PSLARFB(l)

📦 version 1.5, in ubuntu jammy

✨ NAME ✨

okay so bestie, meet PSLARFB 💖 — she applies a real block reflector Q or its transpose Q**T to a real distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1). (Yes, "applie" — that's literally what the page says, we love a typo that ate 💅)

🎀 SYNOPSIS 🎀

SUBROUTINE PSLARFB( SIDE, TRANS, DIRECT, STOREV, M, N, K, V, IV, JV,
                    DESCV, T, C, IC, JC, DESCC, WORK )

    CHARACTER       SIDE, TRANS, DIRECT, STOREV

    INTEGER         IC, IV, JC, JV, K, M, N

    INTEGER         DESCC( * ), DESCV( * )

    REAL            C( * ), T( * ), V( * ), WORK( * )

🦋 PURPOSE 🦋

PSLARFB applies a real block reflector Q or its transpose Q**T to a real distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) — and she can do it from the left or from the right. main character energy fr fr 👑

💫 Notes 💫

Every global data object is described by an associated description vector. That vector stores the info you need to establish the mapping between an object element and its corresponding process and memory location. no cap, that's the whole point of it 🫶

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

💅 ARGUMENTS 💅

  • SIDE — (global input) CHARACTER. Which side we're coming from, bestie: = 'L': apply Q or Q**T from the Left; = 'R': apply Q or Q**T from the Right.
  • TRANS — (global input) CHARACTER. = 'N': No transpose, apply Q; = 'T': Transpose, apply Q**T.
  • DIRECT — (global input) CHARACTER. Indicates how Q is formed from a product of elementary reflectors. = 'F': Q = H(1) H(2) . . . H(k) (Forward); = 'B': Q = H(k) . . . H(2) H(1) (Backward).
  • STOREV — (global input) CHARACTER. Indicates how the vectors which define the elementary reflectors are stored: = 'C': Columnwise; = 'R': Rowwise.
  • M — (global input) INTEGER. The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( C ). 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( C ). N >= 0.
  • K — (global input) INTEGER. The order of the matrix T (= the number of elementary reflectors whose product defines the block reflector).
  • V — (local input) REAL pointer into the local memory to an array of dimension ( LLD_V, LOCc(JV+K-1) ) if STOREV = 'C', ( LLD_V, LOCc(JV+M-1)) if STOREV = 'R' and SIDE = 'L', ( LLD_V, LOCc(JV+N-1) ) if STOREV = 'R' and SIDE = 'R'. It contains the local pieces of the distributed vectors V representing the Householder transformation. See further details. If STOREV = 'C' and SIDE = 'L', LLD_V >= MAX(1,LOCr(IV+M-1)); if STOREV = 'C' and SIDE = 'R', LLD_V >= MAX(1,LOCr(IV+N-1)); if STOREV = 'R', LLD_V >= LOCr(IV+K-1).
  • IV — (global input) INTEGER. The row index in the global array V indicating the first row of sub( V ).
  • JV — (global input) INTEGER. The column index in the global array V indicating the first column of sub( V ).
  • DESCV — (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V.
  • T — (local input) REAL array, dimension MB_V by MB_V if STOREV = 'R' and NB_V by NB_V if STOREV = 'C'. The triangular matrix T in the representation of the block reflector.
  • C — (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the M-by-N distributed matrix sub( C ). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q or sub( C )*Q'.
  • IC — (global input) INTEGER. The row index in the global array C indicating the first row of sub( C ).
  • JC — (global input) INTEGER. The column index in the global array C indicating the first column of sub( C ).
  • DESCC — (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C.
  • WORK — (local workspace) REAL array, dimension (LWORK). Sizing rules, verbatim, because we do not play with workspace math 🫠:
If STOREV = 'C', if SIDE = 'L', LWORK >= ( NqC0 + MpC0 ) * K
else if SIDE = 'R', LWORK >= ( NqC0 + MAX( NpV0 + NUMROC(
NUMROC( N+ICOFFC, NB_V, 0, 0, NPCOL ), NB_V, 0, 0, LCMQ ), MpC0
) ) * K end if else if STOREV = 'R', if SIDE = 'L', LWORK >= (
MpC0 + MAX( MqV0 + NUMROC( NUMROC( M+IROFFC, MB_V, 0, 0, NPROW
), MB_V, 0, 0, LCMP ), NqC0 ) ) * K else if SIDE = 'R', LWORK >=
( MpC0 + NqC0 ) * K end if end if

where:

LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ),

IROFFV = MOD( IV-1, MB_V ), ICOFFV = MOD( JV-1, NB_V ), IVROW =
INDXG2P( IV, MB_V, MYROW, RSRC_V, NPROW ), IVCOL = INDXG2P( JV,
NB_V, MYCOL, CSRC_V, NPCOL ), MqV0 = NUMROC( M+ICOFFV, NB_V,
MYCOL, IVCOL, NPCOL ), NpV0 = NUMROC( N+IROFFV, MB_V, MYROW,
IVROW, NPROW ),

IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW =
INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC,
NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C,
MYROW, ICROW, NPROW ), NpC0 = NUMROC( N+ICOFFC, MB_C, MYROW,
ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL,
NPCOL ),

ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW,
MYCOL, NPROW and NPCOL can be determined by calling the
subroutine BLACS_GRIDINFO.

🔮 Alignment requirements 🔮

The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true:

If STOREV = 'Columnwise' If SIDE = 'Left', ( MB_V.EQ.MB_C .AND.
IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', (
MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if STOREV = 'Rowwise'
If SIDE = 'Left', ( NB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If
SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND.
IVCOL.EQ.ICCOL ) end if
bestie if your offsets are misaligned the vibe check fails and so does the routine — double-check those conditions before you ship 💀

okay that's the whole page, every single flag and formula intact, nothing skipped, you're welcome 🎀✨

This legend came straight out of LAPACK version 1.5, dated 12 May 1997 — shoutout to the ScaLAPACK girlies who wrote it, love that for us 💖 bye bestie 🦋

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