Matrix Layout for LAPACK Routines

Developer Reference for Intel® oneAPI Math Kernel Library for C

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Date 3/22/2024

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Matrix Layout for LAPACK Routines

There are two general methods of storing a two dimensional matrix in linear (one dimensional) memory: column-wise (column major order) or row-wise (row major order). Consider an M-by-N matrix A:

Column Major Layout

In column major layout the first index, i, of matrix elements a_i,j changes faster than the second index when accessing sequential memory locations. In other words, for 1 ≤i < M, if the element a_i,j is stored in a specific location in memory, the element a_i+1,j is stored in the next location, and, for 1 ≤j < N, the element a_M,j is stored in the location previous to element a_1,j+1. So the matrix elements are located in memory according to this sequence:

{a_1,1a_2,1 ... a_M,1a_1,2a_2,2 ... a_M,2 ... ... a_1,Na_2,N ... a_M,N}

Row Major Layout

In row major layout the second index, j, of matrix elements a_i,j changes faster than the first index when accessing sequential memory locations. In other words, for 1 ≤j < N, if the element a_i,j is stored in a specific location in memory, the element a_i,j+1 is stored in the next location, and, for 1 ≤i < M, the element a_i,N is stored in the location previous to element a_i+1,1. So the matrix elements are located in memory according to this sequence:

{a_1,1a_1,2 ... a_1,Na_2,1a_2,2 ... a_2,N ... ... a_N,1a_N,2 ... a_M,N}

Leading Dimension Parameter

A leading dimension parameter allows use of LAPACK routines on a submatrix of a larger matrix. For example, the submatrix B can be extracted from the original matrix A defined previously:

B is formed from rows with indices i₀ + 1 to i₀ + K and columns j₀ + 1 to j₀ + L of matrix A. To specify matrix B, LAPACK routines require four parameters:

the number of rows K;
the number of columns L;
a pointer to the start of the array containing elements of B;
the leading dimension of the array containing elements of B.

The leading dimension depends on the layout of the matrix:

Column major layout

Leading dimension ldb=M, the number of rows of matrix A.

Starting address: offset by i₀ + j₀*ldb from a_1,1.
Row major layout

Leading dimension ldb=N, the number of columns of matrix A.

Starting address: offset by i₀*ldb + j₀ from a_1,1.

Parent topic: LAPACK Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

Matrix Layout for LAPACK Routines

Column Major Layout

Row Major Layout

Leading Dimension Parameter