • l=1

aik · bkl · clj is not SUE.

  • Conclusion: we can only guarantee that the straightforward interval compu-

tation leads to an enclosure – i.e., the result may not be always exact.

  • Our first (simple) result: The problem of computing the exact product of

three interval matrices is NP-hard.

  • Idea of the proof: it is NP-hard, given a square matrix B = (bij)i,j, to

compute the range of xT By, where xi = yj = [−1, 1].

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Why Intervals Why Interval Matrices Why Products of . . . The Problem of . . . Interval Computations Computing the . . . Why Power of a Matrix Feasible Algorithm for . . . Interval Matrix . . . Computing the Cube . . . Acknowledgments Proof Proof (cont-d) Proof (final part) Title Page ◭◭ ◮◮ ◭ ◮ Page 8 of 15 Go Back Full Screen Close Quit

7. Why Power of a Matrix

  • Situation: in many practical situations, we know that the system is station-

ary.

  • This means that the transition from each moment of time to the next is

described by the same matrix A.

  • Then:

– transition t → t + 2 is described by A2, – transition t → t + 3 is described by A3, etc.

  • In case of interval uncertainty, we only know that A ∈ A for a given interval

matrix A.

  • Problem: compute, for every i and j, the set (interval) of possible values of

A2 and/or A3: (A2)ij

def

= {(A2)ij; A ∈ A}; (A3)ij

def

= {(A3)ij; A ∈ A}.

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Why Intervals Why Interval Matrices Why Products of . . . The Problem of . . . Interval Computations Computing the . . . Why Power of a Matrix Feasible Algorithm for . . . Interval Matrix . . . Computing the Cube . . . Acknowledgments Proof Proof (cont-d) Proof (final part) Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 15 Go Back Full Screen Close Quit

8. Feasible Algorithm for Computing the Square of an Interval Matrix

  • Situation: for B = A2, the expression

bij =

n

  • k=1

aik · akj is not SUE.

  • Example: for i = j, we have two occurrences of aij: aij · ajj (when k = j)

and aii · aij (when k = j).

  • Idea: reformulate into SUE:

bij =

  • k:k=i,k=j

aik · akj + aij · (aii + ajj) (i = j); bii =

  • k:k=i

aik · aki + a2

ii.

  • Solution: a feasible algorithm for computing A2:

bij =

  • k:k=i,k=j

aik · akj + aij · (aii + ajj) (i = j); bii =

  • k:k=i

aik · aki + a2

ii.

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Why Intervals Why Interval Matrices Why Products of . . . The Problem of . . . Interval Computations Computing the . . . Why Power of a Matrix Feasible Algorithm for . . . Interval Matrix . . . Computing the Cube . . . Acknowledgments Proof Proof (cont-d) Proof (final part) Title Page ◭◭ ◮◮ ◭ ◮ Page 10 of 15 Go Back Full Screen Close Quit

9. Interval Matrix Product Is Not Associative: Example

A = 1 [0, 1] 1 −1

  • ; then A ∗s A =

[1, 2] [−1, 1] [1, 2]

  • ;

A ∗s (A ∗s A) = [1, 2] [−1, 3] [1, 2] [−3, 0]

  • ;

(A ∗s A) ∗s A = [0, 3] [−1, 3] [1, 2] [−2, −1]

  • = A ∗s (A ∗s A).

Here, A = 1 a12 1 −1

  • ; so A2 =

1 + a12 1 + a12

  • ;

A3 = 1 + a12 a12 + a2

12

1 + a12 −(1 + a12)

  • ; hence

A2 = [1, 2] [1, 2]

  • ;

A3 = [1, 2] [0, 2] [1, 2] [−2, −1]

  • .
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Why Intervals Why Interval Matrices Why Products of . . . The Problem of . . . Interval Computations Computing the . . . Why Power of a Matrix Feasible Algorithm for . . . Interval Matrix . . . Computing the Cube . . . Acknowledgments Proof Proof (cont-d) Proof (final part) Title Page ◭◭ ◮◮ ◭ ◮ Page 11 of 15 Go Back Full Screen Close Quit

10. Computing the Cube of an Interval Matrix Is NP- Hard

  • Result: in general, computing A3 is NP-hard.
  • Conclusions:

– Computing the product of interval matrices is important in many appli- cations. – For two matrices, the corresponding problems are computationally fea- sible: ∗ computing the exact range for the product of two interval matrices; ∗ computing the square of an interval matrix. – The following 3-matrix problems are NP-hard: ∗ computing the exact range for the product of three matrices; ∗ computing the third power of a matrix.

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Why Intervals Why Interval Matrices Why Products of . . . The Problem of . . . Interval Computations Computing the . . . Why Power of a Matrix Feasible Algorithm for . . . Interval Matrix . . . Computing the Cube . . . Acknowledgments Proof Proof (cont-d) Proof (final part) Title Page ◭◭ ◮◮ ◭ ◮ Page 12 of 15 Go Back Full Screen Close Quit

11. Acknowledgments

  • This work was supported:

– by the German Research Council DFG, – by NASA grant NCC5-209, – by USAF grant F49620-00-1-0365, – by NSF grants EAR-0112968, EAR-0225670, and EIA-0321328, – by Army Research Laboratories grant DATM-05-02-C-0046, and – by the NIH grant 3T34GM008048-20S1.

  • The authors are thankful:

– to the participants of the International Dagstuhl Seminar “Numerical Software with Result Verification” and – to the anonymous referees.

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Why Intervals Why Interval Matrices Why Products of . . . The Problem of . . . Interval Computations Computing the . . . Why Power of a Matrix Feasible Algorithm for . . . Interval Matrix . . . Computing the Cube . . . Acknowledgments Proof Proof (cont-d) Proof (final part) Title Page ◭◭ ◮◮ ◭ ◮ Page 13 of 15 Go Back Full Screen Close Quit

12. Proof

  • Idea: use the same known NP-hard problem:

– given a square matrix B = (bij), – compute the range of xT By, where xi = yj = [−1, 1].

  • Specifically, for each n × n matrix B, we will consider the following (2n +

2) × (2n + 2) interval matrix: A = U L

  • ,

where L =     . . . . . . . . . B . . .     ; U =     x1 . . . xn y1 . . . yn    

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Why Intervals Why Interval Matrices Why Products of . . . The Problem of . . . Interval Computations Computing the . . . Why Power of a Matrix Feasible Algorithm for . . . Interval Matrix . . . Computing the Cube . . . Acknowledgments Proof Proof (cont-d) Proof (final part) Title Page ◭◭ ◮◮ ◭ ◮ Page 14 of 15 Go Back Full Screen Close Quit

13. Proof (cont-d)

  • For every matrix

A = U L

  • ∈ A =

U L

  • ,

we have A2 = U L U L

  • =

UL LU

  • ,
  • Hence

A3 = A2A = UL LU U L

  • =
  • ULU

LUL

  • .
  • Here,

UL =     xT y         0T B     =     xT B     .

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Why Intervals Why Interval Matrices Why Products of . . . The Problem of . . . Interval Computations Computing the . . . Why Power of a Matrix Feasible Algorithm for . . . Interval Matrix . . . Computing the Cube . . . Acknowledgments Proof Proof (cont-d) Proof (final part) Title Page ◭◭ ◮◮ ◭ ◮ Page 15 of 15 Go Back Full Screen Close Quit

14. Proof (final part)

  • Hence

ULU =     xT B         xT y     =     z     , where z = xT By.

  • We have shown that

A3 =

  • ULU

LUL

  • .
  • So, (ULU)11 = (A3)1,n+2 = xT By.
  • We know that computing the range of xT By is NP-hard.
  • We conclude that computing the range A3 is also an NP-hard problem.