Compressed Row Storage

最新推荐文章于 2024-10-30 15:16:10 发布

wushiheibing

最新推荐文章于 2024-10-30 15:16:10 发布

阅读量674

点赞数

本文介绍了稀疏矩阵的压缩行存储(CRS)格式，该格式通过连续存储矩阵每一行的非零元素来节省空间，并使用三个数组：浮点数数组存储非零元素值，整数数组存储列索引及每行的起始位置。此方法适用于非对称稀疏矩阵，对于对称矩阵则只需存储上三角或下三角部分。

摘要生成于 C知道，由 DeepSeek-R1 满血版支持，前往体验 >

The compressed row and column (in the next section) storage formats are the most general: they make absolutely no assumptions about the sparsity structure of the matrix, and they don't store any unnecessary elements. On the other hand, they are not very efficient, needing an indirect addressing step for every single scalar operation in a matrix-vector product or preconditioner solve.

The compressed row storage (CRS) format puts the subsequent nonzeros of the matrix rows in contiguous memory locations. Assuming we have a nonsymmetric sparse matrix , we create three vectors: one for floating point numbers (val) and the other two for integers (col_ind, row_ptr). The val vector stores the values of the nonzero elements of the matrix as they are traversed in a row-wise fashion. The col_ind vector stores the column indexes of the elements in the val vector. That is, if ${\tt val(k)}=a_{i,j}$ , then ${\tt col\_ind(k)}=j$ . The row_ptr vector stores the locations in the val vector that start a row; that is, if ${\tt val(k)}=a_{i,j}$ , then ${\tt row\_ptr(i)}\leq k<{\tt row\_ptr(i+1)}$ . By convention, we define ${\tt row\_ptr(n+1)} = nnz+1$ , where is the number of nonzeros in the matrix . The storage savings for this approach is significant. Instead of storing elements, we need only storage locations.

As an example, consider the nonsymmetric matrix defined by

$\begin{displaymath}A =\left[\begin{array}{rrrrrr}10 & 0 & 0 & 0 &-2 & 0 \\...... 9 & 9 & 13 \\0 & 4 & 0 & 0 & 2 & -1\end{array}\right] ~.\end{displaymath}$

(269)

The CRS format for this matrix is then specified by the arrays {val, col_ind, row_ptr} given below:

`val`	10	-2	3	9	3	7	8	7	3 $\cdots$ 9	13	4	2	-1
`col_ind`	1	5	1	2	6	2	3	4	1 $\cdots$ 5	6	2	5	6

row_ptr 1 3 6 9 13 17 20

If the matrix

is symmetric, we need only store the upper (or lower) triangular portion of the matrix. The tradeoff is a more complicated algorithm with a somewhat different pattern of data access.