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#83 suggests to reformulate the matrix power with explicit intervals.

When interpreting an interval matrix M as a set of matrices Aᵢ, and we only require that

M^k ⊇ ⋃ᵢ Aᵢ^k

holds, we can do better: Perform the multiplications symbolically: e.g., in 2D, consider M = [a b; c d] and write M^k as a matrix whose entries are polynomials in the original intervals a,b,c,d. Then in the end evaluate the final polynomials with interval arithmetic.

This issue is related to #88. Proposal: add a new IntervalMatrixPower wrapper which holds the power k of the matrix power. The purpose of this type is to experiment with lazy construction of the matrix power for evaluation using different methods. The new type can also have a cache for the representation of the power using symbols, eg. polynomials.

Builds fail due to insufficient access rights. This is probably another case where we integrated the key at the wrong Travis URL.

• introduce notation for interval matrices [A]

• revise example from the Quickstart about squaring interval matrices (ref. #80, Kosheleva et al's paper).

• non-associativity of matrix products

• SUE

Actually, what we need is, given an interval matrix M, return the conventional matrix C and the conventional matrix S such that M = C + [-S, S].

The term "expm" comes from pre-1.0 Julia, which used expm for the matrix exponential. It's now called exp, so i propose to rename expm_overapproximation to exp_overapproximation and similarly to the other methods that use expm.

As a reminder, the new method can be applied here:

Originally posted by @mforets in #73 (comment)

This is F from [Theorem 3, 1].

[1] https://mediatum.ub.tum.de/doc/1289985/728488.pdf

Problem

The k-th integer power of an interval matrix M, M^k, is not well-defined. The reason is that interval-matrix multiplication is not associative. Furthermore, even for a fixed order, the multiplication induces an error due to the intervals. The only safe case is squaring (i.e., k = 2) (see #79).

Idea

Decompose the power k such that many squares are used. In other words: use as few non-square multiplications as possible.

Example

In the example below, we compute M^9.

• A and B use the naive power operation, which are the worst options. (Interestingly, ^ does something slightly more intelligent. ?> ^ says equivalent to \exp(p\log(A))).
• C, D, E use the fact that 9 = 2² + 2² + 1.
• F, G, H use the fact that 9 = (2 + 1)³ + 2 + 1 (I only tried three permutations).
• As can be seen, the order of the multiplications plays a role.

julia> M = rand(IntervalMatrix)
2×2 IntervalMatrix{Float64,Interval{Float64},Array{Interval{Float64},2}}:
 [-1.88661, 1.81318]     [-0.483837, 0.892705]
 [-0.893564, -0.447015]  [-0.10882, 1.85449]

julia> A = M*M*M*M*M*M*M*M*M  # 9 multiplications
2×2 IntervalMatrix{Float64,Interval{Float64},Array{Interval{Float64},2}}:
 [-4394.68, 4388.85]  [-4300.39, 4320.52]
 [-3599.78, 3582.38]  [-3495.86, 3545.53]

julia> B = M^9
2×2 IntervalMatrix{Float64,Interval{Float64},Array{Interval{Float64},2}}:
 [-3734.67, 3685.31]  [-3311.99, 3430.62]
 [-3602.63, 3458.46]  [-3008.63, 3290.32]

julia> M2 = square(M); M4 = square(M2);

julia> C = M4*M4*M  # 2 multiplications
2×2 IntervalMatrix{Float64,Interval{Float64},Array{Interval{Float64},2}}:
 [-2606.43, 2225.51]  [-1907.27, 2779.07]
 [-2700.44, 2863.45]  [-2624.69, 2006.86]

julia> D = M4*M*M4  # 2 multiplications
2×2 IntervalMatrix{Float64,Interval{Float64},Array{Interval{Float64},2}}:
 [-3126.19, 2073.73]  [-2410.01, 2737.56]
 [-2740.19, 2372.8]   [-2575.55, 2563.49]

julia> E = M*M4*M4  # 2 multiplications
2×2 IntervalMatrix{Float64,Interval{Float64},Array{Interval{Float64},2}}:
 [-3077.05, 2643.18]  [-2871.06, 2697.84]
 [-2781.74, 1880.41]  [-2624.69, 2006.86]

julia> F = square(M2*M) * M2*M  # 3 multiplications
2×2 IntervalMatrix{Float64,Interval{Float64},Array{Interval{Float64},2}}:
 [-3310.01, 2898.24]  [-2427.69, 3330.15]
 [-2954.44, 2930.17]  [-2664.27, 2640.69]

julia> G = M2*M * square(M2*M)  # 3 multiplications
2×2 IntervalMatrix{Float64,Interval{Float64},Array{Interval{Float64},2}}:
 [-3432.39, 2898.24]  [-2646.09, 2878.82]
 [-3104.07, 2437.67]  [-2388.99, 2603.88]

julia> H = M2 * square(M2*M) * M  # 3 multiplications
2×2 IntervalMatrix{Float64,Interval{Float64},Array{Interval{Float64},2}}:
 [-3211.14, 2877.44]  [-2416.47, 3160.68]
 [-3121.46, 3105.47]  [-2767.25, 2662.37]

Heuristics

Presumably there is no optimal way to decompose the power, even when ignoring the order of the multiplications. A heuristics could be to decompose into the largest square numbers.

Sharing results

There is another issue: In the above algorithm/heuristics we may compute certain matrix powers several times. It might be better to have some symbolic representation of the decomposition (as we had in the example) and then compute each term only once. However, that sounds more complicated.

• give formula saying what is t and what is p

Currently the computation of the infinity norm uses left and right:

However, these methods allocate one new vector (of the size of the matrix) each. Since in the end we are only interested in a single value, this is not necessary.

function infinity_norm(A)
    res = abs(A[1, 1].left)
    for itv in A
        res = max(res, abs(itv.left), itv.right)
    end
    return res
end

(Note that I omitted the abs on itv.right for efficiency.)

We should also evaluate if the following is faster (rationale: write only when necessary).

function infinity_norm2(A)
    res = abs(A[1, 1].left)
    for itv in A
        m = max(abs(itv.left), itv.right)
        if m > res
            res = m
        end
    end
    return res
end

Write a function that computes the sum \sum_{i=i0}^{p} A^i * t^i / i!}. Note that if i0=0 one should use the exact computation of At + 1/2A^2 t^2 (matrix W).

• outsource existing code to new function (#123)
• make code parametric in i0

julia> K = copy(Pint.s.A)
4×4 IntervalMatrix{Float64,IntervalArithmetic.Interval{Float64},Array{IntervalArithmetic.Interval{Float64},2}}:
     [0, 0]    [0, 0]  [-2500, -2500]    [2500, 2500]
     [0, 0]    [0, 0]          [0, 0]  [-2500, -2500]
   [10, 10]    [0, 0]    [-100, -100]          [0, 0]
 [-10, -10]  [10, 10]          [0, 0]      [-10, -10]

julia> expm_overapproximation(K, 0.002, 12)
4×4 IntervalMatrix{Float64,IntervalArithmetic.Interval{Float64},Array{IntervalArithmetic.Interval{Float64},2}}:
 [-5619.76, 5621.58]  [-5620.62, 5620.72]  …  [-5615.96, 5625.38]
 [-5620.62, 5620.72]  [-5619.72, 5621.62]     [-5625.46, 5615.88]
 [-5620.65, 5620.69]  [-5620.67, 5620.67]     [-5620.62, 5620.72]
 [-5620.69, 5620.65]  [-5620.65, 5620.69]     [-5619.79, 5621.55]

julia> expm_overapproximation_old(K, 0.002, 12)
4×4 IntervalMatrix{Float64,IntervalArithmetic.Interval{Float64},Array{IntervalArithmetic.Interval{Float64},2}}:
  [0.600556, 1.21044]   [-0.256488, 0.353395]  …   [4.40467, 5.01456]  
 [-0.256488, 0.353395]   [0.64621, 1.2561]        [-5.09231, -4.48241] 
 [-0.287424, 0.322458]  [-0.30463, 0.305252]      [-0.259597, 0.350285]
 [-0.32378, 0.286103]   [-0.285792, 0.324091]      [0.578608, 1.18849] 

julia> exp(mid(K) * 0.002)
4×4 Array{Float64,2}:
  0.905497   0.0484535    -4.37927     4.70961  
  0.0484535  0.951152     -0.0777469  -4.78736  
  0.0175171  0.000310987   0.77567     0.0453436
 -0.0188385  0.0191494     0.0453436   0.883549 

where expm_overapproximation_old is using the function expm_overapproximation before we made the changes that entered in v0.2.0, so it could be in PR#18 or PR#19, or it could be something else.

Note that in this example the intervals in K have width zero. The huge intervals [-5619.76, 5621.58] seem like a bug.

Rump has a test for invertibility of an interval matrix.

This is E(r) in [Theorem 1, 1].

[1] https://mediatum.ub.tum.de/doc/1289985/728488.pdf

Here are some optimizations for the function quadratic_expansion (below):

• Swap iteration over i/j (iterate over columns in Julia).
• Compute t^2/2 only once.
• Compute S first (avoids double access to each cell).
• Inline k ≠ i conditions (have two loops from 1 to i-1 and from i+1 to n) (ifs in loops are performance killers); the version for k ≠ i && k ≠ j needs three loops.
  I think there was some neat Julia trick to make this elegant, but I forgot; the body can be written as a macro to avoid code duplication.
• Generalize to numeric type of A (no performance improvement, but nice to have).

Implement Proposition 1 in https://mediatum.ub.tum.de/doc/1287218/1287218.pdf

See also https://ieeexplore.ieee.org/document/1446499/

Given an interval matrix A, we can compute A² exactly. This is not true for higher powers. See [1, Section 6] for an algorithm.

[1] Olga Kosheleva, Vladik Kreinovich, Günter Mayer, Hung T. Nguyen: Computing the cube of an interval matrix is NP-Hard. SAC 2005. [PDF]

[Theorem 4.1.11, N90] yeilds an enclosure for the inverse of a strongly regular interval matrix.

[N90] - A. Neumaier. Interval Methods for Systems of Equations. Cambridge Univ. Press,
1990.

• https://www.jstor.org/stable/2158215?seq=1

• https://eudml.org/doc/233362

It would be great to implement Rump’s method for efficiently multiplying interval matrices using only floating-point matrix operations (with rounding mode changes).

https://link.springer.com/chapter/10.1007/978-3-642-15956-5_2

Rump has a test for positive definiteness of interval matrices.

https://mediatum.ub.tum.de/doc/1287218/1287218.pdf

Here are some optimizations for the function expm_overapproximation (below):

• Do not explicitly allocate the identity matrices (2x) resp. the [-1, 1] matrix but rather modify the matrices where they are used later.
• Introduce a variable for t^i/factorial(i) in the loop and then just multiply with t/i instead of recomputing from scratch.
• Swap the order in the last loop (saves one multiplication of A)
• Put the assertion to the top.
• Generalize to numeric type of A (no performance improvement, but nice to have).

Write a function that computes the sum \sum_{i=i0}^{p} A^i * t^{i+1} / (i+1)!}.

It is handy to have a function for creating an identity matrix.

Maybe Diagonal(one_interval, n) "just works" (not tested).

We can offer the in-place version and the out-of-place version.

In this line

https://github.com/JuliaReach/IntervalMatrices.jl/blob/master/src/exponential.jl#L91

both W and E are just of type Matrix{Interval} instead of IntervalMatrix.

If we call several functions that compute _expm_remainder (like in Reachability), it would be better to instead compute it only once and pass the result around.

See the Codecov report, which can be easily brought close to 100%.

Implement W* from Prop. 2 in [Reachability Analysis of Linear Systems
with Uncertain Parameters and Inputs] using SUE.

A - B does not return an IntervalMatrix.

julia> rand(IntervalMatrix) - rand(IntervalMatrix)
2×2 Array{Interval{Float64},2}:
  [0.16067, 1.02546]    [1.04121, 1.29847] 
 [-2.29271, -1.06139]  [-0.32639, 0.903041]

The current behavior is rather arbitrary:

julia> [size(rand(IntervalMatrix, i), 2) for i in 2:5]

4-element Array{Int64,1}:
 2
 2
 2
 2

It would be arguably less arbitrary if rand(IntervalMatrix, i) returns a square matrix of order i.

There are several references, see e.g.

• Bounding the eigenvectors of symmetric interval matrices. Dr. N. P. Seif S. Hashem Prof. Dr. A. S. Deif.

• On the range of eigenvalues of an interval matrix. Jiri Rohn Assem Deif Assem Deif.

• Eigenvalues of symmetric interval matrices. Nikolay Stoyanov Kaleyski

• Computing the range of real eigenvalues of an interval
  matrix. Milan Hladik & David Daney.

• Bounds on Eigenvalues of Interval Matrices. Rohn, J.

• Hartman, David, Milan Hladík, and David Říha. "Computing the spectral decomposition of interval matrices and a study on interval matrix power." arXiv preprint arXiv:1912.05275 (2019). pdf

See also the package: https://github.com/dpsanders/IntervalEigenvalues.jl

• https://arxiv.org/pdf/0908.3954.pdf
• https://www.mat.univie.ac.at/~neum/ms/intExp.pdf