Bug: axpy!/axpby! fail with MissingPrimalError (iszero check) #317

Description

LinearAlgebra.axpy! and LinearAlgebra.axpby! fail with TracerSparsityDetector (global sparsity) when the scalar coefficient is a tracer. This is because the generic implementation checks iszero(α) to short-circuit when the coefficient is zero.

MWE

using SparseConnectivityTracer
using LinearAlgebra

detector = TracerSparsityDetector()

f(x) = begin
    y = copy(x[1:3])
    axpy!(x[4], x[5:7], y)  # y = y + x[4] * x[5:7]
    y
end

jacobian_sparsity(f, rand(7), detector)
# ERROR: Function iszero requires primal value(s).

Function iszero requires primal value(s).
A dual-number tracer for local sparsity detection can be used via `TracerLocalSparsityDetector`.
Stacktrace:
  [1] iszero(t::SparseConnectivityTracer.GradientTracer{Int64, BitSet})
    @ SparseConnectivityTracer ~/dev/SparseConnectivityTracer.jl/src/overloads/is_functions.jl:13
  [2] axpy!(α::SparseConnectivityTracer.GradientTracer{Int64, BitSet}, x::Vector{SparseConnectivityTracer.GradientTracer{Int64, BitSet}}, y::Vector{SparseConnectivityTracer.GradientTracer{Int64, BitSet}})
    @ LinearAlgebra /nix/store/.../LinearAlgebra/src/generic.jl:1636

Root Cause

LinearAlgebra.axpy! in generic.jl calls iszero(α) on the scalar coefficient to optimize the case when α=0. This fails with global tracers that don't have primal values.

The same issue affects axpby! which also checks iszero.

Workaround

Use TracerLocalSparsityDetector instead:

local_detector = TracerLocalSparsityDetector()
jacobian_sparsity(f, rand(7), local_detector)  # Works!

Potential Fix

Add overloads for axpy! and axpby! that work with tracers, avoiding the iszero check. The overload could simply perform the operation unconditionally since the sparsity pattern shouldn't depend on whether the coefficient happens to be zero.

Environment

julia> using Pkg; Pkg.status()
Project SparseConnectivityTracer v1.2.1
Status `~/dev/SparseConnectivityTracer.jl/Project.toml`
  [47edcb42] ADTypes v1.22.0
  [37e2e46d] LinearAlgebra v1.12.0
  ...

julia> versioninfo()
Julia Version 1.12.5
Commit 5fe89b8ddc1 (2026-02-09 16:05 UTC)
Platform Info:
  OS: Linux (x86_64-linux-gnu)