本文只记录cumsum的实现逻辑的CUDA部分，也即底层调用了CUDA的什么实现算子。

void launch_cumsum_cuda_kernel(const TensorBase& result, const TensorBase& self, int64_t dim) {AT_DISPATCH_ALL_TYPES_AND_COMPLEX_AND2(ScalarType::Half, ScalarType::BFloat16,self.scalar_type(), "cumsum_cuda",[&]() {scalar_t init = 0;scan_dim<scalar_t>(self,result,dim,init,std::plus<scalar_t>());});
}

通过定位源码，找到了执行kernel的关键代码，可以看到，此代码内部调用了Pytorch定义的宏，核心调用是pytorch定义的名为scan_dim的模板函数。
该模板函数的定义位于：aten/src/ATen/native/cuda/ScanUtils.cuh
代码如下：

template<typename scalar_t, typename BinaryFunction>
void scan_dim(const TensorBase& self, const TensorBase& result,int64_t dim, scalar_t init, BinaryFunction binary_op) {int ndim = self.dim();auto self_ = self.expect_contiguous();TORCH_INTERNAL_ASSERT(result.is_contiguous());if (self.numel() == self.size(dim)) {cuda::cub::inclusive_scan(self_->const_data_ptr<scalar_t>(), result.mutable_data_ptr<scalar_t>(), binary_op, self.numel());} else if (dim == ndim - 1) {scan_innermost_dim<scalar_t>(*self_, result, init, binary_op);} else {scan_outer_dim<scalar_t>(*self_, result, dim, init, binary_op);}
}

该函数内部最重要的是后面的条件结构，首先如果元素的总数和当前维度的元素个数相同，也即tensor是一维的，直接利用cub的前缀扫描方法，如果元素的总数和当前维度的元素个数不同，又分为最内层的维度，也即最后一维，以及其他情况。

template<typename scalar_t, class BinaryFunction>
__host__ void scan_outer_dim(const TensorBase& self, const TensorBase& result,int dim, scalar_t init, BinaryFunction binary_op) {const int64_t row_size = self.size(dim);auto sizes = self.sizes();// Treat all outer dimensions (i.e. dim_ < dim) as one.const int64_t num_orows = c10::multiply_integers(sizes.begin(), sizes.begin() + dim);// Treat all inner dimensions (i.e. dim > dimension) as one.const int64_t num_irows = c10::multiply_integers(sizes.begin() + dim + 1, sizes.end());dim3 threads(std::min(512, int(num_irows)));int64_t maxGridDim = at::cuda::getCurrentDeviceProperties()->maxGridSize[1];dim3 grid(std::min(maxGridDim, num_orows), std::min(maxGridDim, ceil_div(num_irows, int64_t{threads.x})));check_fits_in_unsigned(num_irows, "num_irows");check_fits_in_unsigned(num_orows, "num_orows");check_fits_in_unsigned(row_size, "row_size");tensor_kernel_scan_outer_dim<scalar_t><<<grid, threads, 0, at::cuda::getCurrentCUDAStream()>>>(result.mutable_data_ptr<scalar_t>(), self.const_data_ptr<scalar_t>(),num_orows, num_irows, row_size, init, binary_op);C10_CUDA_KERNEL_LAUNCH_CHECK();
}template <typename scalar_t, class BinaryFunction>
void scan_innermost_dim(const TensorBase& self, const TensorBase& result,scalar_t init, BinaryFunction binary_op) {int64_t ndim = self.dim();// Treat all outer dimensions as a single dimension.int64_t row_size = self.size(ndim - 1);int64_t num_rows = self.numel() / row_size;// assuming max_num_threads per block is 512const uint32_t num_threads = 512;const uint32_t log_num_threads_x = get_log_num_threads_x_inner_scan<uint32_t>(num_rows, row_size);const uint32_t num_threads_x = (1 << log_num_threads_x);const uint32_t num_threads_y = num_threads / num_threads_x;dim3 threads(num_threads_x, num_threads_y);int64_t maxGridDim = at::cuda::getCurrentDeviceProperties()->maxGridSize[0];dim3 grid(std::min(maxGridDim, ceil_div(num_rows, int64_t{threads.y})));check_fits_in_unsigned(num_rows, "Number of rows (self.numel()/self.size(self.dim()-1))");check_fits_in_unsigned(row_size, "row_size");tensor_kernel_scan_innermost_dim<scalar_t><<<grid, threads, num_threads * 2 * sizeof(scalar_t),at::cuda::getCurrentCUDAStream()>>>(result.mutable_data_ptr<scalar_t>(), self.const_data_ptr<scalar_t>(),num_rows, row_size, log_num_threads_x, init, binary_op);C10_CUDA_KERNEL_LAUNCH_CHECK();
}

可以看到Pytorch针对上述两种情况进行了自定义，因为cub的inclusive_scan针对的是一维张量而非多维张量。

在调用核函数前，首先要定义调用核函数的网络结构和线程块结构，pytorch默认的线程块大小是512的，那么如何将512个线程块进行二维切分以满足合适的比例呢，pytorch中的做法是像下面这样：

template <typename integer>
constexpr inline integer get_log_num_threads_x_inner_scan(integer num_rows, integer row_size) {integer log_num_threads_x = 0;integer log_num_threads_y = 0;while (((integer)1 << log_num_threads_x) < row_size) {++log_num_threads_x;}while (((integer)1 << log_num_threads_y) < num_rows) {++log_num_threads_y;}// we want to keep the ratio between the x-threads and y-threads about the same as// the ratio between the row_size and num_rows, but the total number of threads in// a block should be about 512integer diff = log_num_threads_x - log_num_threads_y;// 9 is from log2(512)log_num_threads_x = ((integer)9 + diff) / (integer)2;// I found that in having larger log_num_threads_x can give significant speed up in some cases,// but detrimental in another case, so just keep the lower bound to be log2(16) == 4 to make it// similar to the previous implementation// Keeping the upper bound to be log2(512) == 9 as the maximum number of threads in a block.log_num_threads_x = std::min(std::max((integer)4, log_num_threads_x), (integer)9);return log_num_threads_x;
}

使用对数进行计算是便于计算出的x的结果可以整除，关键点在于最后平衡二者的比例的那行代码。可以预见，在某些情况下由于待处理数据的大小超过512造成线程块不能够完全分配的情况，此时就需要顾及线程块的比例，那么如果两个维度上线程块的对数值分别为x和y，对应的线程数分别为X，Y，也即$X=2^x$。此时X与Y的比例$X/Y$ 的结果也即$2^{x-y}$ ，其实也就是$2^{diff}$。那么如果将x变为（diff+9) / 2, y也就是 (9 - diff) / 2，二者相减也就是diff，因此保证了变换前后的比例。