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column-major access and other perf improvements for generic triangular solves #14475

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Jan 10, 2016
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column-major access and other perf improvements for generic triangular solves #14475

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69 changes: 31 additions & 38 deletions base/linalg/triangular.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -763,69 +763,62 @@ function A_mul_Bc!(A::StridedMatrix, B::UnitLowerTriangular)
end

#Generic solver using naive substitution
function naivesub!(A::UpperTriangular, b::AbstractVector, x::AbstractVector=b)
# manually hoisting x[j] significantly improves performance as of Dec 2015
# manually eliding bounds checking significantly improves performance as of Dec 2015
# directly indexing A.data rather than A significantly improves performance as of Dec 2015
# replacing repeated references to A.data with [Adata = A.data and references to Adata] does not significantly impact performance as of Dec 2015
# replacing repeated references to A.data[j,j] with [Ajj = A.data[j,j] and references to Ajj] does not significantly impact performance as of Dec 2015
function naivesub!(A::UpperTriangular, b::AbstractVector, x::AbstractVector = b)
n = size(A, 2)
if !(n == length(b) == length(x))
throw(DimensionMismatch("second dimension of left hand side A, $n, length of output x, $(length(x)), and length of right hand side b, $(length(b)) must be equal"))
throw(DimensionMismatch("second dimension of left hand side A, $n, length of output x, $(length(x)), and length of right hand side b, $(length(b)), must be equal"))
end
for j = n:-1:1
xj = b[j]
for k = j+1:1:n
xj -= A[j,k] * x[k]
end
Ajj = A[j,j]
if Ajj == zero(Ajj)
throw(SingularException(j))
else
x[j] = Ajj\xj
@inbounds for j in n:-1:1
A.data[j,j] == zero(A.data[j,j]) && throw(SingularException(j))
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might help to do Ajj = A.data[j,j] once instead of 3x

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I thought so as well. So I benchmarked. No significant performance impact. I suppose the compiler is intelligent enough to perform that optimization automagically?

For benefit of future readers, alongside the other optimization notes I added sentences addressing this optimization and manually hoisting A.data out of the loops.

Thanks!

xj = x[j] = A.data[j,j] \ b[j]
for i in j-1:-1:1 # counterintuitively 1:j-1 performs slightly better
b[i] -= A.data[i,j] * xj
end
end
x
end
function naivesub!(A::UnitUpperTriangular, b::AbstractVector, x::AbstractVector=b)
function naivesub!(A::UnitUpperTriangular, b::AbstractVector, x::AbstractVector = b)
n = size(A, 2)
if !(n == length(b) == length(x))
throw(DimensionMismatch("second dimension of left hand side A, $n, length of output x, $(length(x)), and length of right hand side b, $(length(b)) must be equal"))
throw(DimensionMismatch("second dimension of left hand side A, $n, length of output x, $(length(x)), and length of right hand side b, $(length(b)), must be equal"))
end
for j = n:-1:1
xj = b[j]
for k = j+1:1:n
xj -= A[j,k] * x[k]
@inbounds for j in n:-1:1
xj = x[j] = b[j]
for i in j-1:-1:1 # counterintuitively 1:j-1 performs slightly better
b[i] -= A.data[i,j] * xj
end
x[j] = xj
end
x
end
function naivesub!(A::LowerTriangular, b::AbstractVector, x::AbstractVector=b)
function naivesub!(A::LowerTriangular, b::AbstractVector, x::AbstractVector = b)
n = size(A, 2)
if !(n == length(b) == length(x))
throw(DimensionMismatch("second dimension of left hand side A, $n, length of output x, $(length(x)), and length of right hand side b, $(length(b)) must be equal"))
throw(DimensionMismatch("second dimension of left hand side A, $n, length of output x, $(length(x)), and length of right hand side b, $(length(b)), must be equal"))
end
for j = 1:n
xj = b[j]
for k = 1:j-1
xj -= A[j,k] * x[k]
end
Ajj = A[j,j]
if Ajj == zero(Ajj)
throw(SingularException(j))
else
x[j] = Ajj\xj
@inbounds for j in 1:n
A.data[j,j] == zero(A.data[j,j]) && throw(SingularException(j))
xj = x[j] = A.data[j,j] \ b[j]
for i in j+1:n
b[i] -= A.data[i,j] * xj
end
end
x
end
function naivesub!(A::UnitLowerTriangular, b::AbstractVector, x::AbstractVector=b)
function naivesub!(A::UnitLowerTriangular, b::AbstractVector, x::AbstractVector = b)
n = size(A, 2)
if !(n == length(b) == length(x))
throw(DimensionMismatch("second dimension of left hand side A, $n, length of output x, $(length(x)), and length of right hand side b, $(length(b)) must be equal"))
throw(DimensionMismatch("second dimension of left hand side A, $n, length of output x, $(length(x)), and length of right hand side b, $(length(b)), must be equal"))
end
for j = 1:n
xj = b[j]
for k = 1:j-1
xj -= A[j,k] * x[k]
@inbounds for j in 1:n
xj = x[j] = b[j]
for i in j+1:n
b[i] -= A.data[i,j] * xj
end
x[j] = xj
end
x
end
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