[devmotion]

Fixes #243, fixes #794.

Problem

The generic implementations of norm (LinearAlgebra.generic_norm2/generic_normp and StaticArrays' _norm/_norm_scaled) rescale by the maximum absolute value to avoid overflow and underflow. Their branches (iszero, isfinite, <) take partials into account for Duals (< since #791), so Duals end up in the wrong branch:

input	before	after
gradient(norm, [0.0, 0.0])	[NaN, NaN]	[0.0, 0.0]
gradient(x -> norm(x)^2, [0.0, 0.0])	[NaN, NaN]	[0.0, 0.0]
gradient(x -> norm(x, 3), [0.0, 0.0])	[NaN, NaN]	[0.0, 0.0]
norm([Dual(1e-200, 1e-100), Dual(0.0, 0.0)])	Dual(0.0, Inf) (wrong primal)	Dual(1e-200, 1e-100)
gradient(norm, [1e-200, 1e-200])	[Inf, Inf]	[0.707, 0.707]
norm(SA[Dual(0, 1), Dual(0, 1), Dual(0, 1)])	Dual(NaN, NaN)	Dual(0.0, 0.0)
derivative(p -> norm([3.0, 4.0], p), 3.0)	MethodError	-0.304
hypot(Dual(0.0, 1.0), Dual(0.0, 1.0))	Dual(0.0, NaN)	Dual(0.0, 0.0)

All of these fail with NaN-safe mode enabled as well.

Approach

Similar to the rules in ChainRules, the norm of the primal values is computed with the existing (scale-robust) implementations and the chain rule is applied with weights that are bounded by 1 in absolute value (no overflow/underflow):

• with respect to x: w_i = |u_i|^(p-1) * sign(u_i) with u_i = x_i / norm(x, p) (u_i for p = 2; only the first maximum for p = Inf); p ∈ {0, 1, -Inf} keep the generic implementation, which is not affected by the scaling issues
• with respect to p: norm(x, p) / p * sum(|u_i|^p * log(|u_i|)), which is bounded and avoids the cancellation of sum(|x_i|^p * log(|x_i|)) - norm(x, p) * log(norm(x, p))
• at norm(x, p) == 0, the subgradient zero is used (ChainRules uses the same convention via pinv)

Complex arrays, nested differentiation (with all tag orderings of x and p), StaticArrays, Adjoint/Transpose, and sparse arrays are supported. Method ambiguities with LinearAlgebra, StaticArrays, and SparseArrays (new weak dependency/extension) are resolved explicitly.

hypot uses the same convention at zero. Additionally, the three-argument hypot no longer drops the partials of outer tags in nested differentiation (it used the tag-agnostic value/partials).

Limitations

Second-order derivatives through norm at zero remain incorrect (e.g., hessian(x -> norm(x)^2, zeros(2)) returns zeros instead of 2I): Since norm is not differentiable at zero, no choice of partials of norm(x) can recover the correct result. This is documented in the limitations section of the docs; sum(abs2, x) should be used instead.

🤖 Generated with Claude Code

[codecov]

Codecov Report

❌ Patch coverage is 96.25000% with 3 lines in your changes missing coverage. Please review.
✅ Project coverage is 91.78%. Comparing base (70ef06c) to head (50ba6b3).

Files with missing lines	Patch %	Lines
ext/SparseArraysExt.jl	66.66%	2 Missing ⚠️
ext/ForwardDiffStaticArraysExt.jl	66.66%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #872      +/-   ##
==========================================
+ Coverage   91.24%   91.78%   +0.54%     
==========================================
  Files          11       12       +1     
  Lines        1085     1157      +72     
==========================================
+ Hits          990     1062      +72     
  Misses         95       95              

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