feat: add fft/base/fftpack/rffti

[gunjjoshi]

Resolves None.

Description

What is the purpose of this pull request?

This pull request:

• adds fft/base/fftpack/rffti, which is a part of fftpack.
• is an extension of feat: add initial implementation of fft/base/fftpack #4121.

Related Issues

Does this pull request have any related issues?

This pull request has the following related issues:

None.

Questions

Any questions for reviewers of this pull request?

No.

Other

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No.

Checklist

Please ensure the following tasks are completed before submitting this pull request.

• Read, understood, and followed the contributing guidelines.

AI Assistance

When authoring the changes proposed in this PR, did you use any kind of AI assistance?

• Yes
• No

If you answered "yes" above, how did you use AI assistance?

• Code generation (e.g., when writing an implementation or fixing a bug)
• Test/benchmark generation
• Documentation (including examples)
• Research and understanding

Disclosure

If you answered "yes" to using AI assistance, please provide a short disclosure indicating how you used AI assistance. This helps reviewers determine how much scrutiny to apply when reviewing your contribution. Example disclosures: "This PR was written primarily by Claude Code." or "I consulted ChatGPT to understand the codebase, but the proposed changes were fully authored manually by myself.".

Used AI to understand how we generate C fixtures using FFTPACK.

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@stdlib-js/reviewers

[stdlib-bot]

Coverage Report

Package	Statements	Branches	Functions	Lines
fft/base/fftpack/rffti	$\color{green}412/412$ $\color{green}+0.00\%$	$\color{green}12/12$ $\color{green}+0.00\%$	$\color{green}2/2$ $\color{green}+0.00\%$	$\color{green}412/412$ $\color{green}+0.00\%$

The above coverage report was generated for the changes in this PR.

[kgryte]

Not fully understanding why this file differs from what we did in https://github.com/gunjjoshi/stdlib/blob/20edeac7d326443708e922a8aa99dc6a93baa3ef/lib/node_modules/%40stdlib/fft/base/fftpack/rffti/lib/rffti1.js. What is the reason for the various discrepancies?

[gunjjoshi]

The major reasons for these included our discussion to change the signature of the decompose function to include more parameters:
Before:

decompose( N, initial, out, stride, offset );

After:

decompose( N, M, initial, strideInitial, offsetInitial, out, strideOut, offsetOut );

Reference: #10354 (comment)

And, while comparing the older implementation with the test fixtures generated by FFTPACK, we were getting the results being stored off by one place.

I have restored the older implementation as given here: https://github.com/stdlib-js/stdlib/pull/4121/changes#diff-50a196b71ed9215f6c0559fab1caf4289f1d0c4b8cd17e50c0503caf39605ea9, and it now has only the following changes, as compared to the older implementation:

1. Update in the FFTPACK reference implementation link in the license header. (Link)
2. Update in the function call of decompose, given the change in its signature. (Link)
3. Adjustment in the index while calculating it, where we change im+2 to im+1, after which the tests pass. (Link)

[gunjjoshi]

The fixtures are correctly regenerated after merging with the develop branch (to pick the changes from #11819). I deleted the fixtures locally and then regenerated them after doing a make install-deps-fftpack, which generated the same fixtures upon invoking runner.c. Shouldn't the generated fixtures be different in this case, and should changed after each invocation of the runner script, given that we have random number generation functions in runner.c? Or is this an expected behavior?

If this is expected, this PR would be ready for a review.