You can pass matrix arguments to BLAS and LAPACK routines in one- or two- dimensional arrays, using the following storage schemes:

• full storage (in a two-dimensional array)
• packed storage for Hermitian, symmetric, or triangular matrices (in a one-dimensional array)
• band storage for band matrices (in a two-dimensional array).

Full storage is the following obvious scheme: a matrix A is stored in a two-dimensional array a, with the matrix element a_ij stored in the array element a(i,j). If a matrix is triangular (upper or lower, as specified by the argument uplo), only the elements of the relevant triangle are stored; the remaining elements of the array need not be set. Routines that handle symmetric or Hermitian matrices allow for either the upper or lower triangle of the matrix to be stored in the corresponding elements of the array:

Packed storage allows you to store symmetric, Hermitian, or triangular matrices more compactly: the relevant triangle (again, as specified by the argument uplo) is packed by columns in a one-dimensional array ap:

Band storage is as follows: an m by n band matrix with kl non-zero sub-diagonals and ku non-zero super-diagonals is stored compactly in a two-dimensional array ab with kl+ku+1 rows and n columns. Columns of the matrix are stored in the corresponding columns of the array, and diagonals of the matrix are stored in rows of the array.

Thus, a_ij is stored in ab(ku+1+i-j,j) for max(n,j-ku) < i < min(n,j+kl).

Use the band storage scheme only when kl and ku are much less than the matrix size n. (Although the routines work correctly for all values of kl and ku, it's inefficient to use the band storage if your matrices are not really banded.)

When you supply a general band matrix for LU factorization, allow space to store kl additional super-diagonals generated by fill-in as a result of row interchanges. This means that the matrix is stored according to the above scheme, but with kl+ku super-diagonals.

Triangular band matrices are stored in the same format, with either kl=0 if upper triangular, or ku=0 if lower triangular.

For symmetric or Hermitian band matrices with k sub-diagonals or super-diagonals, you need to store only the upper or lower triangle, as specified by the argument uplo: