Create SimpleNonlinearSolve_Kernel_Tutorial.md

docs/src/tutorials/SimpleNonlinearSolve_Kernel_Tutorial.md

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+# Using Nonlinear Solvers inside GPU Kernels
+
+We'll demonstrate how to leverage [SimpleNonlinearSolve.jl](https://github.com/SciML/SimpleNonlinearSolve.jl) inside kernels using [KernelAbstractions.jl](https://github.com/JuliaGPU/KernelAbstractions.jl). This allows for efficient solving of very small nonlinear systems on GPUs by avoiding allocations and dynamic dispatch overhead. We'll use the generalized Rosenbrock problem as an example and solve it for multiple initial conditions on various GPU architectures.
+
+### Prerequisites
+Ensure the following packages are installed:
+- Julia (v1.10 or later)
+- NonlinearSolve.jl
+- StaticArrays.jl
+- KernelAbstractions.jl
+- CUDA.jl (for NVIDIA GPUs)
+- AMDGPU.jl (for AMD GPUs)
+
+## Writing the Kernel
+Define a kernel using **'@kernel'** from **'KernelAbstractions.jl'** to solve a single initial condition.
+
+```@example kernel
+using NonlinearSolve, StaticArrays
+using KernelAbstractions, CUDA, AMDGPU
+
+@kernel function parallel_nonlinearsolve_kernel!(result, @Const(prob), @Const(alg))
+    i = @index(Global)
+    prob_i = remake(prob; u0 = prob.u0[i])
+    sol = solve(prob_i, alg)
+    @inbounds result[i] = sol.u
+    return nothing
+end
+```
+
+## Vectorized Solving
+Define a function to solve the problem for multiple initial conditions in parallel across GPU threads.
+
+```@example kernel
+function vectorized_solve(prob, alg; backend = CPU())
+    result = KernelAbstractions.allocate(backend, eltype(prob.u0), length(prob.u0))
+    groupsize = min(length(prob.u0), 1024)
+    kernel! = parallel_nonlinearsolve_kernel!(backend, groupsize, length(prob.u0))
+    kernel!(result, prob, alg)
+    KernelAbstractions.synchronize(backend)
+    return result
+end
+```
+
+## Define the Rosenbrock Function
+Define the generalized Rosenbrock function.
+
+```@example kernel
+@generated function generalized_rosenbrock(x::SVector{N}, p) where {N}
+    vals = ntuple(i -> gensym(string(i)), N)
+    expr = []
+    push!(expr, :($(vals[1]) = oneunit(x[1]) - x[1]))
+    for i in 2:N
+        push!(expr, :($(vals[i]) = 10.0 * (x[$i] - x[$i - 1] * x[$i - 1])))
+    end
+    push!(expr, :(@SVector [$(vals...)]))
+    return Expr(:block, expr...)
+end
+```
+
+## Define the Problem
+Create the nonlinear problem using the generalized Rosenbrock function and multiple initial conditions.
+
+```@example kernel
+u0 = @SVector [@SVector(rand(10)) for _ in 1:1024]
+prob = NonlinearProblem(generalized_rosenbrock, u0)
+```
+
+## Solve the Problem
+Solve the problem using **SimpleNonlinearSolve.jl** on different GPU architectures.
+
+```@example kernel
+# Threaded CPU
+vectorized_solve(prob, SimpleNewtonRaphson(); backend = CPU())
+
+# AMD ROCM GPU
+# vectorized_solve(prob, SimpleNewtonRaphson(); backend = ROCBackend())
+
+# NVIDIA CUDA GPU
+# vectorized_solve(prob, SimpleNewtonRaphson(); backend = CUDABackend())
+```
+
+## Conclusion
+This tutorial illustrated how to utilize **SimpleNonlinearSolve.jl** inside kernels using **KernelAbstractions.jl**, enabling efficient solving of small nonlinear systems on GPUs for applications requiring parallel processing and high performance.