interval linear algebra and computational...

Interval Linear Algebra and ComputationalComplexity

Jaroslav Horacek

Department of Applied MathematicsCharles University, Prague

MatTriadSeptember 2015

The journey of Pi . . .

Our goal

Journey through interval linear algebra

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Interval linear algebra

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I The key notion is an interval (real closed)

I We use the intervals instead of real coefficients

I Many reasons:I Floating point rounding (

√2, π)

I Data uncertaintyI Verification, computer assisted proofs

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√2, π)

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√2, π)

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Interval matrixA = [A,A] = {A | A ≤ A ≤ A}

I Can be also defined using midpoint matrix Ac and radiusmatrix ∆

A = [Ac −∆, Ac + ∆]

I A vector is a special type of the matrix

I Denoted by boldface!

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Computational complexity

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PContains problems that can be solved in polynomial time.

I It is a class of tractable problems

I Examples: linear programming, sorting numbers, . . .

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NPContains problems where the solution can be verified inpolynomial time

I P ⊆ NPI Solving usually needs exponential many computationsI Examples: SAT, graph 3-coloring, . . .

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Philosophical meaning of NP?

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NP-hardI A problem is NP-hard if it is at least as hard as any

problem in NP

I Any problem in NP can be solved using some NP-hardproblem

I A problem is NP-complete if it is the hardest problem in NP

I All NP-complete problems can be reduced to each other

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P, NP, NP-complete, NP-hard

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I Problem is in co-NP if its complement is in NP

I Examples: TAUT, . . .

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co-NP-hard

I A problem is co-NP-hard if it is at least as hard as anyproblem in co-NP

I A problem is co-NP-complete if it is the hardest problem inco-NP

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Difference between NP and co-NP?

I SAT = {φ | φ is a boolean satisfiable formula}

(NP-complete)

I TAUT = {φ | φ is a tautology}

(co-NP-complete)

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Before we land . . .

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. . . let’s mention the deeper goal of this talk

I We meet various difficult problems

I All of then have exponential ways to solve them

I Surprising sufficient conditions

I Surprising feasible cases

I Reducibility between problems

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The map of interval linear algebra

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Regularity and singularity

Regularity

A square interval matrix A is called regular if every A ∈ A isregular.

Singularity

Otherwise, it is called singular.

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I Regularity and singularity are complement problems.

I Checking matrix singularity is NP-complete

I Checking matrix regularity is co-NP-complete

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Regularity sufficient conditions

1. %(|A−1c |∆) < 1,

2. σmax(∆) < σmin(Ac),

3. ATc Ac − ‖∆T∆‖I is positive definite for some consistent

matrix norm ‖ · ‖.

Singularity sufficient conditions

1. maxj(∆|A−1c |)jj ≥ 1,

2. (∆− |Ac|)−1 ≥ 0,

3. ∆T∆ −ATc Ac is positive semidefinite.

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Regularity sufficient conditions

1. %(|A−1c |∆) < 1,

3. ATc Ac − ‖∆T∆‖I is positive definite for some consistent

matrix norm ‖ · ‖.

Singularity sufficient conditions

1. maxj(∆|A−1c |)jj ≥ 1,

2. (∆− |Ac|)−1 ≥ 0,

3. ∆T∆ −ATc Ac is positive semidefinite.

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Special classes

I interval M-matrices

I interval H-matrices

I If A,A are regular and A−1 ≥ 0, A−1 ≥ 0 then A is regular

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Special classes

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Special classes

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Map of interval linear algebra

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Full column rank

Full column rankAn m× n interval matrix A has full column rank if every A ∈ Ahas full column rank.

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Full column rank

I For square matrices this becomes regularityI Checking full column rank is co-NP-complete?

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Full column rank

An interval m× n matrix A = [Ac −∆, Ac + ∆] has full columnrank if some of these conditions holds

Sufficient conditions

1. Ac has full column rank and %(|(Ac)+|A∆) < 1,

2. ‖I −RA‖ < 1,

4. ‖∆‖ < ‖A+c ‖−1.

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Solving linear systems

Linear system

A is an interval matrix, b an interval vector.

Ax = b,

Solution set

Σ = {x | Ax = b for some A ∈ A, b ∈ b}.

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HullThe tightest possible interval vector (box) x such that

Σ ⊆ x.

EnclosureAn interval vector (box) x such that

Σ ⊆ x.

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HullThe tightest possible interval vector (box) x such that

Σ ⊆ x.

EnclosureAn interval vector (box) x such that

Σ ⊆ x.

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Solution set 2D

[5, 10]x + [−20,−5] y = [50, 100],[10, 15]x + [5, 10] y = [−50, 280].

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Solution set 3D

[−5, 10]x + 10 y + [15, 20] z = [50, 100],10x + −5 y + [5, 15] z = [−50, 50],10x + [10, 25] y + [−10,−5] z = [50, 100].

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I Computing the exact hull is NP-hardI Computing arbitrary ε-approximation of the hull is NP-hardI Even if we limit the widths of intervals of a matrix in a

system.

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MethodsSquare systems

1. Gaussian elimination2. Jacobi, Gauss-Seidel, Krawczyk method3. Hansen-Bliek-Rohn method4. Hladik shaving method

Overdetermined systems

1. Gaussian elimination2. Rohn method3. Subsquares method4. Least squares5. Popova method

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MethodsSquare systems

1. Gaussian elimination2. Jacobi, Gauss-Seidel, Krawczyk method3. Hansen-Bliek-Rohn method4. Hladik shaving method

Overdetermined systems

1. Gaussian elimination2. Rohn method3. Subsquares method4. Least squares5. Popova method

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Special linear systems

I When A has full column rank and Ac is a diagonal matrixwith positive diagonal entries then HBR returns the exacthull

I When A is M-matrix Gauss-Seidel iteration converges tothe hull

I When A is M-matrix and b nonnegative, Gaussianelimination yields the hull

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Matrix inverse

An interval inverse to A is a matrix A−1 = [B,B], where

B = min{A−1, A ∈ A}

B = max{A−1, A ∈ A}

Enclosure of A−1

B, where A−1 ⊆ B

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Matrix inverse

An interval inverse to A is a matrix A−1 = [B,B], where

B = min{A−1, A ∈ A}

B = max{A−1, A ∈ A}

Enclosure of A−1

B, where A−1 ⊆ B

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Matrix inverse

I Interval matrix inverse can be computed by computing thehull of the systems Ax = ei

I Computing exact bounds of the interval inverse is NP-hardI Checking whether a matrix has nonnegative inverse is in P

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Matrix inverse

I Enclosures of the systems Ax = ei

I Gaussian elimination on the matrix [A | I]

I Rohn formula for inverse enclosure

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Matrix inverse

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Matrix inverse

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Matrix inverse

Special classes

If A,A are regular and A−1, A−1 ≥ 0, then A is regular and

A−1 = [A−1, A−1] ≥ 0.

I M-matrices

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Checking solvability

Solvability

An interval linear system Ax = b is (weakly) solvable if itssolution set Σ 6= ∅.

Strong solvability

An interval linear system Ax = b is strongly solvable if everyAx = b, A ∈ A, b ∈ b is solvable.

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Solvability

An interval linear system Ax = b is (weakly) solvable if itssolution set Σ 6= ∅.

Strong solvability

An interval linear system Ax = b is strongly solvable if everyAx = b, A ∈ A, b ∈ b is solvable.

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I Ax = b is not solvable if the matrix [A b] has full columnrank

I Checking solvability NP-completeI Checking strong solvability is in NP-complete

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I Use sufficient full column rank conditions

I Gaussian elimination, Jacobi method, Gauss-Seidel,subsquares method.

I If we have some enclosure x then clearly a system Ax = bis unsolvable if Ax ∩ b = ∅.

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Interesting paradox

I In systems of linear equations

I Checking weak solvability NP-complete

I Checking strong solvability P!

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Positive definiteness

Symmetric matrix

Given A, with Ac,∆ are symmetric. ThenAS = {symmetric A ∈ A}

Positive definitenessA symmetric interval matrix AS is positive definite if xTAx > 0for each A ∈ AS and x 6= 0.

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Symmetric matrix

Given A, with Ac,∆ are symmetric. ThenAS = {symmetric A ∈ A}

Positive definitenessA symmetric interval matrix AS is positive definite if xTAx > 0for each A ∈ AS and x 6= 0.

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I Checking positive definiteness is co-NP-completeI Checking positive semi-definiteness is co-NP-complete

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Sufficient and necessary condition

A symmetric interval matrix AS is positive definite if and only ifit is regular and contains at least one positive definite matrix.

I Equivalent to AS being regular and Ac positive definite

I Can be transformed to regularity checking

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Sufficient conditions

1. Check regularity and positive definiteness of some matrix(e.g., Ac)

2. %(∆) < λmin(Ac)

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Eigenvalues

Eigenvalue

λ is an eigenvalue of A if it is an eigenvalue of some A ∈ A

Eigenvector

x 6= 0 is an eigenvector of A if it is an eigenvector of someA ∈ A

Eigenpair

(λ, x) is an eigenpair of A if Ax = λx for some some A ∈ A

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Eigenvalues

Eigenvalue

Eigenvector

Eigenpair

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Eigenvalues

Eigenvalue

Eigenvector

Eigenpair

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Eigenvalues

I Checking whether λ = 0 is an eigenvalue of A is equal tochecking singularity

I Checking whether λ is an eigenvalue of A is NP-complete

I Checking whether x is an eigenvector of A is in P

I Checking whether (λ, x) is an eigenpair of A is in P

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Eigenvalues

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Eigenvalues

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Eigenvalues

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Eigenvalues

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Eigenvalues of symmetric matrices

Eigenvalues of a symmetric A are realλ1(A) ≥ λ2(A) ≥ . . . λn(A)

Eigenvalues

λi(AS) = {λi(A) : A ∈ AS}

I Set of compact intervals

I Computing λ1(AS), λn(AS) is NP-hard

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Special cases

I [λi(AS), λi(A

S)] ⊆ [λi(Ac)− %(∆), λi(Ac) + %(∆)]

I If Ac is nonnegative then λ1(AS) = λmax(A)

I If ∆ is diagonal

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Special cases

I [λi(AS), λi(A

S)] ⊆ [λi(Ac)− %(∆), λi(Ac) + %(∆)]

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Special cases

I [λi(AS), λi(A

S)] ⊆ [λi(Ac)− %(∆), λi(Ac) + %(∆)]

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Our journey ends

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interval linear algebra and computational...

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