Columnmajor

Column-major order is a method of storing multi-dimensional arrays in linear memory in which the elements of each column are stored contiguously. This layout is widely used in scientific computing and in languages that emphasize mathematical column vectors.

In a two-dimensional array with m rows and n columns, the elements are laid out so that all elements of the first column come first, followed by the second column, and so on. For example, a 3x3 matrix A would be stored as A11, A21, A31, A12, A22, A32, A13, A23, A33. With zero-based indexing, the linear index of element A[i][j] is i + j*m. This convention is intrinsic to languages such as Fortran, MATLAB, and R, and is supported by some array libraries in other languages.

The column-major layout has trade-offs. It yields good locality when iterating down a column, which aligns with many linear-algebra routines and with BLAS/LAPACK implementations. It can be advantageous for operations that operate on column vectors. However, accessing across rows tends to cause cache misses, and interoperability with row-major languages (like C) often requires transposition or copying to convert layouts.

Common examples and contexts include Fortran’s native arrays, MATLAB variables, and the preference of many numerical libraries for column-major storage. Some languages and libraries offer a choice between column-major and row-major or provide ways to view data in Fortran-ordered form.

See also: row-major order, Fortran, MATLAB, BLAS/LAPACK.