Packed storage matrix

A packed storage matrix, also known as packed matrix, is a term used in programming for representing an matrix. It is a more compact way than an m-by-n rectangular array by exploiting a special structure of the matrix.

Typical examples of matrices that can take advantage of packed storage include: symmetric or hermitian matrix Triangular matrix Banded matrix.

Triangular packed matrices

The packed storage matrix allows a matrix to be converted to an array, shrinking the matrix significantly. In doing so, a square matrix is converted to an array of length .

Consider the following upper matrix:

which can be packed into the one array:

Similarly the lower matrix:

can be packed into the following one dimensional array:

Code examples (Fortran) Both of the following storage schemes are used extensively in BLAS and LAPACK.

An example of packed storage for Hermitian matrix: complex :: A(n,n) ! a hermitian matrix complex :: AP(n*(n+1)/2) ! packed storage for A ! the lower triangle of A is stored column-by-column in AP. ! unpacking the matrix AP to A do j=1,n k = j*(j-1)/2 A(1:j,j) = AP(1+k:j+k) A(j,1:j-1) = conjg(AP(1+k:j-1+k)) end do

An example of packed storage for banded matrix: real :: A(m,n) ! a banded matrix with kl subdiagonals and ku superdiagonals real :: AP(-kl:ku,n) ! packed storage for A ! the band of A is stored column-by-column in AP. Some elements of AP are unused. ! unpacking the matrix AP to A do j = 1, n forall(i=max(1,j-kl):min(m,j+ku)) A(i,j) = AP(i-j,j) end do print *,AP(0,:) ! the diagonal

See also Sparse matrix Skyline matrix band matrix

Further reading https://www.netlib.org/lapack/lug/ https://www.netlib.org/blas/ https://github.com/numericalalgorithmsgroup/LAPACK_Examples

References