## source code

#=
The Computer Language Benchmarks Game
https://salsa.debian.org/benchmarksgame-team/benchmarksgame/
contributed by Adam Beckmeyer with help from Vincent Yu
=#
A(i, j) = (i + j - 2) * (i + j - 1) / 2 + i
At(i, j) = A(j, i)
# Multiply vector v by the matrix represented by function f (f(i, j) returns the element in
# i-th row and j-th column) and store result in out
function mul_by_f!(f, v, out)
n = length(v)
# Threads.@threads has lower overhead than Threads.@spawn
Threads.@threads :static for i=1:n
x1 = x2 = 0.0
# If we used the @simd macro instead of manually iterating by 2s, the compiler would
# emit instructions using the ymm registers instead of xmm which appears to be
# slower on ivybridge cpus
@inbounds for j=1:2:n
# Manually convert indices to Float64 so that arithmetic in function f can be
# carried out using vectorized floating point arithmetic
x1 += v[j] / f(Float64(i), Float64(j))
x2 += v[j+1] / f(Float64(i), Float64(j+1))
end
@inbounds out[i] = x1 + x2
end
end
# Multiply vector v by matrix A and store result in out
mul_by_A!(v, out) = mul_by_f!(A, v, out)
# Multiply vector v by matrix A' and store result in out
mul_by_At!(v, out) = mul_by_f!(At, v, out)
# Multiply v by (A' * A) and store result in out using w as a temporary workspace
function mul_by_AtA!(v, out, w)
mul_by_A!(v, w)
mul_by_At!(w, out)
end
function main(n)
# This program is not compatible with odd values of n
isodd(n) && (n += 1)
u = ones(Float64, n)
v = Vector{Float64}(undef, n)
# temporary working vector w
w = Vector{Float64}(undef, n)
for _=1:10
mul_by_AtA!(u, v, w)
mul_by_AtA!(v, u, w)
end
uv = vv = 0.0
@inbounds for i=1:n
uv += u[i] * v[i]
vv += v[i] * v[i]
end
sqrt(uv / vv)
end
# The expanded form of Printf.@printf macro takes a significant time to compile, accounting
# for 25% of the total program runtime. Base.Ryu.writefixed(::Float64, ::Int) should already
# be compiled into the default system image.
isinteractive() || println(Base.Ryu.writefixed(main(parse(Int, ARGS[1])), 9))

## notes, command-line, and program output

NOTES:
64-bit Ubuntu quad core
julia version build 19.0.1+10-21
Wed, 25 Jan 2023 03:32:09 GMT
MAKE:
printenv JULIA_NUM_THREADS
4
0.10s to complete and log all make actions
COMMAND LINE:
/opt/src/julia-1.8.5/bin/julia -O3 --cpu-target=ivybridge --math-mode=ieee -- spectralnorm.julia-4.julia 5500
PROGRAM OUTPUT:
1.274224153