sparse matrix

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https://en.wikipedia.org/wiki/Sparse_matrix

稀疏矩阵存储格式总结+存储效率对比:COO,CSR,DIA,ELL,HYB - Bin的专栏 - 博客园
http://www.cnblogs.com/xbinworld/p/4273506.html

稀疏矩阵的存储格式（Sparse Matrix Storage Formats） - Donkey Vision - 博客频道 - CSDN.NET
http://blog.csdn.net/anshan1984/article/details/8580952

对于很多元素为零的稀疏矩阵，仅存储非零元素可使矩阵操作效率更高。现有许多种稀疏矩阵的存储方式，但是多数采用相同的基本技术，即存储矩阵所有的非零元素到一个线性数组中，并提供辅助数组来描述原数组中非零元素的位置。

稀疏矩阵是指矩阵中的元素大部分是0的矩阵，事实上，实际问题中大规模矩阵基本上都是稀疏矩阵，很多稀疏度在90%甚至99%以上。因此我们需要有高效的稀疏矩阵存储格式。本文总结几种典型的格式：COO,CSR,DIA,ELL,HYB。

下面摘自[2]

6. Skyline Storage Format

The skyline storage format is important for the direct sparse solvers, and it is well suited for Cholesky or LU decomposition when no pivoting is required.

The skyline storage format accepted in Intel MKL can store only triangular matrix or triangular part of a matrix. This format is specified by two arrays:values andpointers. The following table describes these arrays:

values

A scalar array. For a lower triangular matrix it contains the set of elements from each row of the matrix starting from the first non-zero element to and including the diagonal element. For an upper triangular matrix it contains the set of elements from each column of the matrix starting with the first non-zero element down to and including the diagonal element. Encountered zero elements are included in the sets.

pointers

An integer array with dimension(m+1), where m is the number of rows for lower triangle (columns for the upper triangle).pointers(i) -pointers(1)+1gives the index of element invalues that is first non-zero element in row (column)i. The value ofpointers(m+1)is set tonnz+pointers(1), wherennz is the number of elements in the arrayvalues.

7. Block Compressed Sparse Row Format (BSR)

The Intel MKL block compressed sparse row (BSR) format for sparse matrices is specified by four arrays:values,columns,pointerB, andpointerE. The following table describes these arrays.

values

A real array that contains the elements of the non-zero blocks of a sparse matrix. The elements are stored block-by-block in row-major order. A non-zero block is the block that contains at least one non-zero element. All elements of non-zero blocks are stored, even if some of them is equal to zero. Within each non-zero block elements are stored in column-major order in the case of one-based indexing, and in row-major order in the case of the zero-based indexing.

columns

Element i of the integer array columns is the number of the column in the block matrix that contains thei-th non-zero block.

pointerB

Element j of this integer array gives the index of the element in thecolumns array that is first non-zero block in a rowj of the block matrix.

pointerE

Element j of this integer array gives the index of the element in thecolumns array that contains the last non-zero block in a rowj of the block matrix plus 1.

[1] Sparse Matrix Representations & Iterative Solvers, Lesson 1 by Nathan Bell. http://www.bu.edu/pasi/files/2011/01/NathanBell1-10-1000.pdf

[2] http://blog.csdn.net/anshan1984/article/details/8580952

[3] http://zhangjunhd.github.io/2014/09/29/sparse-matrix.html

[4] http://www.360doc.com/content/09/0204/17/96202_2458312.shtml

[5] Implementing Sparse Matrix-Vector Multiplication on Throughput-Oriented Processors, Nathan Bell and Michael Garland, Proceedings of Supercomputing '09

[6] Efficient Sparse Matrix-Vector Multiplication on CUDA, Nathan Bell and Michael Garland, NVIDIA Technical Report NVR-2008-004, December 2008