PZLARZB(l)

📦 version 1.5, in ubuntu jammy

💖 PZLARZB 💖

okay bestie, come closer, we're doing PZLARZB today. it's a LAPACK auxiliary routine, it lives in the l manual, and honestly? it's giving distributed-linear-algebra main character energy. let's get into it. 🦋

✨ NAME ✨

PZLARZB — applies a complex block reflector Q or its conjugate transpose Q**H to a complex M-by-N distributed matrix sub( C ) denoting C(IC:IC+M-1,JC:JC+N-1), from the left or the right. periodt. 💅

🎀 SYNOPSIS 🎀

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

    CHARACTER       DIRECT, SIDE, STOREV, TRANS

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

    INTEGER         DESCC( * ), DESCV( * )

    COMPLEX*16      C( * ), T( * ), V( * ), WORK( * )

🌸 PURPOSE 🌸

so like, PZLARZB applies a complex block reflector Q — or its conjugate transpose Q**H — to a complex M-by-N distributed matrix sub( C ) denoting C(IC:IC+M-1,JC:JC+N-1), from the left or from the right. that's the whole job. she's focused. 💅

and Q itself? it's a product of k elementary reflectors, as returned by PZTZRZF. so if you got your reflectors from there, this is your girl. 🫶

⚠️ heads up bestie, this is important: currently, only STOREV = 'R' and DIRECT = 'B' are supported. the rest? not yet. don't come for me, that's just what the page says. 💀

📝 Notes

every global data object gets described by an associated description vector, and that vector stores the info you need to map an object element to its corresponding process and memory location. very "everyone gets a name tag at the party" 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. and in the comments below, read that underscore _ as "of the global array".

here's the whole notation situation, laid out like a little cheat sheet:

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)).

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

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

and 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 girlies rise 💅

📐 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

no cap, if these aren't true, things get weird. respect the alignment. 🙈

💅 ARGUMENTS 💅

okay this is the long part, grab a snack 🍓

SIDE — (global input) CHARACTER
- = 'L': apply Q or Q**H from the Left;
- = 'R': apply Q or Q**H from the Right.
TRANS — (global input) CHARACTER
- = 'N': No transpose, apply Q;
- = 'C': Conjugate transpose, apply Q**H.
DIRECT — (global input) CHARACTER Indicates how H is formed from a product of elementary reflectors:
- = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
- = 'B': H = H(k) . . . H(2) H(1) (Backward)
STOREV — (global input) CHARACTER Indicates how the vectors which define the elementary reflectors are stored:
- = 'C': Columnwise (not supported yet)
- = '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).
L — (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0; if SIDE = 'R', N >= L >= 0.
V — (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_V, LOCc(JV+M-1)) if SIDE = 'L', (LLD_V, LOCc(JV+N-1)) if SIDE = 'R'. It contains the local pieces of the distributed vectors V representing the Householder transformation as returned by PZTZRZF. 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) COMPLEX*16 array, dimension MB_V by MB_V The lower triangular matrix T in the representation of the block reflector.
C — (local input/local output) COMPLEX*16 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) COMPLEX*16 array, dimension (LWORK)

and here's the LWORK situation, in all its glory — read it slowly bestie, it's a whole branching tree 😭:

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.

okay that's literally everything, you ate that. 🫶 this whole thing is from LAPACK version 1.5, dated 12 May 1997 — yes she's an icon, she's been iconic, and she's still out here aligning submatrices. 💖✨ go forth and call your reflectors responsibly. 💅

📦 where this page comes from

ubuntu jammy: scalapack-doc 1.5-11
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-10-14: a language model's rewrite of the original, under its terms