Constructing quaternion from matrix causes infinite loop about nalgebra HOT 5 OPEN

As an occasional user of this method I'd appreciate a guaranteed-stable variant, and shuffling the names around makes sense. The iterative version should have a scarier name to indicate it's fallible, though. Maybe also take a max iteration count and return Option so it's possible to use safely?

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I have been able to reproduce this.

https://play.rust-lang.org/?version=nightly&mode=debug&edition=2021&gist=828c7f0da360ba36aed2062aa65c90f0

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Related: #627, #1187.

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So we've had plenty of problems with these construction methods. The crux of the issue, in my opinion, is that they rely on an iterative algorithm with - to my understanding - no guaranteed convergence (don't get fooled by the paper title containing the word "Robust", it is very loosely used).

I think a method with the name from_matrix should use a provably reliable method. We can have a separate method for a faster iterative variant which users can opt in to. I also think that from_matrix is not a good name in the first place, because it's not "just" a "from conversion": it attempts to find the closest rotation to the given matrix. I'd suggest something like the following:

• Introduce a method closest_rotation_to_matrix which uses a rotationally proper Polar decomposition or similar to guarantee a robust result (barring problems with our SVD ...).
• Deprecate from_matrix and from_matrix_eps, and instead move the functionality here to separate methods closest_rotation_to_matrix_iterative or something like that.

I didn't think too much about names here. I also don't have a personal stake in this problem so I might have missed some important considerations apparent to users who are running into the problem. A 5 minute literature study turned up this recent paper which upon a cursory scan seems to feature a decent discussion of the problem:

Sarabandi et al., On Closed-Form Formulas for the 3-D Nearest Rotation Matrix Problem

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There's some more in-depth discussion of the current implementation in this merged PR from earlier this year: #1101.

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