逆迭代期间求解线性方程
我正在使用 OpenCL 计算矩阵的特征向量。 AMD 有一个特征值计算示例，所以我决定使用逆迭代得到特征向量。
我遵循此处描述的算法，我注意到按顺序要求解步骤 4，我需要求解线性方程组（或计算矩阵的逆）。
使用 OpenCL 在 GPU 上执行此操作的最佳方法是什么？有没有我应该研究的示例/参考文献？
编辑：对不起，我应该提到我的矩阵是对称三对角的。从我读到的内容来看，这可能很重要，并且可能会大大简化整个过程
I am using OpenCL to calculate the eigenvectors of a matrix. AMD has an example of eigenvalue calculation so I decided to use inverse iteration to get the eigenvectors.
I was following the algorithm described here and I noticed that in order to solve step 4 I need to solve a system of linear equations (or calculate the inverse of a matrix).
What is the best way to do this on a GPU using OpenCL? Are there any examples/references that I should look into?
EDIT: I'm sorry, I should have mentioned that my matrix is symmetric tridiagonal. From what I have been reading this could be important and maybe simplifies the whole process a lot
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矩阵是三对角的事实非常重要 - 这将问题的复杂性从 O(N^3) 降低到 O(N)。您可能可以从它也是对称的事实中获得一些加速，但这不会那么引人注目。
求解三对角系统的方法在这里：http://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm。
另请注意，您不需要存储矩阵的所有 N^2 个元素，因为几乎所有元素都为零。您只需要一个长度为 N 的向量（对于对角线）和两个长度为 N-1 的向量作为下对角线和上对角线。由于矩阵是对称的，因此下对角线和上对角线是相同的。
希望这有帮助...
The fact that the matrix is tridiagonal is VERY important - that reduces the complexity of the problem from O(N^3) to O(N). You can probably get some speedup from the fact that it's symmetric too, but that won't be as dramatic.
The method for solving a tridiagonal system is here: http://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm.
Also note that you don't need to store all N^2 elements of the matrix, since almost all of them will be zeroes. You just need one vector of length N (for the diagonal) and two of length N-1 for the sub- and superdiagonals. And since your matrix is symmetric, the sub- and superdiagonals are the same.
Hope that's helpful...
我建议使用LU分解。
这是 示例。
它是用 CUDA 编写的，但我认为，用 OpenCL 重写它并不难。
I suggest using LU decomposition.
Here's example.
It's written in CUDA, but I think, it's not so hard to rewrite it in OpenCL.