CUDA Dot Product
Problem Statement
Compute the dot product of two 1D arrays, sum(a * b), by multiplying elementwise and reducing. Use a shared-memory tree reduction per block plus one atomic add per block into the global result.
Background
This fuses an elementwise multiply with a reduction: each thread loads a[i] * b[i] into shared memory, the block tree-reduces, and thread 0 atomically adds the block's partial into out[0].
Your Task
Implement dot_kernel and run(n=8192) returning whether out[0] matches np.dot(a, b).
How it is tested
Your solution must define a top-level function run(...) that allocates the inputs, copies them to the GPU, launches your @cuda.jit kernel, and returns a Python bool from np.allclose(gpu_result, reference). The grader prints run(...); the expected output is True.
Example:
n = 8192
True
- The input
n = 8192determines the size of the 1D arraysaandb, which are used to compute the dot product. - Each element of
ais multiplied by the corresponding element ofb, resulting in an array of products: ai⋅bi. - The products are then reduced using a tree reduction per block, followed by an atomic add per block into the global result, to compute the dot product: ∑i=0n−1ai⋅bi.
- The computed dot product is compared to the reference result obtained using
np.dot(a, b), and the function returnsTrueif the two results are close, indicating a successful computation.
Constraints:
- Load s[t] = a[i] * b[i] (0.0 for out-of-range threads)
- Tree-reduce within the block, then thread 0 atomic-adds the partial
- Initialize the device accumulator to 0
Background Knowledge
The problem involves computing the dot product of two 1D arrays using CUDA, a parallel computing platform developed by NVIDIA. To solve this problem, it's essential to understand the basics of CUDA programming, including kernel launches, thread blocks, and shared memory. In CUDA, a kernel is a function that runs on the GPU, and it's launched with a specified number of thread blocks, each containing a certain number of threads. Shared memory is a small, on-chip memory space that's shared among threads within a block, allowing for efficient communication and data exchange.
The problem also requires knowledge of reduction operations, which involve combining multiple values into a single output. In this case, the reduction operation is a tree reduction, where each thread loads a value into shared memory, and then the block reduces these values using a tree-like pattern. This approach helps to minimize the number of operations required to compute the final result. Additionally, the problem involves using atomic operations, such as atomic adds, to update the global result variable.
To understand the problem, it's also important to be familiar with NumPy and Numba, which are used to create and manipulate arrays, as well as to launch CUDA kernels. Numba is a just-in-time compiler that allows you to write CUDA kernels in Python, making it easier to develop and test CUDA applications.
Algorithm/Approach
The general approach to solving this problem involves the following steps:
- Launch a CUDA kernel with multiple thread blocks, each containing a certain number of threads.
- Within each block, perform an element-wise multiplication of the input arrays and store the results in shared memory.
- Perform a tree reduction within each block to compute the partial dot product.
- Use atomic operations to update the global result variable with the partial dot product from each block.
- Compare the final result with the reference result computed using NumPy.
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