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Add Halley and Householder to SimpleNonlinearSolve #507

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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

Note that I didn't use AutoTaylorDiff() as the ADType, since TaylorDiff hasn't become a full-featured AD like ForwardDiff, therefore it's still faster to compute Jacobian with ForwardDiff or something else. Instead, an additional field Taylor_mode is added to control whether to use TaylorDiff or not.

@tansongchen tansongchen requested a review from avik-pal November 18, 2024 23:02
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Would be fine once SciML/ADTypes.jl#99 is ready

@tansongchen tansongchen requested a review from avik-pal November 21, 2024 21:07
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Compared to 83f8c3d , it's not introducing new failures. Could @avik-pal review this again?

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