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Use numba.jit: The first step to increasing the speed of Poincare Section code is to decorate the function with numba.jit. This will compile the function to machine code and run it faster.

Use numba.cuda.jit for GPU acceleration: If you have a CUDA-enabled GPU, you can use numba.cuda.jit to accelerate the computation further. This will allow you to take advantage of the parallel processing power of your GPU.

Use numpy.ndarray instead of lists: Numpy arrays are much faster than lists in Python. So, if your code uses lists, try to convert them to numpy arrays.

Use numpy vectorized functions: If possible, try to use numpy vectorized functions instead of loops. Vectorized functions can perform computations on entire arrays at once, which can be much faster than performing computations on individual elements in a loop.

Remove unnecessary computations: Evaluate the code and identify if there are any unnecessary computations performed. Remove those computations to save time.

Reduce memory usage: If your code is using a lot of memory, try to reduce memory usage by reusing variables and arrays instead of creating new ones.

Parallelize loops: If there are any loops in your code that can be parallelized, consider using numba.prange to parallelize them.

Use the parallel setting: You can enable the parallel setting in numba.jit to further optimize your code. This will allow the function to use multiple threads to perform computations in parallel.