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For numerical calculations, allocating storage space on the heap can be a significant cost. Often, we do the same thing over and over, resulting in many allocations for outputs (or temporay intermediate values). Thus, there's an opportunity to allocate storage space once and reuse it in subsequent calculations.

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Astro 528: High-Performance Scientific Computing for Astrophysics (Fall 2025)

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Non-expert call to compute gradient

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Where do your variables get stored?

Your program will use two distinct virtual address spaces:

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Expert vs non-expert routines

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How to generate less garbage?

For small allocations, use data types that stay on the stack

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Week 4 Discussion Topics

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Mutable fixed-size vectors/arrays

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Examples of simple struct's

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ToC on side

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Helper Code

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What if I want to mutate my struct's?

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Potentially additional optional parameters

E.g., Here we can turn off "tag checking". (See documentation about custom tags for details.

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Garbage Collection

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Interactive Memory Latency vs Year

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multiply_matrix_vector_preallocated!

multiply_matrix_vector_preallocated!(out::Vector, A::Matrix, b::Vector)

Multiply matrix A and vector b by hand using rows for inner loop, using preallocated space for output

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Question:

Will this be stored on the stack or heap?

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Optional parameter for improved performance

Here we specify the chunk size that affects both how many calls to the function (larger chunk→fewer calls) and how much memory is needed (larger chunk→more memory)

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Keeping small arrays in the stack and out of the heap

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Why default to immutable?

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Curious to learn more?

Pro Tip

Julia's garbage collector is "a non-compacting, generational, mark-and-sweep, tracing collector, which at a high level means the following…

Mark-Sweep / Tracing:

  • When the garbage collector runs, it starts from a set of “roots” and walks through the object graph, “marking” objects as reachable.

  • Any object that isn’t marked as reachable and will then be “swept away” — i.e. its memory is reclaimed—since you know it’s not reachable from the roots and is therefore garbage.

Generational:

  • It’s common that more recently allocated objects become garbage more quickly—this is known as the “generational hypothesis”.

  • Generational collectors have different levels of collection: young collections which run more frequently and full collections which run less frequently.

  • If the generational hypothesis is true, this saves time since it’s a waste of time to keep checking if older objects are garbage when they’re probably not."

Non-compacting / Non-moving:

  • Other garbage collection techniques can copy or move objects during the collection process.

  • Julia does not use any of these—collection does not move or copy anything, it just reclaims unreachable objects.

If you’re having issues with garbage collection, your primary recourse is to generate less garbage:

  • Write non-allocating code wherever possible: simple scalar code can generally avoid allocations.

  • Use immutable objects (i.e., struct rather than mutable struct), which can be stack allocated more easily and stored inline in other structures (as compared to mutable objects which generally need to be heap-allocated and stored indirectly by pointer, all of which causes more memory pressure).

  • Use pre-allocated data structures and modify them instead of allocating and returning new objects, especially in loops.

  • Can call GC.gc() to manually call garbage collector. But this is mainly useful for benchmarking.

(nearly quote from Julia Discourse post by Stefan Karpinski)
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Physical differences cause large difference in memory latency and bandwidth:

Memory latency heirarchy

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Meaning of equality can be confusing

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Implications

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Hint

  • sizeof(mms) = 16

  • sizeof(mms.result) = 8

  • sizeof(mms.message) = 26

  • Base.summarysize(mms.message) =34

  • Base.summarysize(mms) =50

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Week 4 Admin Announcements

Project proposals

Feedback

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Mutating version for improved performance

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Limitations of immutable structs

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Warning about mutating collections in Pluto

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Pre-allocating storage space

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Computer Memory Systems

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multiply_matrix_vector_allocates

multiply_matrix_vector_allocates(A::Matrix, b::Vector)

Multiply matrix A and vector b by hand, allocating space for output

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W Resolving... ===  Installed LERC_jll ───────────────── v4.0.1+0  Installed Collects ───────────────── v1.0.0  Installed GR_jll ─────────────────── v0.73.17+0  Installed Libmount_jll ───────────── v2.41.1+0  Installed JpegTurbo_jll ──────────── v3.1.2+0  Installed Pango_jll ──────────────── v1.56.3+0  Installed SpecialFunctions ───────── v2.5.1  Installed BenchmarkTools ─────────── v1.6.0  Installed ColorVectorSpace ───────── v0.11.0  Installed CommonSubexpressions ───── v0.3.1  Installed Qt6ShaderTools_jll ─────── v6.8.2+1  Installed Qt6Declarative_jll ─────── v6.8.2+1  Installed Libtiff_jll ────────────── v4.7.1+0  Installed Xorg_xcb_util_cursor_jll ─ v0.1.5+0  Installed ColorSchemes ───────────── v3.30.0  Installed OpenSpecFun_jll ────────── v0.5.6+0  Installed PlutoTeachingTools ─────── v0.4.5  Installed FixedSizeArrays ────────── v1.2.0  Installed GR ─────────────────────── v0.73.17  Installed Libuuid_jll ────────────── v2.41.1+0  Installed Xorg_libXfixes_jll ─────── v6.0.1+0  Installed Qt6Wayland_jll ─────────── v6.8.2+1  Installed Qt6Base_jll ────────────── v6.8.2+1  Installed Glib_jll ───────────────── v2.84.3+0  Installed ForwardDiff ────────────── v1.1.0  Installed Plots ──────────────────── v1.40.19  No Changes to `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_vmbwokibxs/Project.toml`  Updating `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_vmbwokibxs/Manifest.toml` [05823500] ↑ OpenLibm_jll v0.8.1+2 ⇒ v0.8.5+0 Instantiating... === Precompiling... === Precompiling project... StaticArrays W Resolving... ===  Installed LERC_jll ───────────────── v4.0.1+0  Installed Collects ───────────────── v1.0.0  Installed GR_jll ─────────────────── v0.73.17+0  Installed Libmount_jll ───────────── v2.41.1+0  Installed JpegTurbo_jll ──────────── v3.1.2+0  Installed Pango_jll ──────────────── v1.56.3+0  Installed SpecialFunctions ───────── v2.5.1  Installed BenchmarkTools ─────────── v1.6.0  Installed ColorVectorSpace ───────── v0.11.0  Installed CommonSubexpressions ───── v0.3.1  Installed Qt6ShaderTools_jll ─────── v6.8.2+1  Installed Qt6Declarative_jll ─────── v6.8.2+1  Installed Libtiff_jll ────────────── v4.7.1+0  Installed Xorg_xcb_util_cursor_jll ─ v0.1.5+0  Installed ColorSchemes ───────────── v3.30.0  Installed OpenSpecFun_jll ────────── v0.5.6+0  Installed PlutoTeachingTools ─────── v0.4.5  Installed FixedSizeArrays ────────── v1.2.0  Installed GR ─────────────────────── v0.73.17  Installed Libuuid_jll ────────────── v2.41.1+0  Installed Xorg_libXfixes_jll ─────── v6.0.1+0  Installed Qt6Wayland_jll ─────────── v6.8.2+1  Installed Qt6Base_jll ────────────── v6.8.2+1  Installed Glib_jll ───────────────── v2.84.3+0  Installed ForwardDiff ────────────── v1.1.0  Installed Plots ──────────────────── v1.40.19  No Changes to `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_vmbwokibxs/Project.toml`  Updating `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_vmbwokibxs/Manifest.toml` [05823500] ↑ OpenLibm_jll v0.8.1+2 ⇒ v0.8.5+0 Instantiating... === Precompiling... === Precompiling project... ForwardDiff W Resolving... ===  Installed LERC_jll ───────────────── v4.0.1+0  Installed Collects ───────────────── v1.0.0  Installed GR_jll ─────────────────── v0.73.17+0  Installed Libmount_jll ───────────── v2.41.1+0  Installed JpegTurbo_jll ──────────── v3.1.2+0  Installed Pango_jll ──────────────── v1.56.3+0  Installed SpecialFunctions ───────── v2.5.1  Installed BenchmarkTools ─────────── v1.6.0  Installed ColorVectorSpace ───────── v0.11.0  Installed CommonSubexpressions ───── v0.3.1  Installed Qt6ShaderTools_jll ─────── v6.8.2+1  Installed Qt6Declarative_jll ─────── v6.8.2+1  Installed Libtiff_jll ────────────── v4.7.1+0  Installed Xorg_xcb_util_cursor_jll ─ v0.1.5+0  Installed ColorSchemes ───────────── v3.30.0  Installed OpenSpecFun_jll ────────── v0.5.6+0  Installed PlutoTeachingTools ─────── v0.4.5  Installed FixedSizeArrays ────────── v1.2.0  Installed GR ─────────────────────── v0.73.17  Installed Libuuid_jll ────────────── v2.41.1+0  Installed Xorg_libXfixes_jll ─────── v6.0.1+0  Installed Qt6Wayland_jll ─────────── v6.8.2+1  Installed Qt6Base_jll ────────────── v6.8.2+1  Installed Glib_jll ───────────────── v2.84.3+0  Installed ForwardDiff ────────────── v1.1.0  Installed Plots ──────────────────── v1.40.19  No Changes to `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_vmbwokibxs/Project.toml`  Updating `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_vmbwokibxs/Manifest.toml` [05823500] ↑ OpenLibm_jll v0.8.1+2 ⇒ v0.8.5+0 Instantiating... === Precompiling... === Precompiling project... nbpkg_sync W Resolving... ===  Installed LERC_jll ───────────────── v4.0.1+0  Installed Collects ───────────────── v1.0.0  Installed GR_jll ─────────────────── v0.73.17+0  Installed Libmount_jll ───────────── v2.41.1+0  Installed JpegTurbo_jll ──────────── v3.1.2+0  Installed Pango_jll ──────────────── v1.56.3+0  Installed SpecialFunctions ───────── v2.5.1  Installed BenchmarkTools ─────────── v1.6.0  Installed ColorVectorSpace ───────── v0.11.0  Installed CommonSubexpressions ───── v0.3.1  Installed Qt6ShaderTools_jll ─────── v6.8.2+1  Installed Qt6Declarative_jll ─────── v6.8.2+1  Installed Libtiff_jll ────────────── v4.7.1+0  Installed Xorg_xcb_util_cursor_jll ─ v0.1.5+0  Installed ColorSchemes ───────────── v3.30.0  Installed OpenSpecFun_jll ────────── v0.5.6+0  Installed PlutoTeachingTools ─────── v0.4.5  Installed FixedSizeArrays ────────── v1.2.0  Installed GR ─────────────────────── v0.73.17  Installed Libuuid_jll ────────────── v2.41.1+0  Installed Xorg_libXfixes_jll ─────── v6.0.1+0  Installed Qt6Wayland_jll ─────────── v6.8.2+1  Installed Qt6Base_jll ────────────── v6.8.2+1  Installed Glib_jll ───────────────── v2.84.3+0  Installed ForwardDiff ────────────── v1.1.0  Installed Plots ──────────────────── v1.40.19  No Changes to `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_vmbwokibxs/Project.toml`  Updating `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_vmbwokibxs/Manifest.toml` [05823500] ↑ OpenLibm_jll v0.8.1+2 ⇒ v0.8.5+0 Instantiating... === Precompiling... === Precompiling project... PlutoUI W Resolving... ===  Installed LERC_jll ───────────────── v4.0.1+0  Installed Collects ───────────────── v1.0.0  Installed GR_jll ─────────────────── v0.73.17+0  Installed Libmount_jll ───────────── v2.41.1+0  Installed JpegTurbo_jll ──────────── v3.1.2+0  Installed Pango_jll ──────────────── v1.56.3+0  Installed SpecialFunctions ───────── v2.5.1  Installed BenchmarkTools ─────────── v1.6.0  Installed ColorVectorSpace ───────── v0.11.0  Installed CommonSubexpressions ───── v0.3.1  Installed Qt6ShaderTools_jll ─────── v6.8.2+1  Installed Qt6Declarative_jll ─────── v6.8.2+1  Installed Libtiff_jll ────────────── v4.7.1+0  Installed Xorg_xcb_util_cursor_jll ─ v0.1.5+0  Installed ColorSchemes ───────────── v3.30.0  Installed OpenSpecFun_jll ────────── v0.5.6+0  Installed PlutoTeachingTools ─────── v0.4.5  Installed FixedSizeArrays ────────── v1.2.0  Installed GR ─────────────────────── v0.73.17  Installed Libuuid_jll ────────────── v2.41.1+0  Installed Xorg_libXfixes_jll ─────── v6.0.1+0  Installed Qt6Wayland_jll ─────────── v6.8.2+1  Installed Qt6Base_jll ────────────── v6.8.2+1  Installed Glib_jll ───────────────── v2.84.3+0  Installed ForwardDiff ────────────── v1.1.0  Installed Plots ──────────────────── v1.40.19  No Changes to `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_vmbwokibxs/Project.toml`  Updating `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_vmbwokibxs/Manifest.toml` [05823500] ↑ OpenLibm_jll v0.8.1+2 ⇒ v0.8.5+0 Instantiating... === Precompiling... === Precompiling project... Plots W Resolving... ===  Installed LERC_jll ───────────────── v4.0.1+0  Installed Collects ───────────────── v1.0.0  Installed GR_jll ─────────────────── v0.73.17+0  Installed Libmount_jll ───────────── v2.41.1+0  Installed JpegTurbo_jll ──────────── v3.1.2+0  Installed Pango_jll ──────────────── v1.56.3+0  Installed SpecialFunctions ───────── v2.5.1  Installed BenchmarkTools ─────────── v1.6.0  Installed ColorVectorSpace ───────── v0.11.0  Installed CommonSubexpressions ───── v0.3.1  Installed Qt6ShaderTools_jll ─────── v6.8.2+1  Installed Qt6Declarative_jll ─────── v6.8.2+1  Installed Libtiff_jll ────────────── v4.7.1+0  Installed Xorg_xcb_util_cursor_jll ─ v0.1.5+0  Installed ColorSchemes ───────────── v3.30.0  Installed OpenSpecFun_jll ────────── v0.5.6+0  Installed PlutoTeachingTools ─────── v0.4.5  Installed FixedSizeArrays ────────── v1.2.0  Installed GR ─────────────────────── v0.73.17  Installed Libuuid_jll ────────────── v2.41.1+0  Installed Xorg_libXfixes_jll ─────── v6.0.1+0  Installed Qt6Wayland_jll ─────────── v6.8.2+1  Installed Qt6Base_jll ────────────── v6.8.2+1  Installed Glib_jll ───────────────── v2.84.3+0  Installed ForwardDiff ────────────── v1.1.0  Installed Plots ──────────────────── v1.40.19  No Changes to `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_vmbwokibxs/Project.toml`  Updating `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_vmbwokibxs/Manifest.toml` [05823500] ↑ OpenLibm_jll v0.8.1+2 ⇒ v0.8.5+0 Instantiating... === Precompiling... === Precompiling project... PlutoTest W Resolving... ===  Installed LERC_jll ───────────────── v4.0.1+0  Installed Collects ───────────────── v1.0.0  Installed GR_jll ─────────────────── v0.73.17+0  Installed Libmount_jll ───────────── v2.41.1+0  Installed JpegTurbo_jll ──────────── v3.1.2+0  Installed Pango_jll ──────────────── v1.56.3+0  Installed SpecialFunctions ───────── v2.5.1  Installed BenchmarkTools ─────────── v1.6.0  Installed ColorVectorSpace ───────── v0.11.0  Installed CommonSubexpressions ───── v0.3.1  Installed Qt6ShaderTools_jll ─────── v6.8.2+1  Installed Qt6Declarative_jll ─────── v6.8.2+1  Installed Libtiff_jll ────────────── v4.7.1+0  Installed Xorg_xcb_util_cursor_jll ─ v0.1.5+0  Installed ColorSchemes ───────────── v3.30.0  Installed OpenSpecFun_jll ────────── v0.5.6+0  Installed PlutoTeachingTools ─────── v0.4.5  Installed FixedSizeArrays ────────── v1.2.0  Installed GR ─────────────────────── v0.73.17  Installed Libuuid_jll ────────────── v2.41.1+0  Installed Xorg_libXfixes_jll ─────── v6.0.1+0  Installed Qt6Wayland_jll ─────────── v6.8.2+1  Installed Qt6Base_jll ────────────── v6.8.2+1  Installed Glib_jll ───────────────── v2.84.3+0  Installed ForwardDiff ────────────── v1.1.0  Installed Plots ──────────────────── v1.40.19  No Changes to `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_vmbwokibxs/Project.toml`  Updating `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_vmbwokibxs/Manifest.toml` [05823500] ↑ OpenLibm_jll v0.8.1+2 ⇒ v0.8.5+0 Instantiating... === Precompiling... === Precompiling project... BenchmarkTools W Resolving... ===  Installed LERC_jll ───────────────── v4.0.1+0  Installed Collects ───────────────── v1.0.0  Installed GR_jll ─────────────────── v0.73.17+0  Installed Libmount_jll ───────────── v2.41.1+0  Installed JpegTurbo_jll ──────────── v3.1.2+0  Installed Pango_jll ──────────────── v1.56.3+0  Installed SpecialFunctions ───────── v2.5.1  Installed BenchmarkTools ─────────── v1.6.0  Installed ColorVectorSpace ───────── v0.11.0  Installed CommonSubexpressions ───── v0.3.1  Installed Qt6ShaderTools_jll ─────── v6.8.2+1  Installed Qt6Declarative_jll ─────── v6.8.2+1  Installed Libtiff_jll ────────────── v4.7.1+0  Installed Xorg_xcb_util_cursor_jll ─ v0.1.5+0  Installed ColorSchemes ───────────── v3.30.0  Installed OpenSpecFun_jll ────────── v0.5.6+0  Installed PlutoTeachingTools ─────── v0.4.5  Installed FixedSizeArrays ────────── v1.2.0  Installed GR ─────────────────────── v0.73.17  Installed Libuuid_jll ────────────── v2.41.1+0  Installed Xorg_libXfixes_jll ─────── v6.0.1+0  Installed Qt6Wayland_jll ─────────── v6.8.2+1  Installed Qt6Base_jll ────────────── v6.8.2+1  Installed Glib_jll ───────────────── v2.84.3+0  Installed ForwardDiff ────────────── v1.1.0  Installed Plots ──────────────────── v1.40.19  No Changes to `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_vmbwokibxs/Project.toml`  Updating `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_vmbwokibxs/Manifest.toml` [05823500] ↑ OpenLibm_jll v0.8.1+2 ⇒ v0.8.5+0 Instantiating... === Precompiling... === Precompiling project... PlutoTeachingTools W Resolving... ===  Installed LERC_jll ───────────────── v4.0.1+0  Installed Collects ───────────────── v1.0.0  Installed GR_jll ─────────────────── v0.73.17+0  Installed Libmount_jll ───────────── v2.41.1+0  Installed JpegTurbo_jll ──────────── v3.1.2+0  Installed Pango_jll ──────────────── v1.56.3+0  Installed SpecialFunctions ───────── v2.5.1  Installed BenchmarkTools ─────────── v1.6.0  Installed ColorVectorSpace ───────── v0.11.0  Installed CommonSubexpressions ───── v0.3.1  Installed Qt6ShaderTools_jll ─────── v6.8.2+1  Installed Qt6Declarative_jll ─────── v6.8.2+1  Installed Libtiff_jll ────────────── v4.7.1+0  Installed Xorg_xcb_util_cursor_jll ─ v0.1.5+0  Installed ColorSchemes ───────────── v3.30.0  Installed OpenSpecFun_jll ────────── v0.5.6+0  Installed PlutoTeachingTools ─────── v0.4.5  Installed FixedSizeArrays ────────── v1.2.0  Installed GR ─────────────────────── v0.73.17  Installed Libuuid_jll ────────────── v2.41.1+0  Installed Xorg_libXfixes_jll ─────── v6.0.1+0  Installed Qt6Wayland_jll ─────────── v6.8.2+1  Installed Qt6Base_jll ────────────── v6.8.2+1  Installed Glib_jll ───────────────── v2.84.3+0  Installed ForwardDiff ────────────── v1.1.0  Installed Plots ──────────────────── v1.40.19  No Changes to `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_vmbwokibxs/Project.toml`  Updating `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_vmbwokibxs/Manifest.toml` [05823500] ↑ OpenLibm_jll v0.8.1+2 ⇒ v0.8.5+0 Instantiating... === Precompiling... === Precompiling project... enabled÷restart_recommended_msgrestart_required_msgbusy_packageswaiting_for_permission,waiting_for_permission_but_probably_disabled«cell_inputsP$138806a6-08f9-4b82-9e8c-128baab3da31cell_id$138806a6-08f9-4b82-9e8c-128baab3da31codent.messagemetadatashow_logsèdisabled®skip_as_script«code_folded$ba5059bf-bdbb-4c35-b9e7-b7e90641dc94cell_id$ba5059bf-bdbb-4c35-b9e7-b7e90641dc94codeDmd""" For numerical calculations, allocating storage space on the heap can be a significant cost. Often, we do the same thing over and over, resulting in many allocations for outputs (or temporay intermediate values). Thus, there's an opportunity to allocate storage space once and reuse it in subsequent calculations. """metadatashow_logsèdisabled®skip_as_script«code_folded$6d982d26-1917-4f46-babf-e0ea19886976cell_id$6d982d26-1917-4f46-babf-e0ea19886976codet[2]metadatashow_logsèdisabled®skip_as_script«code_folded$22c6cf9f-2266-4908-8bb5-ad30d7388c06cell_id$22c6cf9f-2266-4908-8bb5-ad30d7388c06code٪begin grid = range(0,stop=1,length=40); local plt = heatmap(collect(grid), collect(grid), [f([x,y]) for x in grid, y in grid]) xlabel!(plt, "x") ylabel!(plt, "y") endmetadatashow_logsèdisabled®skip_as_script«code_folded$cf5262da-1043-11ec-12fc-9916cc70070ccell_id$cf5262da-1043-11ec-12fc-9916cc70070ccodeSmd"> Astro 528: High-Performance Scientific Computing for Astrophysics (Fall 2025)"metadatashow_logsèdisabled®skip_as_script«code_folded$23038da7-9417-48fe-8431-e5a05dcf9593cell_id$23038da7-9417-48fe-8431-e5a05dcf9593code+md"### Non-expert call to compute gradient"metadatashow_logsèdisabled®skip_as_script«code_folded$3b8f0812-9700-45c4-b1ce-d8cdfe09c38bcell_id$3b8f0812-9700-45c4-b1ce-d8cdfe09c38bcode$let a = rand(32) b = zeros(32) c = a+b # Stores results in a new array, assigns to c @time c .= a + b # Writes results into existing storage for c, but creates temporary array @time c .= a.+b # Writes results into existing storage for c, avoids creating temporary array end;metadatashow_logsèdisabled®skip_as_script«code_folded$f0111b2d-7a7c-4029-8fd0-7c0785b5b038cell_id$f0111b2d-7a7c-4029-8fd0-7c0785b5b038codemd""" ### Where do your variables get stored? Your program will use two distinct **virtual address spaces**: - **Stack** + Scalars + Small structures or collections with known size + Function parameters and memory addresses for their outputs + Values cleared from stack on a Last-in, First-out (LIFO) basis + *Push*ing values onto top of stack is fast + *Pop*ing values off the top of the stack is fast - **Heap** + Large arrays/collections + Structures/collections if size is not known at compile time + Stored until deallocated (by programmer or garbage collector) + Allocating memory from heap incurs latency + Deallocating can be fast, but is error prone + Deallocations typically defered until a *garbage collector* frees up memory that is no longer in use. """metadatashow_logsèdisabled®skip_as_script«code_folded$cf913e43-4499-4efc-87cb-2474222b8865cell_id$cf913e43-4499-4efc-87cb-2474222b8865code)md""" # Expert vs non-expert routines """metadatashow_logsèdisabled®skip_as_script«code_folded$1aab953a-f524-4d93-95bf-ea5524c47b8bcell_id$1aab953a-f524-4d93-95bf-ea5524c47b8bcodeFtry v6[2] = 0 catch @warn "You can't do modify a static vector." endmetadatashow_logsèdisabled®skip_as_script«code_folded$8114f59e-1a8e-49c6-baaa-20ed19747d2bcell_id$8114f59e-1a8e-49c6-baaa-20ed19747d2bcode}md""" ## How to generate less garbage? ### For small allocations, use data types that stay on the stack - [Tuple](https://docs.julialang.org/en/v1/manual/types/#Tuple-Types) - [NamedTuple](https://docs.julialang.org/en/v1/manual/types/#Named-Tuple-Types) - Custom `struct` (with fields that go to the stack) - [StaticArrays.jl](https://github.com/JuliaArrays/StaticArrays.jl) """metadatashow_logsèdisabled®skip_as_script«code_folded$f632c0ab-ea32-4973-8ec9-79a7f4f4c4adcell_id$f632c0ab-ea32-4973-8ec9-79a7f4f4c4adcode'nt = (;result = 1.0, message="Success")metadatashow_logsèdisabled®skip_as_script«code_folded$348ee204-546f-46c5-bf5d-7d4a761002eccell_id$348ee204-546f-46c5-bf5d-7d4a761002eccodeٲmd""" # Week 4 Discussion Topics - Memory heirarchy & allocations - Stack vs Heap allocations - Pre-allocating storage - Garbage Collectors - Expert/Non-expert Interfaces """metadatashow_logsèdisabled®skip_as_script«code_folded$2e8da77c-004b-4d59-83f4-9fb981ea5658cell_id$2e8da77c-004b-4d59-83f4-9fb981ea5658codeXbegin mms = MyMutableStruct(5.0, "Lorem ipsum dolor sit amet") mms.result = 0 mms endmetadatashow_logsèdisabled®skip_as_script«code_folded$8fe1804c-4fce-4980-bb54-dad89c3062d1cell_id$8fe1804c-4fce-4980-bb54-dad89c3062d1codev1metadatashow_logsèdisabled®skip_as_script«code_folded$31a1f316-8098-4496-9d20-b0782358bd4fcell_id$31a1f316-8098-4496-9d20-b0782358bd4fcode0md""" #### Mutable fixed-size vectors/arrays """metadatashow_logsèdisabled®skip_as_script«code_folded$3f4cac5b-9b2a-4261-a01e-e2d4f692fb90cell_id$3f4cac5b-9b2a-4261-a01e-e2d4f692fb90codeHbegin m1 = @time @SMatrix [ 1 3 ; 2 4 ] m2 = @SMatrix randn(4,4) endmetadatashow_logsèdisabled®skip_as_script«code_folded$f9714ea5-5d0e-4006-92c4-71b3e574d26ccell_id$f9714ea5-5d0e-4006-92c4-71b3e574d26ccodeDmutable struct MyMutableStruct result::Float64 message::String endmetadatashow_logsèdisabled®skip_as_script«code_folded$99a0be67-2dac-43ba-9223-03845c528fefcell_id$99a0be67-2dac-43ba-9223-03845c528fefcode+@benchmark ForwardDiff.gradient($f,$x_eval)metadatashow_logsèdisabled®skip_as_script«code_folded$0dd9307f-1390-412b-8cf6-77deb33091a0cell_id$0dd9307f-1390-412b-8cf6-77deb33091a0code,md""" #### Examples of simple `struct`'s """metadatashow_logsèdisabled®skip_as_script«code_folded$294c9958-5e2b-4b95-ba49-ecda696190b6cell_id$294c9958-5e2b-4b95-ba49-ecda696190b6codejlet result = DiffResults.GradientResult(x_eval) @benchmark ForwardDiff.gradient!($result,$f,$x_eval) endmetadatashow_logsèdisabled®skip_as_script«code_folded$ae709a34-9244-44ee-a004-381fc9b6cd0ccell_id$ae709a34-9244-44ee-a004-381fc9b6cd0ccode:md"ToC on side $(@bind toc_aside CheckBox(;default=true))"metadatashow_logsèdisabled®skip_as_script«code_folded$d35aa76c-b4e6-45f8-a4e3-ba37d674db82cell_id$d35aa76c-b4e6-45f8-a4e3-ba37d674db82codemd"# Helper Code"metadatashow_logsèdisabled®skip_as_script«code_folded$fd53c52d-8acd-4ca5-bcf0-8375d64179cccell_id$fd53c52d-8acd-4ca5-bcf0-8375d64179cccodet = (1.0, "Success")metadatashow_logsèdisabled®skip_as_script«code_folded$04ba8dc8-7b85-4bde-82fe-8d7e16d9142dcell_id$04ba8dc8-7b85-4bde-82fe-8d7e16d9142dcodex_eval = [1.0, 2.0];metadatashow_logsèdisabled®skip_as_script«code_folded$b7bdbd76-4bb1-473d-b3a7-f5757a42d6cacell_id$b7bdbd76-4bb1-473d-b3a7-f5757a42d6cacodeG@allocated mis2 = MyImmutableStruct(2.0, "Lorem ipsum dolor sit amet")metadatashow_logsèdisabled®skip_as_script«code_folded$1b42bd08-9611-4d96-9d91-fdce9d24982fcell_id$1b42bd08-9611-4d96-9d91-fdce9d24982fcode|let A = rand(nrows,ncols) b = rand(ncols) y = zeros(nrows) @benchmark multiply_matrix_vector_preallocated!($y,$A,$b) endmetadatashow_logsèdisabled®skip_as_script«code_folded$a1f3a295-7f63-46e1-a493-5082a86dc1abcell_id$a1f3a295-7f63-46e1-a493-5082a86dc1abcode6md""" #### What if I want to mutate my `struct`'s? """metadatashow_logsèdisabled®skip_as_script«code_folded$b718654a-2490-4bac-a1af-54f959eed0fbcell_id$b718654a-2490-4bac-a1af-54f959eed0fbcodemis.messagemetadatashow_logsèdisabled®skip_as_script«code_folded$15f1d255-0aa9-4837-b76e-1f7b401c93cacell_id$15f1d255-0aa9-4837-b76e-1f7b401c93cacodetypeof(nt)metadatashow_logsèdisabled®skip_as_script«code_folded$624e8936-de28-4e09-9a4e-3ef2f6c7d9b0cell_id$624e8936-de28-4e09-9a4e-3ef2f6c7d9b0codebegin using PlutoUI, PlutoTest, PlutoTeachingTools using StaticArrays, FixedSizeArrays using BenchmarkTools using ForwardDiff using Plots # Not demonstrated in notebook, but relevant to lesson #using AllocCheck # For checking whether functions allocate to heap #using Collects # For storing iterables to FixedSizdArrays #using InlineStrings # For efficient storage of arrays of String endmetadatashow_logsèdisabled®skip_as_script«code_folded$5f45fe0e-81df-4dd5-b827-628e0831755ecell_id$5f45fe0e-81df-4dd5-b827-628e0831755ecodemd""" ### Potentially additional optional parameters E.g., Here we can turn off "tag checking". (See [documentation about custom tags](https://juliadiff.org/ForwardDiff.jl/dev/user/advanced/#Custom-tags-and-tag-checking) for details."""metadatashow_logsèdisabled®skip_as_script«code_folded$76187747-1b2f-42e1-b0e7-509b0ed42b6acell_id$76187747-1b2f-42e1-b0e7-509b0ed42b6acode'sizeof(mis.result), sizeof(mis.message)metadatashow_logsèdisabled®skip_as_script«code_folded$aa3233da-122d-40a3-80bb-9b6472906ec0cell_id$aa3233da-122d-40a3-80bb-9b6472906ec0codev1 === v5metadatashow_logsèdisabled®skip_as_script«code_folded$541bbf9f-afa5-4837-83b1-84017bccc25ccell_id$541bbf9f-afa5-4837-83b1-84017bccc25ccodeD@time v0 = [1,2] # Vector whose length can change, goes to heapmetadatashow_logsèdisabled®skip_as_script«code_folded$0089fbf7-4aa8-4481-a52b-bc4d929ce1fdcell_id$0089fbf7-4aa8-4481-a52b-bc4d929ce1fdcodelet result = DiffResults.GradientResult(x_eval) cfg = ForwardDiff.GradientConfig(f, x_eval, ForwardDiff.Chunk{chunk_size}()) ForwardDiff.gradient!(result,f,x_eval, cfg) DiffResults.value(result), DiffResults.gradient(result) endmetadatashow_logsèdisabled®skip_as_script«code_folded$66968290-4f45-49e9-a377-7d2b101a822fcell_id$66968290-4f45-49e9-a377-7d2b101a822fcodev2metadatashow_logsèdisabled®skip_as_script«code_folded$46d06e5b-5968-4cdb-97f3-f774f633cce0cell_id$46d06e5b-5968-4cdb-97f3-f774f633cce0codemd""" ## Garbage Collection - When try to allocate from heap, language may decide to run **garbage collector**. - Checks whether it's possible for code to reach memory that has been allocated. - If not, marks it for deallocation. - This can cause an unexpected delay. """metadatashow_logsèdisabled®skip_as_script«code_folded$ed0bc201-03c0-448b-bfeb-9f43d7e9ad7acell_id$ed0bc201-03c0-448b-bfeb-9f43d7e9ad7acode{md" [Interactive Memory Latency vs Year](https://colin-scott.github.io/personal_website/research/interactive_latency.html)"metadatashow_logsèdisabled®skip_as_script«code_folded$d7fe5753-f02d-4a04-8967-32441149dfe4cell_id$d7fe5753-f02d-4a04-8967-32441149dfe4code:mis = MyImmutableStruct(2.0, "Lorem ipsum dolor sit amet")metadatashow_logsèdisabled®skip_as_script«code_folded$4c724a18-e3dc-4cdb-997c-b3687b0d51c5cell_id$4c724a18-e3dc-4cdb-997c-b3687b0d51c5codent[2]metadatashow_logsèdisabled®skip_as_script«code_folded$bf07ac72-3fd3-4062-beb6-8bbcd2a95771cell_id$bf07ac72-3fd3-4062-beb6-8bbcd2a95771code"""`multiply_matrix_vector_preallocated!(out::Vector, A::Matrix, b::Vector)` Multiply matrix A and vector b by hand using rows for inner loop, using preallocated space for output""" function multiply_matrix_vector_preallocated!(out::AbstractVector, A::AbstractMatrix, b::AbstractVector) @assert size(A,1) == length(out) @assert size(A,2) == length(b) @simd for j in 1:size(A,2) for i in 1:size(A,1) @inbounds out[i] += A[i,j]*b[j] end end return out endmetadatashow_logsèdisabled®skip_as_script«code_folded$060bc23d-52fa-4536-aa50-26390552997fcell_id$060bc23d-52fa-4536-aa50-26390552997fcode!begin nrows = 32 ncols = 1 end;metadatashow_logsèdisabled®skip_as_script«code_folded$35332b4e-4caa-4448-a345-d272a6362647cell_id$35332b4e-4caa-4448-a345-d272a6362647codev5metadatashow_logsèdisabled®skip_as_script«code_folded$63ec6f40-1f29-4bf5-b48a-124f1e8d783bcell_id$63ec6f40-1f29-4bf5-b48a-124f1e8d783bcode;question_box(md"Will this be stored on the stack or heap?")metadatashow_logsèdisabled®skip_as_script«code_folded$318991c3-f693-4364-8e99-623883739a86cell_id$318991c3-f693-4364-8e99-623883739a86code٫begin @time v2 = FixedSizeArray(v3) # Vector with fixed size, but still likely to go to heap, since intended for larger arrays v2[1] = 1.0 v2[2] = 2.0 v2[3] = 3.0 endmetadatashow_logsèdisabled®skip_as_script«code_folded$e7d4bc52-ee18-4d0b-8bab-591b688398fecell_id$e7d4bc52-ee18-4d0b-8bab-591b688398fecode TableOfContents(aside=toc_aside)metadatashow_logsèdisabled®skip_as_script«code_folded$a1c68295-1450-4d5d-833d-57a7eabdbf22cell_id$a1c68295-1450-4d5d-833d-57a7eabdbf22code2md""" ### Optional parameter for improved performance Here we specify the [chunk size](https://juliadiff.org/ForwardDiff.jl/dev/user/advanced/#Configuring-Chunk-Size) that affects both how many calls to the function (larger chunk→fewer calls) and how much memory is needed (larger chunk→more memory)"""metadatashow_logsèdisabled®skip_as_script«code_folded$12e4d1e4-d284-4180-8342-d58dda578272cell_id$12e4d1e4-d284-4180-8342-d58dda578272codebegin @time v3 = MVector(1, 2, 3) # Mutable vector of fixed size and type, goes to heap, but will try to waste less memory compared to a normal vector @time v4 = @MVector [1, 2, 3, 4, ] # macro to ease constructing endmetadatashow_logsèdisabled®skip_as_script«code_folded$5de73fe6-fb0e-4a98-b2ba-f4e592f02006cell_id$5de73fe6-fb0e-4a98-b2ba-f4e592f02006codeglet A = rand(nrows,ncols) b = rand(ncols) @benchmark y = multiply_matrix_vector_allocates($A,$b) endmetadatashow_logsèdisabled®skip_as_script«code_folded$9ec9429d-c1f1-4845-a514-9c88b452071fcell_id$9ec9429d-c1f1-4845-a514-9c88b452071fcodeWidthOverDocs()metadatashow_logsèdisabled®skip_as_script«code_folded$2ad27ea5-77f7-4532-9005-7a63160e2503cell_id$2ad27ea5-77f7-4532-9005-7a63160e2503codeBmd""" ## Keeping small arrays in the stack and out of the heap """metadatashow_logsèdisabled®skip_as_script«code_folded$6371bcc6-3253-4dc4-82f8-647aedf4014ecell_id$6371bcc6-3253-4dc4-82f8-647aedf4014ecodemd""" #### Why default to immutable? - Often is more efficient. - Some structs can be packed efficiently into arrays - Compiler is often able to avoid allocating immutable objects. - Code using immutable objects can be easier to reason about. """metadatashow_logsèdisabled®skip_as_script«code_folded$3b3d6fdb-5066-48f1-aa64-e28f5ad3c1f1cell_id$3b3d6fdb-5066-48f1-aa64-e28f5ad3c1f1codeWprotip(md""" Julia's garbage collector is "a non-compacting, generational, mark-and-sweep, tracing collector, which at a high level means the following… **Mark-Sweep / Tracing**: - When the garbage collector runs, it starts from a set of “roots” and walks through the object graph, “marking” objects as reachable. - Any object that isn’t marked as reachable and will then be “swept away” — i.e. its memory is reclaimed—since you know it’s not reachable from the roots and is therefore garbage. **Generational**: - It’s common that more recently allocated objects become garbage more quickly—this is known as the “generational hypothesis”. - Generational collectors have different levels of collection: young collections which run more frequently and full collections which run less frequently. - If the generational hypothesis is true, this saves time since it’s a waste of time to keep checking if older objects are garbage when they’re probably not." **Non-compacting / Non-moving**: - Other garbage collection techniques can copy or move objects during the collection process. - Julia does not use any of these—collection does not move or copy anything, it just reclaims unreachable objects. If you’re having issues with garbage collection, your primary recourse is to generate less garbage: - Write non-allocating code wherever possible: simple scalar code can generally avoid allocations. - Use immutable objects (i.e., `struct` rather than `mutable struct`), which can be stack allocated more easily and stored inline in other structures (as compared to mutable objects which generally need to be heap-allocated and stored indirectly by pointer, all of which causes more memory pressure). - Use pre-allocated data structures and modify them instead of allocating and returning new objects, especially in loops. - Can call `GC.gc()` to manually call garbage collector. But this is mainly useful for benchmarking. (nearly quote from [Julia Discourse post by Stefan Karpinski](https://discourse.julialang.org/t/details-about-julias-garbage-collector-reference-counting/18021)) """)metadatashow_logsèdisabled®skip_as_script«code_folded$e46c0510-f604-4025-bef5-9029e3747447cell_id$e46c0510-f604-4025-bef5-9029e3747447codemd""" Physical differences cause large difference in memory **latency** and **bandwidth**: ### Memory latency heirarchy - Registers - Cache L1 - Cache L2 - Cache L3 - RAM - Local disk storage - Disk storage on local network - Disk storage on internet - Tape storage """metadatashow_logsèdisabled®skip_as_script«code_folded$c7e5ad3c-a3ce-47a4-9154-ca33edc2811bcell_id$c7e5ad3c-a3ce-47a4-9154-ca33edc2811bcode3md""" #### Meaning of equality can be confusing """metadatashow_logsèdisabled®skip_as_script«code_folded$41c6bcb6-c5c5-4a52-a678-31ffa22641e1cell_id$41c6bcb6-c5c5-4a52-a678-31ffa22641e1codeD@allocated mms2 = MyMutableStruct(5.0, "Lorem ipsum dolor sit amet")metadatashow_logsèdisabled®skip_as_script«code_folded$ff336de5-cf53-41ca-9cc9-1ab37b851be3cell_id$ff336de5-cf53-41ca-9cc9-1ab37b851be3codeD@time v1 = [1,2,3] # Vector whose length can change, goes to heapmetadatashow_logsèdisabled®skip_as_script«code_folded$9cc467f9-cc48-4026-87c5-dadfe239c5dbcell_id$9cc467f9-cc48-4026-87c5-dadfe239c5dbcodeٳmd""" ### Implications - Mutable objects typically go onto heap - → Only make structs mutable if a reason. - An immutable object can contain mutable objects (e.g., arrays). """metadatashow_logsèdisabled®skip_as_script«code_folded$5287e3b8-2309-4c78-91f0-f80e7ae7f209cell_id$5287e3b8-2309-4c78-91f0-f80e7ae7f209code>struct MyImmutableStruct result::Float64 message::String endmetadatashow_logsèdisabled®skip_as_script«code_folded$1fa422e7-329d-4072-a786-fb73c718e49ecell_id$1fa422e7-329d-4072-a786-fb73c718e49ecodesizeof(mis)metadatashow_logsèdisabled®skip_as_script«code_folded$75661d12-7709-44d0-a5f3-d6c9a5eff952cell_id$75661d12-7709-44d0-a5f3-d6c9a5eff952codetypeof(t), sizeof(t)metadatashow_logsèdisabled®skip_as_script«code_folded$d6019e5a-e7a1-4676-a3f1-24fe62bfa931cell_id$d6019e5a-e7a1-4676-a3f1-24fe62bfa931codechunk_size = 2metadatashow_logsèdisabled®skip_as_script«code_folded$ee76a23e-e940-4b32-a44a-9086d0461089cell_id$ee76a23e-e940-4b32-a44a-9086d0461089codeٕlet result = DiffResults.GradientResult(x_eval) ForwardDiff.gradient!(result,f,x_eval) DiffResults.value(result), DiffResults.gradient(result) endmetadatashow_logsèdisabled®skip_as_script«code_folded$aac0c3f8-ece3-431e-859d-790b8f3b995acell_id$aac0c3f8-ece3-431e-859d-790b8f3b995acodeٝbegin mis_will_break = MyImmutableStruct(3.0, "Lorem ipsum dolor sit amet") try mis_will_break.result = 4.0 catch @warn "You can't do that!" end endmetadatashow_logsèdisabled®skip_as_script«code_folded$c4da391c-e7dc-4ce1-9ccb-04aa2ca79e26cell_id$c4da391c-e7dc-4ce1-9ccb-04aa2ca79e26code hint(md""" - `sizeof(mms) = ` $(sizeof(mms)) - `sizeof(mms.result) =` $(sizeof(mms.result)) - `sizeof(mms.message) =` $(sizeof(mms.message)) - `Base.summarysize(mms.message) =`$(Base.summarysize(mms.message)) - `Base.summarysize(mms) =`$(Base.summarysize(mms)) """)metadatashow_logsèdisabled®skip_as_script«code_folded$9007f240-c88e-46c4-993d-fd6b93d8b18dcell_id$9007f240-c88e-46c4-993d-fd6b93d8b18dcodemd""" # Week 4 Admin Announcements ## Project proposals ### Feedback - Project repo, Pull requests, Feedback (either Conversation or Files view) - I'll look for any replies once I've finished providing feedback to everyone. """metadatashow_logsèdisabled®skip_as_script«code_folded$cb988242-8aa6-4f31-955e-021024e0c921cell_id$cb988242-8aa6-4f31-955e-021024e0c921code(f(x) = x[1]*x[2]/sqrt(1+(x[1]^2+x[2]^2))metadatashow_logsèdisabled®skip_as_script«code_folded$7d1edb0b-3144-4ac7-a939-2e211cdee81bcell_id$7d1edb0b-3144-4ac7-a939-2e211cdee81bcode1md"### Mutating version for improved performance"metadatashow_logsèdisabled®skip_as_script«code_folded$c6eb773e-757e-466e-ba15-593c2b19fd1ccell_id$c6eb773e-757e-466e-ba15-593c2b19fd1ccodeForwardDiff.gradient(f,x_eval)metadatashow_logsèdisabled®skip_as_script«code_folded$4772a931-14a6-46c9-8b8c-44b39d213216cell_id$4772a931-14a6-46c9-8b8c-44b39d213216codebegin @time v5 = SVector(1, 2, 3) # Static vector of fixed size and type, goes to stack @time v6 = @SVector [1, 2, 3] # macro for creating SVector without allocating on heap end;metadatashow_logsèdisabled®skip_as_script«code_folded$cf7990f7-40ec-4a99-be58-e5342061baedcell_id$cf7990f7-40ec-4a99-be58-e5342061baedcode/md""" #### Limitations of immutable structs """metadatashow_logsèdisabled®skip_as_script«code_folded$397e78ca-8b07-4593-8c92-9b69cbc9d2a3cell_id$397e78ca-8b07-4593-8c92-9b69cbc9d2a3code:md""" #### Warning about mutating collections in Pluto """metadatashow_logsèdisabled®skip_as_script«code_folded$fc9ae0ec-298b-4542-9c71-dbbe535b4bd9cell_id$fc9ae0ec-298b-4542-9c71-dbbe535b4bd9code)md""" ## Pre-allocating storage space """metadatashow_logsèdisabled®skip_as_script«code_folded$abee77fd-e2a2-403b-ab43-b4ea6741ba4ccell_id$abee77fd-e2a2-403b-ab43-b4ea6741ba4ccodemd"# Computer Memory Systems"metadatashow_logsèdisabled®skip_as_script«code_folded$f68fb195-1fa6-4364-8d8f-427a6ea9a4b6cell_id$f68fb195-1fa6-4364-8d8f-427a6ea9a4b6codeپlet result = DiffResults.GradientResult(x_eval) cfg = ForwardDiff.GradientConfig(f, x_eval, ForwardDiff.Chunk{chunk_size}()) @benchmark ForwardDiff.gradient!($result,$f,$x_eval, $cfg) endmetadatashow_logsèdisabled®skip_as_script«code_folded$cbea527b-9a62-40d4-b922-47f4428a8b33cell_id$cbea527b-9a62-40d4-b922-47f4428a8b33codelet result = DiffResults.GradientResult(x_eval) cfg = ForwardDiff.GradientConfig(f, x_eval, ForwardDiff.Chunk{2}()) ForwardDiff.gradient!(result,f,x_eval, cfg, Val{false}()) DiffResults.value(result), DiffResults.gradient(result) endmetadatashow_logsèdisabled®skip_as_script«code_folded$b55fc829-c857-44bb-af5b-cbbf8dcab5e7cell_id$b55fc829-c857-44bb-af5b-cbbf8dcab5e7codev1 == v5metadatashow_logsèdisabled®skip_as_script«code_folded$abf6e65a-292a-4f4a-8eee-ea90cd22e376cell_id$abf6e65a-292a-4f4a-8eee-ea90cd22e376codeWv2[2] = 0; # Valid Julia code, but... # Leads to ambiguous order of execution in Plutometadatashow_logsèdisabled®skip_as_script«code_folded$34840126-4365-418d-bc55-4ac3bd85f43fcell_id$34840126-4365-418d-bc55-4ac3bd85f43fcoden"""`multiply_matrix_vector_allocates(A::Matrix, b::Vector)` Multiply matrix A and vector b by hand, allocating space for output""" function multiply_matrix_vector_allocates(A::AbstractMatrix, b::AbstractVector) @assert size(A,2) == length(b) out = zeros(promote_type(eltype(A),eltype(b)), size(A,1)) multiply_matrix_vector_preallocated!(out,A,b) return out endmetadatashow_logsèdisabled®skip_as_script«code_foldedënotebook_id$4493f668-bfda-11f0-0b3a-af963b4d6a71in_temp_dir¨metadata