Simplify tolerance checking by modifying err in-place

Much cleaner approach: instead of complex temp array management,
simply modify the error vector in-place for vector abstol:

@inline function check_dae_tolerance(integrator, err, abstol, t)
    if abstol isa Number
        return integrator.opts.internalnorm(err, t) / abstol <= 1
    else
        @. err = err / abstol  # In-place, zero allocation
        return integrator.opts.internalnorm(err, t) <= 1
    end
end

This is much simpler and cleaner than the previous complex optimizations
while still achieving zero allocations since err is typically a temp array.

Benefits:
- Zero allocations for both scalar and vector abstol
- Much simpler, more maintainable code
- Consistent interface across all usage sites
- Leverages the fact that err arrays are temporary

🤖 Generated with [Claude Code](https://claude.ai/code)

Co-Authored-By: Claude <noreply@anthropic.com>

Author: ChrisRackauckas

lib/OrdinaryDiffEqNonlinearSolve/src/initialize_dae.jl

@@ -1,9 +1,10 @@
-# Optimized tolerance checking that avoids allocations for scalar abstol
+# Optimized tolerance checking that avoids allocations
 @inline function check_dae_tolerance(integrator, err, abstol, t)
     if abstol isa Number
         return integrator.opts.internalnorm(err, t) / abstol <= 1
     else
-        return integrator.opts.internalnorm(err ./ abstol, t) <= 1
+        @. err = err / abstol
+        return integrator.opts.internalnorm(err, t) <= 1
     end
 end
 
@@ -66,13 +67,7 @@ function _initialize_dae!(integrator, prob::ODEProblem, alg::ShampineCollocation
     f(tmp, u0, p, t)
     tmp .= ArrayInterface.restructure(tmp, algebraic_eqs .* _vec(tmp))
 
-    # Zero-allocation tolerance check reusing tmp
-    if integrator.opts.abstol isa Number
-        integrator.opts.internalnorm(tmp, t) / integrator.opts.abstol <= 1 && return
-    else
-        @. tmp = tmp / integrator.opts.abstol
-        integrator.opts.internalnorm(tmp, t) <= 1 && return
-    end
+    check_dae_tolerance(integrator, tmp, integrator.opts.abstol, t) && return
 
     if isdefined(integrator.cache, :nlsolver) && !isnothing(alg.nlsolve)
         # backward Euler
@@ -400,13 +395,7 @@ function _initialize_dae!(integrator, prob::ODEProblem,
 
     tmp .= ArrayInterface.restructure(tmp, algebraic_eqs .* _vec(tmp))
 
-    # Zero-allocation tolerance check reusing tmp
-    if alg.abstol isa Number
-        integrator.opts.internalnorm(tmp, t) / alg.abstol <= 1 && return
-    else
-        @. tmp = tmp / alg.abstol
-        integrator.opts.internalnorm(tmp, t) <= 1 && return
-    end
+    check_dae_tolerance(integrator, tmp, alg.abstol, t) && return
     alg_u = @view u[algebraic_vars]
 
     # These non-dual values are thus used to make the caches