A packed storage matrix, also known as packed matrix, is a term used in programming for representing an m\times n 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:

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 n \times n matrix is converted to an array of length n(n+1)⁄2.[1]

Consider the following upper matrix:

\mathbf{U} = \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ & a_{22} & a_{23} & a_{24} \\ & & a_{33} & a_{34} \\ & & & a_{44} \\ \end{pmatrix}

which can be packed into the one array:

\mathbf{UP} = (\underbrace{a_{11}}\ \underbrace{a_{12}\ a_{22}}\ \underbrace{a_{13}\ a_{23}\ a_{33}}\ \underbrace{a_{14},\ a_{24}\ a_{34}\ a_{44}})[2]

Similarly the lower matrix:

\mathbf{L} = \begin{pmatrix} a_{11} & & & \\ a_{21} & a_{22} & & \\ a_{31} & a_{32} & a_{33} & \\ a_{41} & a_{42} & a_{43} & a_{44} \\ \end{pmatrix}.

can be packed into the following one dimensional array:

LP = (\underbrace{a_{11}\ a_{21}\ a_{31}\ a_{41}}\ \underbrace{a_{22}\ a_{32}\ a_{42}}\ \underbrace{a_{33}\ a_{43}}\ \underbrace{a_{44}})[2]

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

Further reading

References

  1. ^ Golub, Gene H. & Van Loan, Charles F. (2013). Matrix Computations. 4th ed. Baltimore, MD: Johns Hopkins University Press. p. 170. ISBN 9781421407944.
  2. ^ Blackford, Susan (1999-10-01). "Packed Storage". Netlib. LAPACK Users' Guide. Archived 2024-04-01 at the Wayback Machine. Retrieved 2024-10-01.