Packed storage matrix

The packed storage matrix allows a matrix to be converted to an array, shrinking the matrix significantly. In doing so, a square $n\; \backslash times\; n$ matrix is converted to an array of length {{math|{{sfrac|n(n+1)|2}}}}.{{cite book |last1=Golub |first1=Gene H. |last2=Van Loan |first2=Charles F. |title=Matrix Computations |date=2013 |edition=4th |publisher=Johns Hopkins University Press |location=Baltimore, MD |page=170 |isbn=9781421407944}}

Consider the following upper matrix:

:$\backslash mathbf\{U\}\; =\; \backslash begin\{pmatrix\}$

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:

:$\backslash mathbf\{UP\}\; =\; (\backslash underbrace\{a\_\{11\}\}\backslash \; \backslash underbrace\{a\_\{12\}\backslash \; a\_\{22\}\}\backslash \; \backslash underbrace\{a\_\{13\}\backslash \; a\_\{23\}\backslash \; a\_\{33\}\}\backslash \; \backslash underbrace\{a\_\{14\},\backslash \; a\_\{24\}\backslash \; a\_\{34\}\backslash \; a\_\{44\}\})$

{{cite web |last=Blackford |first=Susan |title=Packed Storage |url=https://www.netlib.org/lapack/lug/node123.html |website=Netlib |publisher=LAPACK Users' Guide |date=1999-10-01 |access-date=2024-10-01 |archive-url=https://web.archive.org/web/20240401131450/https://www.netlib.org/lapack/lug/node123.html |archive-date=2024-04-01}}

Similarly the lower matrix:

:$\backslash mathbf\{L\}\; =\; \backslash begin\{pmatrix\}$

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}})

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

• Sparse matrix
• Skyline matrix
• band matrix

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

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Category:Arrays

Category:Matrices (mathematics)

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