Fix memory allocation TODO in KenCarp step 2 and 4 by introducing chi…

[Emanuele-Elias]

Checklist

• Appropriate tests were added
• Any code changes were done in a way that does not break public API
• All documentation related to code changes were updated
• The new code follows the contributor guidelines, in particular the SciML Style Guide and COLPRAC.
• Any new documentation only uses public API

Additional context

Hi everyone! This is my first PR to SciML, aiming to prepare for GSoC.

I addressed the TODO left in src/perform_step/kencarp.jl inside perform_step! for SKenCarpCache.
Previously, z₄ was being used as temporary storage for g1 * chi2 during Step 2 and Step 4, which could lead to it being overwritten and failing to properly repeat on a step fail.

Changes made:

• Added a dedicated chi2g1::uType cache variable in the SKenCarpCache struct.
• Pre-allocated chi2g1 = similar(u) inside alg_cache.
• Replaced the temporary usage of z₄ with chi2g1 in kencarp.jl.
• Ran Runic to format the code to adhere to the SciML Style Guide.

Note on checklist: No new tests or docs were added because this is an internal memory/cache optimization and the existing test suite already covers this algorithm. Local tests passed (the only failure was EulerHeun sparse alloc, which seems to be a pre-existing issue on master).

[ChrisRackauckas]

The todo was about g1 though? https://github.com/SciML/StochasticDiffEq.jl/pull/691/changes#diff-455cecbc916082b6b8109009404a720ac2e589ba0c966a92e4f5176e49ef4a85R309-L314

[Emanuele-Elias]

Hi @ChrisRackauckas! Maybe I misunderstood the TODO😅

I read it as referring to z₄ being used as temporary storage for chi2 * g1 on the lines right below it, so I added a cache to replace that temporary z₄ usage.

Maybe what you meant was that in the non-diagonal noise case, g1 is mutated via g1 .-= g4, which permanently overwrites it and ruins it for a repeated step?

To fix this properly, should I revert the chi2g1 changes and instead add a gtmp::rateNoiseType to the cache so we can do @.. gtmp = g1 - g4 and preserve g1? Let me know and I'll push the updated commit

[ChrisRackauckas]

It's about fixing the repeated computation.

[Emanuele-Elias]

I've just pushed a new commit that corrects my previous chi2g1 changes. Instead, I wrapped g(g1, ...) in an if !repeat_step && !integrator.last_stepfail block to prevent the repeated computation. To make this work, I also added a gtmp cache to replace g1 .-= g4 with @.. gtmp = g1 - g4 in Step 4 so g1 isn't destroyed with step failure. let me know if now it's ok and thank you @ChrisRackauckas for the guidance!!

[ChrisRackauckas]

why is this all changed?

[ChrisRackauckas]

To fix the formatting and add an appropriate test, your changes were pulled into #698 and merged. Thanks!