# Ldiv for Cholesky is slower than two substitutions

**URL:** https://discourse.julialang.org/t/ldiv-for-cholesky-is-slower-than-two-substitutions/130092
**Category:** Performance
**Created:** [June 21, 2025, 3:37pm UTC](https://discourse.julialang.org/t/ldiv-for-cholesky-is-slower-than-two-substitutions/130092 "2025-06-21T15:37:26Z")
**Posts on this page:** 1
**Showing post:** 6

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### Author: ![danielwe](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/danielwe/32/35657_2.png) [@danielwe](https://discourse.julialang.org/u/danielwe)
#### Post date: [June 22, 2025, 4:31pm UTC](https://discourse.julialang.org/t/ldiv-for-cholesky-is-slower-than-two-substitutions/130092/6 "2025-06-22T16:31:24Z")

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> [@stepanzh](#):
>
> `julia> @benchmark U \ (L \ $b) setup=(L=$Achol.L; U=$Achol.U)`

To elaborate on what @stevengj said: You’re pulling the most costly part of the computation out into the setup here, so you’re only measuring a small part of the actual cost of the operation. In a fair benchmark, `Achol \ b` wins handily. Here’s what I find:

```julia-repl
julia> @benchmark $Achol \ $b
BenchmarkTools.Trial: 10000 samples with 1 evaluation per sample.
 Range (min … max): 248.917 μs … 388.625 μs ┊ GC (min … max): 0.00% … 0.00%
 Time (median): 261.416 μs ┊ GC (median): 0.00%
 Time (mean ± σ): 263.493 μs ± 9.988 μs ┊ GC (mean ± σ): 0.00% ± 0.00%

   ▁▂ ▃██▄▂▃▃▃▂▁
  ▃██▇▆▆████████████▆▇▇▇▇▆▆▆▇▆▇▇▇▇▇▇▇▇▇▆▅▅▅▆▅▅▄▄▄▄▃▃▂▂▂▃▂▂▂▂▂▂▂ ▄
  249 μs Histogram: frequency by time 289 μs <

 Memory estimate: 8.00 KiB, allocs estimate: 1.

julia> @benchmark U \ (L \ $b) setup=(U = $Achol.U; L = $Achol.L) # Unfair
BenchmarkTools.Trial: 5858 samples with 1 evaluation per sample.
 Range (min … max): 132.750 μs … 227.459 μs ┊ GC (min … max): 0.00% … 0.00%
 Time (median): 145.292 μs ┊ GC (median): 0.00%
 Time (mean ± σ): 145.427 μs ± 5.723 μs ┊ GC (mean ± σ): 0.00% ± 0.00%

             ▁▁▄▄▄▄▄▄▃▂▄▄▄▄▆▅█▆▅▄▆▄▄▅▅▂▃
  ▁▁▁▁▂▃▄▅▄▇▇███████████████████████████▇▇▅▆▄▃▃▃▂▂▂▂▂▂▂▂▂▂▁▁▁▁▁ ▅
  133 μs Histogram: frequency by time 162 μs <

 Memory estimate: 16.00 KiB, allocs estimate: 2.

julia> myldiv(chol, x) = chol.U \ (chol.L \ x)
myldiv (generic function with 1 method)

julia> @benchmark myldiv($Achol, $b) # Fair, but slower
BenchmarkTools.Trial: 5796 samples with 1 evaluation per sample.
 Range (min … max): 659.125 μs … 2.103 ms ┊ GC (min … max): 0.00% … 57.96%
 Time (median): 765.250 μs ┊ GC (median): 0.00%
 Time (mean ± σ): 862.006 μs ± 209.153 μs ┊ GC (mean ± σ): 9.99% ± 14.68%

    ▁▂▂▆██▆▄▄▄▄▃▂▂▂▁▁ ▁▁▂▂▃▃▃▂▂▂▁▁▁ ▂
  ▆▆█████████████████▇▆▇▅▁▄▄▃▃▁▁▁▁▁▁▁▁▄▆████████████████▇█▆▆▆▅▄ █
  659 μs Histogram: log(frequency) by time 1.5 ms <

 Memory estimate: 7.65 MiB, allocs estimate: 4.

```

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