Sparse BLAS CSC Matrix Storage Format

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2025

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Sparse BLAS CSC Matrix Storage Format

The Intel® oneAPI Math Kernel Library (oneMKL) Sparse BLAS compressed sparse column (CSC) format is specified by four arrays:

values
columns
pointerB
pointerE

In addition, each sparse matrix has an associated variable, indexing, which specifies whether the matrix indices are 0-based (indexing=0) or 1-based (indexing=1). The following table describes the arrays in terms of the values, row, and column positions of the non-zero elements in a sparse matrix A:

values: A real or complex array that contains the non-zero elements of A. Values of the non-zero elements of A are mapped into the values array using the column-major layout.
rows: Element i of the integer array rows is the number of the row in A that contains the i-th value in the values array.
pointerB: Element j of this integer array gives the index of the element in the values and rows arrays that is the first non-zero element in column j of A. Note that the 1-based position of this element in the arrays is equal to pointerB(j)-indexing+1.
pointerE: An integer array that contains column indices, such that pointerE(j)-1 is the index of the element in the values and rows arrays that is the last non-zero element in column j of A. Note that the 1-based position of this element in the arrays is equal to pointerE(j)-indexing.

The length of the values and columns arrays is equal to the number of non-zero elements in A. The length of the pointerB and pointerE arrays is equal to the number of columns in A.

The CSC format is similar to the CSR format, but the indices are compressed along the columns instead of the rows. Storing a matrix in CSC format and storing the transpose of that same matrix in CSR format will result in identical arrays.

NOTE:

Note that the Intel® oneAPI Math Kernel Library (oneMKL) Sparse BLAS routines support the CSC format both with one-based indexing and zero-based indexing.

For example, consider matrix B:

It can be represented in the CSC format as:

Storage Arrays for a Matrix in CSC Format
one-based indexing
`values`	=	(1	-2	-4	-1	5	8	4	2	-3	6	7	4	-5)
`rows`	=	(1	2	4	1	2	5	3	4	1	3	4	3	5)
`pointerB`	=	(1	4	7	9	12)
`pointerE`	=	(4	7	9	12	14)
zero-based indexing
`values`	=	(1	-2	-4	-1	5	8	4	2	-3	6	7	4	-5)
`rows`	=	(0	1	3	0	1	4	2	3	0	2	3	2	4)
`pointerB`	=	(0	3	6	8	11)
`pointerE`	=	(3	6	8	11	13)

Parent topic: Sparse Matrix Storage Formats

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Sparse BLAS CSC Matrix Storage Format