Packed storage matrix

A packed storage matrix, also known as packed matrix, is a term used in programming for representing an <math>m\times n</math> 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 <math>n \times n</math> matrix is converted to an array of length Template:Math.<ref>Template:Cite book</ref>

Consider the following upper matrix:

<math>\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}</math> which can be packed into the one array:

<math> \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}})

</math><ref name = blackford>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Similarly the lower matrix:

<math>\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}.</math> can be packed into the following one dimensional array:

<math>

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}}) </math><ref name=blackford/>

Code examples (Fortran)

Both of the following storage schemes are used extensively in BLAS and LAPACK.

An example of packed storage for Hermitian matrix: <syntaxhighlight lang="fortran"> 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 </syntaxhighlight>

An example of packed storage for banded matrix: <syntaxhighlight lang="fortran"> 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 </syntaxhighlight>

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