PDLAPIV(l)

📦 version 1.5, in ubuntu jammy

💖 PDLAPIV(l) 💖

okay bestie, meet PDLAPIV(l) — a LAPACK auxiliary routine, version 1.5. it's giving distributed-matrix permutation realness and honestly? iconic. 🫶

✨ NAME ✨

PDLAPIV — applies either P (the permutation matrix indicated by IPIV) or inv( P ) to a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1), resulting in row or column pivoting.

🦋 SYNOPSIS 🦋

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

           CHARACTER*1     DIREC, PIVROC, ROWCOL

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

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

           DOUBLE          PRECISION A( * )

🌸 PURPOSE 🌸

bestie, PDLAPIV applies either P (the permutation matrix indicated by IPIV) or inv( P ) to a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1), resulting in row or column pivoting. The pivot vector may be distributed across a process row or a column. The pivot vector should be aligned with the distributed matrix A. This routine will transpose the pivot vector if necessary. For example, if the row pivots should be applied to the columns of sub( A ), pass ROWCOL='C' and PIVROC='C'.

Notes

Each global data object is described by an associated description vector. That vector stores the information required to establish the mapping between an object element and its corresponding process and memory location.

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

Restrictions

IPIV must always be a distributed vector (not a matrix). Thus:

IF( ROWPIV .EQ. 'C' ) THEN
   JP must be 1
ELSE
   IP must be 1
END IF

The following restrictions apply when IPIV must be transposed:

IF( ROWPIV.EQ.'C' .AND. PIVROC.EQ.'C') THEN
    DESCIP(MB_) must equal DESCA(NB_)
ELSE IF( ROWPIV.EQ.'R" .AND. PIVROC.EQ.'R') THEN
    DESCIP(NB_) must equal DESCA(MB_)
END IF

👑 ARGUMENTS 👑

  • DIREC — (global input) CHARACTER*1. 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*1. Specifies if the rows or columns are to be permuted: = 'R' Rows will be permuted, = 'C' Columns will be permuted.
  • PIVROC — (global input) CHARACTER*1. Specifies whether IPIV is distributed over a process row or column: = 'R' IPIV distributed over a process row = 'C' IPIV distributed over a process column.
  • 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) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the distributed submatrix sub( A ) to which the row or column interchanges will be applied. On exit, the local pieces of the permuted distributed submatrix.
  • 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 — (local input) INTEGER array, dimension >= LOCr(M_A)+MB_A if ROWCOL='R', otherwise LOCc(N_A)+NB_A. 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. This array is tied to the distributed matrix A.
  • IWORK — (local workspace) INTEGER array, dimension (LDW) where LDW is equal to the workspace necessary for transposition, and the storage of the tranposed IPIV:
Let LCM be the least common multiple of NPROW and NPCOL.  IF( ROWCOL.EQ.'R' .AND. PIVROC.EQ.'R' ) THEN IF( NPROW.EQ.NPCOL ) THEN LDW = LOCr( N_P + MOD(JP-1, NB_P) ) + NB_P ELSE LDW = LOCr( N_P + MOD(JP-1, NB_P) ) + NB_P * CEIL( CEIL(LOCc(N_P)/NB_P) / (LCM/NPCOL) ) END IF ELSE IF( ROWCOL.EQ.'C' .AND. PIVROC.EQ.'C' ) THEN IF( NPROW.EQ.NPCOL ) THEN LDW = LOCc( M_P + MOD(IP-1, MB_P) ) + MB_P ELSE LDW = LOCc( M_P + MOD(IP-1, MB_P) ) + MB_P * CEIL( CEIL(LOCr(M_P)/MB_P) / (LCM/NPROW) ) END IF ELSE IWORK is not referenced.  END IF

LAPACK version 1.5 — 12 May 1997 — PDLAPIV(l) 💖✨ that's the whole vibe, bestie. periodt. 💅

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