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.

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None.

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• Read, understood, and followed the contributing guidelines.

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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]

Suggested change

	declare function rffti( N: number, workspace: Collection<number>, strideW: number, offsetW: number ): void;
	declare function rffti<T extends Collection<number>>( N: number, workspace: T, strideW: number, offsetW: number ): T;

I suggest going ahead and just returning the workspace array. Will require some minor edits here and elsewhere in this PR.

[gunjjoshi]

Thanks for making the updates @kgryte. Should we also update all the examples to show the usage of this as a returned workspace array? For example:

Changing this:

rffti( N, workspace, 1, 0 );

To this:

var out = rffti( N, workspace, 1, 0 );

[kgryte]

Yes, that works. And then you can also add

var bool = ( out === workspace);
// returns true

which is something we do elsewhere in similar situations.

[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)

[kgryte]

I would revisit this file. IMO, the changes in https://github.com/gunjjoshi/stdlib/blob/20edeac7d326443708e922a8aa99dc6a93baa3ef/lib/node_modules/%40stdlib/fft/base/fftpack/rffti/lib/rffti1.js are arguably better, so unless there is some justification for the current implementation, I'd say go ahead and align with the current status in the draft FFT PR.

[gunjjoshi]

Aligned the implementation with the one in the draft FFT PR, with a few changes described here.

[kgryte]

Should add a test confirming that the workspace array is returned.

[kgryte]

Other than the implementation file, this PR is looking good.

[gunjjoshi]

Adding a summary: I have made the changes to make the rffti1 implementation be more aligned with the one in the draft FFT PR in the commit 2eb785a. Now, the following two things need to be addressed in this PR, which I'll do next:

1. Update examples to show that we return the workspace array, as per our discussion here: feat: add fft/base/fftpack/rffti #11467 (comment)
2. Add a test to confirm that the workspace array is returned, as per feat: add fft/base/fftpack/rffti #11467 (comment)

[gunjjoshi]

I've made all the required changes. This PR is now ready for a review.