PZLAPV2(l)

🌸 NAME 🌸

PZLAPV2 — bestie, this one applies either P (the permutation matrix that IPIV is pointing at) or inv( P ) to a M-by-N distributed matrix sub( A ) (which is shorthand for A(IA:IA+M-1,JA:JA+N-1)), and it comes out the other side as row or column pivoting. That's it, that's the whole thing, and honestly? iconic. 💅

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✨ SYNOPSIS ✨

SUBROUTINE PZLAPV2( DIREC, ROWCOL, M, N, A, IA, JA, DESCA, IPIV, IP, JP,
                    DESCIP )

    CHARACTER       DIREC, ROWCOL

    INTEGER         IA, IP, JA, JP, M, N

    INTEGER         DESCA( * ), DESCIP( * ), IPIV( * )

    COMPLEX*16      A( * )

Look at those arguments. So many. We love a thorough queen. 👑

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🦋 PURPOSE 🦋

PZLAPV2 applies either P (that's the permutation matrix indicated by IPIV) or inv( P ) to a M-by-N distributed matrix sub( A ), which means A(IA:IA+M-1,JA:JA+N-1), and the result is row or column pivoting. Slay.

Real talk though: the pivot vector should be aligned with the distributed matrix A. For pivoting the rows of sub( A ), IPIV should be distributed along a process column and replicated over all process rows. Similarly, for column pivoting, IPIV should be distributed along a process row and replicated over all process columns. No cap. 🫶

📝 Notes 📝

Every global data object is described by an associated description vector, periodt. This vector stores all the info you need to establish the mapping between an object element and its corresponding process and memory location. It's giving personal assistant for arrays.

Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array".

• DTYPE_A (global) — stored in DESCA( DTYPE_ ) — the descriptor type. In this case, DTYPE_A = 1.
• CTXT_A (global) — stored in DESCA( CTXT_ ) — the BLACS context handle, indicating the BLACS process grid A is distributed over. The context itself is global, but the handle (the integer value) may vary.
• M_A (global) — stored in DESCA( M_ ) — the number of rows in the global array A.
• N_A (global) — stored in DESCA( N_ ) — the number of columns in the global array A.
• MB_A (global) — stored in DESCA( MB_ ) — the blocking factor used to distribute the rows of the array.
• NB_A (global) — stored in DESCA( NB_ ) — the blocking factor used to distribute the columns of the array.
• RSRC_A (global) — stored in DESCA( RSRC_ ) — the process row over which the first row of the array A is distributed.
• CSRC_A (global) — stored in DESCA( CSRC_ ) — the process column over which the first column of the array A is distributed.
• LLD_A (local) — stored in DESCA( LLD_ ) — the leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)).

Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q.

LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column.

Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row.

The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC:

LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).

An upper bound for these quantities may be computed by:

LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A

math but make it cute ✨

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💜 ARGUMENTS 💜

• DIREC — (global input) CHARACTER. Specifies in which order the permutation is applied:
  • = 'F' (Forward) — applies pivots Forward from top of matrix. Computes P * sub( A ).
  • = 'B' (Backward) — applies pivots Backward from bottom of matrix. Computes inv( P ) * sub( A ).
• ROWCOL — (global input) CHARACTER. Specifies if the rows or columns are to be permuted:
  • = 'R' — Rows will be permuted.
  • = 'C' — Columns will be permuted.
• M — (global input) INTEGER. The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0.
• N — (global input) INTEGER. The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0.
• A — (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this local array contains the local pieces of the distributed matrix sub( A ) to which the row or columns interchanges will be applied. On exit, this array contains the local pieces of the permuted distributed matrix. it goes in one way and comes out the other, that's the whole arc 💖
• IA — (global input) INTEGER. The row index in the global array A indicating the first row of sub( A ).
• JA — (global input) INTEGER. The column index in the global array A indicating the first column of sub( A ).
• DESCA — (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A.
• IPIV — (input) INTEGER array, dimension >= LOCr(M_A)+MB_A if ROWCOL = 'R', LOCc(N_A)+NB_A otherwise. It contains the pivoting information. IPIV(i) is the global row (column), local row (column) i was swapped with. The last piece of the array of size MB_A (resp. NB_A) is used as workspace. IPIV is tied to the distributed matrix A.
• IP — (global input) INTEGER. IPIV's global row index, which points to the beginning of the submatrix which is to be operated on.
• JP — (global input) INTEGER. IPIV's global column index, which points to the beginning of the submatrix which is to be operated on.
• DESCIP — (global and local input) INTEGER array of dimension 8. The array descriptor for the distributed matrix IPIV.

Heads up bestie: IPIV needs that extra MB_A (or NB_A) chunk at the end for workspace, so don't skimp on the dimension or it's giving undefined behaviour. 🙈

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And that's the whole page, babe. This one's from LAPACK version 1.5, dated 12 May 1997 — yes, it's older than us, and it's still serving. 🎀 PZLAPV2(l), out. 💖✨