Data Storage

Full-Storage Format for Vectors

All elements in a vector are stored.

Sparse Matrix Storage Format

The compressed sparse row (CSR) format is specified by the following arrays: value, column, and pointerB. For the LL^T or LL^H factorization method in cluster mode, only the upper or lower triangular part of a matrix is stored. Table 1 describes the three arrays in sparse matrix A in CSR format.

**Table 1** Storage arrays of sparse matrix A in CSR format
Array	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 zero-based indexing, pointerB(j) indicates the index of the element in the value array that is the first non-zero element in the j-th row of matrix A. Array length = Number of rows of matrix A + 1

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

When the LL^T or LL^H factorization method in cluster mode is used, only the upper triangular matrix needs to be stored. See Table 3.

**Table 2** Matrix A in CSR format
Matrix	Indexing Method	CSR Format
	Zero-based indexing	value = [2, -3, 7, -3, 1, -6, 1, -4, 5, 7, -6, 5] column = [0, 1, 3, 0, 2, 3, 1, 2, 3, 0, 1, 2] pointerB = [0, 3, 6, 9, 12]

**Table 3** Upper triangular part of symmetric matrix A in CSR format
Matrix	Indexing Method	CSR Format
	Zero-based indexing	value = [2, -3, 7, 1, -6, -4, 5] column = [0, 1, 3, 2, 3, 2, 3] pointerB = [0, 3, 5, 7, 7]

CSC (Compressed Sparse Column) format, which contains three elements: value, row, and pointerB (column index). The format is similar to the CSR format and is not described here.

COO (Coordinate) format, which contains three elements: value, row, and column. Table 4 describes the three arrays in sparse matrix A in COO format.

**Table 4** Storage arrays of sparse matrix A in COO 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.
row	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.
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.

Matrix A $\text{[math]}$ can be represented in COO format as in Table 5.

**Table 5** Matrix A in COO format
Matrix	COO Format
	value = [2, -3, 7, -3, 1, -6, 1, -4, 5, 7, -6, 5] row = [0, 1, 3, 0, 2, 3, 1, 2, 3, 0, 1, 2] column = [0, 1, 3, 0, 2, 3, 1, 2, 3, 0, 1, 2]

Storage Formats of Dense Matrices

Dense formats are classified into row-major and column-major layouts, which consist of the data values and the leading dimension (ld). If dense matrix A is row-major, ld represents the array length required to store each row; if it is column-major, ld represents the array length required to store each column. Table 6 describes the parameters of dense matrix A.

**Table 6** Dense matrix A
Parameter	Description
value	All elements in matrix A, stored in row-major order (column-major) The array length is equal to the number of rows (columns) multiplied by ld.
ld	An integer representing the stride between elements of two rows (columns), which is greater than or equal to the number of elements in a row (column).

Assume that matrix A is . Table 7 shows the dense storage formats of matrix A.

**Table 7** Matrix A in dense format
Matrix	Dense Format (Row-Major)	Dense Format (Column-Major)
	value = [2, -3, 0, 7, -1, 2, 4, -6, 2, 1, -4, 3, 3, 5, 9, 8] ld = 4	value = [2, -1, 2, 3, -3, 2, 1, 5, 0, 4, -4, 9, 7, -6, 3, 8] ld = 4

Parent topic: Data Types and Data Storage