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Efficiency improvement of exp(::StridedMatrix) with UniformScaling and mul! #40668

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merged 5 commits into from
May 6, 2021
Merged

Efficiency improvement of exp(::StridedMatrix) with UniformScaling and mul! #40668

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28d77af
Reduce nof matmat's in exp(StridedMatrix)
jarlebring Apr 30, 2021
44b9eaa
exp(::StridedMatrix): use Uniformscaling
jarlebring Apr 30, 2021
ae811e8
exp!() In-place mul! for case nA<2.1
jarlebring May 4, 2021
e10d87f
exp!() mul! for allocation economical also for nA > 2.1
jarlebring May 4, 2021
0c3c520
exp!() Preserve original order in U update for nA>2.1
jarlebring May 4, 2021
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42 changes: 31 additions & 11 deletions stdlib/LinearAlgebra/src/dense.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -617,7 +617,6 @@ function exp!(A::StridedMatrix{T}) where T<:BlasFloat
end
ilo, ihi, scale = LAPACK.gebal!('B', A) # modifies A
nA = opnorm(A, 1)
Inn = Matrix{T}(I, n, n)
## For sufficiently small nA, use lower order Padé-Approximations
if (nA <= 2.1)
if nA > 0.95
Expand All @@ -634,17 +633,21 @@ function exp!(A::StridedMatrix{T}) where T<:BlasFloat
C = T[120.,60.,12.,1.]
end
A2 = A * A
P = copy(Inn)
U = C[2] * P
V = C[1] * P
for k in 1:(div(size(C, 1), 2) - 1)
# Compute U and V: Even/odd terms in Padé numerator & denom
# Expansion of k=1 in for loop
P = A2
U = C[2]*I + C[4]*P
V = C[1]*I + C[3]*P
for k in 2:(div(size(C, 1), 2) - 1)
k2 = 2 * k
P *= A2
U += C[k2 + 2] * P
V += C[k2 + 1] * P
mul!(U, C[k2 + 2], P, true, true) # U += C[k2+2]*P
mul!(V, C[k2 + 1], P, true, true) # V += C[k2+1]*P
end

U = A * U
X = V + U
# Padé approximant: (V-U)\(V+U)
LAPACK.gesv!(V-U, X)
else
s = log2(nA/5.4) # power of 2 later reversed by squaring
Expand All @@ -660,10 +663,27 @@ function exp!(A::StridedMatrix{T}) where T<:BlasFloat
A2 = A * A
A4 = A2 * A2
A6 = A2 * A4
U = A * (A6 * (CC[14].*A6 .+ CC[12].*A4 .+ CC[10].*A2) .+
CC[8].*A6 .+ CC[6].*A4 .+ CC[4].*A2 .+ CC[2].*Inn)
V = A6 * (CC[13].*A6 .+ CC[11].*A4 .+ CC[9].*A2) .+
CC[7].*A6 .+ CC[5].*A4 .+ CC[3].*A2 .+ CC[1].*Inn
Ut = CC[4]*A2
Ut[diagind(Ut)] .+= CC[2]
# Allocation economical version of:
#U = A * (A6 * (CC[14].*A6 .+ CC[12].*A4 .+ CC[10].*A2) .+
# CC[8].*A6 .+ CC[6].*A4 .+ Ut)
U = mul!(CC[8].*A6 .+ CC[6].*A4 .+ Ut,
A6,
CC[14].*A6 .+ CC[12].*A4 .+ CC[10].*A2,
true, true)
U = A*U

# Allocation economical version of: Vt = CC[3]*A2 (recycle Ut)
Vt = mul!(Ut, CC[3], A2, true, false)
Vt[diagind(Vt)] .+= CC[1]
# Allocation economical version of:
#V = A6 * (CC[13].*A6 .+ CC[11].*A4 .+ CC[9].*A2) .+
# CC[7].*A6 .+ CC[5].*A4 .+ Vt
V = mul!(CC[7].*A6 .+ CC[5].*A4 .+ Vt,
A6,
CC[13].*A6 .+ CC[11].*A4 .+ CC[9].*A2,
true, true)

X = V + U
LAPACK.gesv!(V-U, X)
Expand Down