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Packed storage matrix

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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:

Triangular packed matrices

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The packed storage matrix allows a matrix to be converted to an array, shrinking the matrix significantly. Where a square matrix is converted to a array of length n(n-1)/2.[1]

Consider the following upper matrix:

which can be packed into the one array:

[2]


Similarly the lower matrix:

can be packed into the following one dimensional array:

[2]

Code examples (Fortran)

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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

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Further reading

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References

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