#75 - Truncated exponential sum

[schillic]

This is the first step for #75. The proposed starting index i0 is missing.

• first commit: just a refactoring of the old code (plus adding assertions p < 2 that were not checked before)
• second commit: use the new IntervalMatrixPower type
• third commit: generalize code to nonnegative p

An example shows that the simple use of square in IntervalMatrixPower already pays off.

julia> A = rand(IntervalMatrix)
2×2 IntervalMatrix{Float64,Interval{Float64},Array{Interval{Float64},2}}:
 [-0.152272, 1.1966]   [-0.340722, 1.00641]
 [-0.688822, 1.30464]  [-1.08989, 2.67171]

julia> exp_overapproximation(A, 1.0, 10)  # master
2×2 IntervalMatrix{Float64,Interval{Float64},Array{Interval{Float64},2}}:
 [-1.39748, 7.5724]   [-3.65965, 9.33968]
 [-6.39121, 12.0942]  [-5.87441, 21.2407]

julia> d1 = diam(exp_overapproximation(A, 1.0, 10))
2×2 Array{Float64,2}:
  8.96987  12.9993
 18.4854   27.115 

julia> exp_overapproximation(A, 1.0, 10)  # this branch
2×2 IntervalMatrix{Float64,Interval{Float64},Array{Interval{Float64},2}}:
 [-1.23017, 7.5724]   [-3.46151, 9.33968]
 [-6.39121, 12.0942]  [-4.36419, 21.2407]

julia> d2 = diam(exp_overapproximation(A, 1.0, 10))
2×2 Array{Float64,2}:
  8.80256  12.8012
 18.4854   25.6048

julia> d1 - d2
2×2 Array{Float64,2}:
 0.167307  0.198139
 0.0       1.51022 

[mforets]

LGTM, very nice!

[mforets]

This is the first step for #75. The proposed starting index i0 is missing.

Can you open an issue (or update #75?) so that we don't forget what remains?