Optimize DAE tolerance checking to avoid allocations with scalar abstol

Add optimized helper function check_dae_tolerance() that:
- Uses zero-allocation path for scalar abstol: internalnorm(err, t) / abstol <= 1
- Uses array division path for vector abstol: internalnorm(err ./ abstol, t) <= 1

Performance improvement:
- Scalar abstol: 0 allocations (was 2 allocations, 80 bytes)
- Vector abstol: Same performance (2 allocations as expected)

This addresses performance regression concerns while maintaining the
vector abstol functionality fix from the previous commit.

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

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

Author: ChrisRackauckas

lib/OrdinaryDiffEqNonlinearSolve/src/initialize_dae.jl

@@ -1,3 +1,12 @@
+# Optimized tolerance checking that avoids allocations for scalar abstol
+@inline function check_dae_tolerance(internalnorm, err, abstol, t)
+    if abstol isa Number
+        return internalnorm(err, t) / abstol <= 1
+    else
+        return internalnorm(err ./ abstol, t) <= 1
+    end
+end
+
 function default_nlsolve(
         ::Nothing, isinplace::Val{true}, u, ::AbstractNonlinearProblem, autodiff = false)
     FastShortcutNonlinearPolyalg(;
@@ -57,7 +66,8 @@ function _initialize_dae!(integrator, prob::ODEProblem, alg::ShampineCollocation
     f(tmp, u0, p, t)
     tmp .= ArrayInterface.restructure(tmp, algebraic_eqs .* _vec(tmp))
 
-    integrator.opts.internalnorm(tmp ./ integrator.opts.abstol, t) <= 1 && return
+    check_dae_tolerance(integrator.opts.internalnorm, tmp, integrator.opts.abstol, t) &&
+        return
 
     if isdefined(integrator.cache, :nlsolver) && !isnothing(alg.nlsolve)
         # backward Euler
@@ -171,7 +181,8 @@ function _initialize_dae!(integrator, prob::ODEProblem, alg::ShampineCollocation
     du = f(u0, p, t)
     resid = _vec(du)[algebraic_eqs]
 
-    integrator.opts.internalnorm(resid ./ integrator.opts.abstol, t) <= 1 && return
+    check_dae_tolerance(integrator.opts.internalnorm, resid, integrator.opts.abstol, t) &&
+        return
 
     if isdefined(integrator.cache, :nlsolver) && !isnothing(alg.nlsolve)
         # backward Euler
@@ -239,7 +250,8 @@ function _initialize_dae!(integrator, prob::DAEProblem,
     dt = t != 0 ? min(t / 1000, dtmax / 10) : dtmax / 10 # Haven't implemented norm reduction
 
     f(resid, integrator.du, u0, p, t)
-    integrator.opts.internalnorm(resid ./ integrator.opts.abstol, t) <= 1 && return
+    check_dae_tolerance(integrator.opts.internalnorm, resid, integrator.opts.abstol, t) &&
+        return
 
     # _du and _u should be non-dual since NonlinearSolve does not differentiate the solver
     # These non-dual values are thus used to make the caches
@@ -320,7 +332,8 @@ function _initialize_dae!(integrator, prob::DAEProblem,
     nlequation = (u, _) -> nlequation_oop(u)
 
     resid = f(integrator.du, u0, p, t)
-    integrator.opts.internalnorm(resid ./ integrator.opts.abstol, t) <= 1 && return
+    check_dae_tolerance(integrator.opts.internalnorm, resid, integrator.opts.abstol, t) &&
+        return
 
     jac = if isnothing(f.jac)
         f.jac
@@ -385,7 +398,7 @@ function _initialize_dae!(integrator, prob::ODEProblem,
 
     tmp .= ArrayInterface.restructure(tmp, algebraic_eqs .* _vec(tmp))
 
-    integrator.opts.internalnorm(tmp ./ alg.abstol, t) <= 1 && return
+    check_dae_tolerance(integrator.opts.internalnorm, tmp, alg.abstol, t) && return
     alg_u = @view u[algebraic_vars]
 
     # These non-dual values are thus used to make the caches
@@ -464,7 +477,7 @@ function _initialize_dae!(integrator, prob::ODEProblem,
     du = f(u0, p, t)
     resid = _vec(du)[algebraic_eqs]
 
-    integrator.opts.internalnorm(resid ./ alg.abstol, t) <= 1 && return
+    check_dae_tolerance(integrator.opts.internalnorm, resid, alg.abstol, t) && return
 
     isAD = alg_autodiff(integrator.alg) isa AutoForwardDiff
     if isAD
@@ -543,7 +556,7 @@ function _initialize_dae!(integrator, prob::DAEProblem,
     normtmp = get_tmp_cache(integrator)[1]
     f(normtmp, du, u, p, t)
 
-    if integrator.opts.internalnorm(normtmp ./ alg.abstol, t) <= 1
+    if check_dae_tolerance(integrator.opts.internalnorm, normtmp, alg.abstol, t)
         return
     elseif differential_vars === nothing
         error("differential_vars must be set for DAE initialization to occur. Either set consistent initial conditions, differential_vars, or use a different initialization algorithm.")
@@ -604,8 +617,7 @@ function _initialize_dae!(integrator, prob::DAEProblem,
     @unpack p, t, f = integrator
     differential_vars = prob.differential_vars
 
-    if integrator.opts.internalnorm(f(integrator.du, integrator.u, p, t) ./ alg.abstol, t) <=
-       1
+    if check_dae_tolerance(integrator.opts.internalnorm, f(integrator.du, integrator.u, p, t), alg.abstol, t)
         return
     elseif differential_vars === nothing
         error("differential_vars must be set for DAE initialization to occur. Either set consistent initial conditions, differential_vars, or use a different initialization algorithm.")