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#131 - Matrix power via square decomposition, v2 #134

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Feb 12, 2020
Merged

#131 - Matrix power via square decomposition, v2 #134

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c36a519
add square-root decomposition
schillic Feb 11, 2020
6811d03
add algorithm to recursive call
schillic Feb 11, 2020
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22 changes: 20 additions & 2 deletions src/power.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -67,11 +67,12 @@ function IntervalMatrixPower(M::IntervalMatrix{T}) where {T}
return IntervalMatrixPower(M, M, 1)
end

function IntervalMatrixPower(M::IntervalMatrix{T}, k::Int) where {T}
function IntervalMatrixPower(M::IntervalMatrix{T}, k::Int;
algorithm::String="power") where {T}
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Should algorithm be a field of the type?

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I was also thinking about that. It is more general to let you choose at every step, but we could at least have a fallback algorithm defined in the type maybe?

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@mforets mforets Feb 11, 2020 •
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At first sight it looks better to me if the algorithm is in the type, since it is a property of how to get to M^k, and also how to get higher powers. (Of course i'm assuming that there is no mixed algorithm, but the "mixed" algorithm can be an algorithm by itself).

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Anyway, we can revisit this later.

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@schillic schillic Feb 12, 2020 •
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(Of course i'm assuming that there is no mixed algorithm, but the "mixed" algorithm can be an algorithm by itself).

Yes and no. If the mixed algorithm follows a strict pattern, it can be added as another algorithm. But adding a new algorithm for any pattern is not sustainable. Furthermore, you can imagine patterns that depend on user knowledge (like "next I need more precision") that you cannot express in a static algorithm. So I conclude that a String as a field is not sufficient. If we make the algorithm a field, it should be its own struct <:IntervalMatrixPowerAlgorithm that you can influence from outside.

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I opened #139.

@assert k >= 1 "matrix powers must be positive"
pow = IntervalMatrixPower(M, M, 1)
@inbounds for i in 1:(k-1)
increment!(pow)
increment!(pow; algorithm=algorithm)
end
return pow
end
Expand All @@ -93,6 +94,7 @@ Increment a matrix power in-place (i.e., storing the result in `pow`).
* `"multiply"` -- fast computation using `*` from the previous result
* `"power"` -- recomputation using `^`
* `"intersect"` -- combination of `"multiply"` and `"power"`
* `"sqrt"` -- decompose `k = a² + b`

### Output

Expand All @@ -113,6 +115,8 @@ function increment!(pow::IntervalMatrixPower; algorithm::String="intersect")
pow.Mᵏ = _eval_power(pow)
elseif algorithm == "intersect"
pow.Mᵏ = _eval_intersect(pow)
elseif algorithm == "sqrt"
pow.Mᵏ = _eval_sqrt(pow)
else
throw(ArgumentError("algorithm $algorithm is not available; choose " *
"from 'multiply', 'power', 'intersect'"))
Expand Down Expand Up @@ -164,6 +168,20 @@ function _eval_intersect(pow::IntervalMatrixPower)
return intersect(_eval_multiply(pow), _eval_power(pow))
end

function _eval_sqrt(pow::IntervalMatrixPower; algorithm::String="power")
# decompose k = a² + b with a, b being integers
k = pow.k
a = floor(Int, sqrt(k))
b = k - a^2

# recursively compute M^a and M^b
Mᵏ = square(get(IntervalMatrixPower(pow.M, a; algorithm=algorithm)))
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@mforets mforets Feb 11, 2020 •
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Is line 178 the same as get(IntervalMatrixPower(pow.M, a*a) ? (Also, this is the functionality that i was asking in this comment. I'll create an issue).

EDIT: This is now #138.

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My question still is why not use get(IntervalMatrixPower(pow.M, a*a; algorithm=algorithm))) here.

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@schillic schillic Feb 12, 2020 •
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why not use get(IntervalMatrixPower(pow.M, a*a; algorithm=algorithm))) here.

Because it is not equivalent. We do not use square for arbitrary square numbers but only for powers of 2.

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Oh, damn, your version was the correct one:
A^(a²) = A^(a*a) = (A^a)^a != (A^a)² for a != 2

if b > 0
Mᵏ *= get(IntervalMatrixPower(pow.M, b; algorithm=algorithm))
end
return Mᵏ
end

"""
base(pow::IntervalMatrixPower)

Expand Down
1 change: 1 addition & 0 deletions test/runtests.jl
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Expand Up @@ -142,4 +142,5 @@ end
increment!(pow) # power of two
increment!(pow, algorithm="power")
increment!(pow, algorithm="intersect")
increment!(pow, algorithm="sqrt")
end