Data Storage

Only non-zero elements in a sparse matrix are stored. Suppose sparse matrix a has nz non-zero elements: a(k₁), a(k₂), ..., a(k_nz)

Two arrays are used to store the information about matrix a:

x(1)=a(k₁), x(2)=a(k₂), ..., x(nz)=a(k_nz)

indx(1)=k₁, indx(2)=k₂, ..., indx(nz)=k_nz

The compressed storage format is memory-saving and time-efficient.

All elements in a sparse matrix are stored.

The compressed sparse row (CSR) format is specified by four arrays: value, column, pointerB, and pointerE,

as described in Table 1.

**Table 1** Storage arrays of sparse matrix A in CSR format
Parameter	Description
value	Non-zero elements in matrix A, stored in row-major order The array length is equal to the number of non-zero elements in matrix A.
column	Number of the column in matrix A that contains the i-th value in the value array The array length is equal to the number of non-zero elements in matrix A.
pointerB	For one-based indexing, pointerB(j)-pointerB(1)+1 indicates the index of the element in the value array that is the first non-zero element in the jth row of matrix A. For zero-based indexing, pointerB(j)-pointerB(0) indicates the index of the element in the value array that is the first non-zero element in the jth row of matrix A. The array length is equal to the number of rows of matrix A.
pointerE	For one-based indexing, pointerE(j)-pointerB(1) indicates the index of the element in the value array that is the last non-zero element in the jth row of matrix A. For zero-based indexing, pointerE(j)-pointerB(0)-1 indicates the index of the element in the value array that is the last non-zero element in the jth row of matrix A. The array length is equal to the number of rows of matrix A.

Matrix A $\text{[math]}$ can be represented in CSR format as in Table 2.

**Table 2** Matrix A in CSR format
Matrix	Indexing Method	CSR Format
$\text{[math]}$	One-based indexing	value = [2, -3, 7, 1, -6, 8, -4, 5, 9] column = [1, 2, 4, 3, 4, 1, 3, 4, 1] pointerB = [1, 4, 6, 9] pointerE = [4, 6, 9, 10]
$\text{[math]}$	Zero-based indexing	value = [2, -3, 7, 1, -6, 8, -4, 5, 9] column = [0, 1, 3, 2, 3, 0, 2, 3, 0] pointerB = [0, 3, 5, 8] pointerE = [3, 5, 8, 9]

The compressed sparse column (CSC) format is specified by four arrays: value, column, pointerB, and pointerE,

as described in Table 3.

**Table 3** Storage arrays of sparse matrix A in CSC format
Parameter	Description
value	Non-zero elements in matrix A, stored in column-major order The array length is equal to the number of non-zero elements in matrix A.
column	Number of the row in matrix A that contains the i-th value in the value array The array length is equal to the number of non-zero elements in matrix A.
pointerB	For one-based indexing, pointerB(j)-pointerB(1)+1 indicates the index of the element in the value array that is the first non-zero element in the jth column of matrix A. For zero-based indexing, pointerB(j)-pointerB(0) indicates the index of the element in the value array that is the first non-zero element in the jth column of matrix A. The array length is equal to the number of columns of matrix A.
pointerE	For one-based indexing, pointerE(j)-pointerB(1) indicates the index of the element in the value array that is the last non-zero element in the jth column of matrix A. For zero-based indexing, pointerE(j)-pointerB(0)-1 indicates the index of the element in the value array that is the last non-zero element in the jth column of matrix A. The array length is equal to the number of columns of matrix A.

Matrix A $\text{[math]}$ can be represented in CSC format as in Table 4.

**Table 4** Matrix A in CSC format
Matrix	Indexing Method	CSC Format
$\text{[math]}$	One-based indexing	value = [2, 8, 9, -3, 1, -4, 7, -6, 5] column = [1, 3, 4, 1, 2, 3, 1, 2, 3] pointerB = [1, 4, 5, 7] pointerE = [4, 5, 7, 10]
$\text{[math]}$	Zero-based indexing	value = [2, 8, 9, -3, 1, -4, 7, -6, 5] column = [0, 2, 3, 0, 1, 2, 0, 1, 2] pointerB = [0, 3, 4, 6] pointerE = [3, 4, 6, 9]

Parent topic: Data Types and Data Storage