PDORMBR(l)

📦 version 1.5, in ubuntu jammy

💅✨ PDORMBR(l) ✨💅

okay bestie, gather round — this is the manual page for PDORMBR(l), a LAPACK routine, version 1.5. No cap, it's giving distributed linear algebra realness. 💖 Lowkey iconic, highkey useful.

🌸 NAME 🌸

PDORMBR — when VECT = 'Q', PDORMBR overwrites the general real distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L', SIDE = 'R', TRANS = 'N'. Periodt. 💅

🦋 SYNOPSIS 🦋

the way this signature ate, no crumbs:

       SUBROUTINE PDORMBR( VECT, SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU,
                           C, IC, JC, DESCC, WORK, LWORK, INFO )

           CHARACTER       SIDE, TRANS, VECT

           INTEGER         IA, IC, INFO, JA, JC, K, LWORK, M, N

           INTEGER         DESCA( * ), DESCC( * )

           DOUBLE          PRECISION A( * ), C( * ), TAU( * ), WORK( * )

👑 PURPOSE 👑

okay so, if VECT = 'Q', PDORMBR overwrites the general real distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T':

  • Q**T * sub( C )
  • sub( C ) * Q**T

If VECT = 'P', PDORMBR overwrites sub( C ) with:

                         SIDE = 'L'           SIDE = 'R'
TRANS = 'N':      P * sub( C )          sub( C ) * P
TRANS = 'T':      P**T * sub( C )       sub( C ) * P**T

Here Q and P**T are the orthogonal distributed matrices determined by PDGEBRD when reducing a real distributed matrix A(IA:*,JA:*) to bidiagonal form: A(IA:*,JA:*) = Q * B * P**T. Q and P**T are defined as products of elementary reflectors H(i) and G(i) respectively. iykyk ✨

Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the order of the orthogonal matrix Q or P**T that is applied.

If VECT = 'Q', A(IA:*,JA:*) is assumed to have been an NQ-by-K matrix:

  • if nq >= k, Q = H(1) H(2) . . . H(k);
  • if nq < k, Q = H(1) H(2) . . . H(nq-1).

If VECT = 'P', A(IA:*,JA:*) is assumed to have been a K-by-NQ matrix:

  • if k < nq, P = G(1) G(2) . . . G(k);
  • if k >= nq, P = G(1) G(2) . . . G(nq-1).

📝 Notes 📝

ngl, this part is important if you wanna understand the descriptor vibes. Each global data object is described by an associated description vector. This 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".

Here's the breakdown of the descriptor vector fields — NOTATION, STORED IN, EXPLANATION:

  • 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 ✨

bestie, grab a snack, because we're about to go through every single argument one by one. no skipping, no skimming. we're doing this together. 💖

  • VECT (global input) CHARACTER — which matrix are we applying, bestie? it's giving choose-your-own-adventure:
    • = 'Q': apply Q or Q**T;
    • = 'P': apply P or P**T.
  • SIDE (global input) CHARACTER — from which side are we applying it? left or right, no in-between:
    • = 'L': apply Q, QT, P or PT from the Left;
    • = 'R': apply Q, QT, P or PT from the Right.
  • TRANS (global input) CHARACTER — transpose or no transpose? the eternal question:
    • = 'N': No transpose, apply Q or P;
    • = 'T': Transpose, apply QT or PT.
  • 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. periodt.
  • 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. slay.
  • K (global input) INTEGER — if VECT = 'Q', the number of columns in the original distributed matrix reduced by PDGEBRD. if VECT = 'P', the number of rows in the original distributed matrix reduced by PDGEBRD. K >= 0. the way that this is literally just the reduction size, iykyk.
  • A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+MIN(NQ,K)-1)) if VECT='Q', and (LLD_A,LOCc(JA+NQ-1)) if VECT = 'P'. NQ = M if SIDE = 'L', and NQ = N otherwise. the vectors which define the elementary reflectors H(i) and G(i), whose products determine the matrices Q and P, as returned by PDGEBRD. if VECT = 'Q', LLD_A >= max(1,LOCr(IA+NQ-1)); if VECT = 'P', LLD_A >= max(1,LOCr(IA+MIN(NQ,K)-1)). no cap, this is the tea. 🫖
  • IA (global input) INTEGER — the row index in the global array A indicating the first row of sub( A ). iconic.
  • JA (global input) INTEGER — the column index in the global array A indicating the first column of sub( A ). also iconic.
  • DESCA (global and local input) INTEGER array of dimension DLEN_. the array descriptor for the distributed matrix A. don't forget it, bestie.
  • TAU (local input) DOUBLE PRECISION array, dimension LOCc(JA+MIN(NQ,K)-1) if VECT = 'Q', LOCr(IA+MIN(NQ,K)-1) if VECT = 'P', TAU(i) must contain the scalar factor of the elementary reflector H(i) or G(i), which determines Q or P, as returned by PDGEBRD in its array argument TAUQ or TAUP. TAU is tied to the distributed matrix A. it's giving inseparable duo energy. 💅
  • C (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). on entry, the local pieces of the distributed matrix sub(C). on exit, if VECT='Q', sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q; if VECT='P, sub( C )is overwritten byP*sub( C )orP'*sub( C )orsub( C )*Porsub( C )*P'`. yes, the matrix is literally getting a glow-up in place. 🔥
  • IC (global input) INTEGER — the row index in the global array C indicating the first row of sub( C ). you know the drill.
  • JC (global input) INTEGER — the column index in the global array C indicating the first column of sub( C ). same vibe.
  • DESCC (global and local input) INTEGER array of dimension DLEN_. the array descriptor for the distributed matrix C. don't sleep on this one either.
  • WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) — on exit, WORK(1) returns the minimal and optimal LWORK. it's giving "I got you, here's the perfect size". ✨
  • LWORK (local or global input) INTEGER — the dimension of the array WORK. LWORK is local input and must be at least:
If SIDE = 'L', NQ = M;
if( (VECT = 'Q' and NQ >= K) or (VECT <> 'Q' and NQ > K) ),
   IAA=IA; JAA=JA; MI=M; NI=N; ICC=IC; JCC=JC;
else
   IAA=IA+1; JAA=JA; MI=M-1; NI=N; ICC=IC+1; JCC=JC;
end if
else if SIDE = 'R', NQ = N;
if( (VECT = 'Q' and NQ >= K) or (VECT <> 'Q' and NQ > K) ),
   IAA=IA; JAA=JA; MI=M; NI=N; ICC=IC; JCC=JC;
else
   IAA=IA; JAA=JA+1; MI=M; NI=N-1; ICC=IC; JCC=JC+1;
end if
end if

then the actual size conditions:

If VECT = 'Q',
   If SIDE = 'L',
      LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A
   else if SIDE = 'R',
      LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC(
      NI+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 )
      )*NB_A ) + NB_A * NB_A
   end if
else if VECT <> 'Q',
   if SIDE = 'L',
      LWORK >= MAX( (MB_A*(MB_A-1))/2, ( MpC0 + MAX( MqA0 +
      NUMROC( NUMROC( MI+IROFFC, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP
      ), NqC0 ) )*MB_A ) + MB_A * MB_A
   else if SIDE = 'R',
      LWORK >= MAX( (MB_A*(MB_A-1))/2, (MpC0 + NqC0)*MB_A ) + MB_A * MB_A
   end if
end if

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

IROFFA = MOD( IAA-1, MB_A ), ICOFFA = MOD( JAA-1, NB_A ), IAROW
= INDXG2P( IAA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P(
JAA, NB_A, MYCOL, CSRC_A, NPCOL ), MqA0 = NUMROC( MI+ICOFFA,
NB_A, MYCOL, IACOL, NPCOL ), NpA0 = NUMROC( NI+IROFFA, MB_A,
MYROW, IAROW, NPROW ),

IROFFC = MOD( ICC-1, MB_C ), ICOFFC = MOD( JCC-1, NB_C ), ICROW
= INDXG2P( ICC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P(
JCC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( MI+IROFFC,
MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( NI+ICOFFC, NB_C,
MYCOL, ICCOL, NPCOL ),

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

if LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. it's giving "just asking for the size, no drama". 💅

  • INFO (global output) INTEGER — the receipts:
    • = 0: successful exit. we love to see it. ✨
    • < 0: if the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. basically, it tells you exactly which argument is misbehaving. not me being called out. 🙈

Alignment requirements ======================

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

If VECT = 'Q',
   If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW )
   If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC )
else
   If SIDE = 'L', ( MB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC )
   If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL )
end if

yeah bestie, you gotta match those block sizes and offsets. it's the price of admission. no cap. 💀

that's literally the whole vibe of PDORMBR(l). this page is from LAPACK version 1.5, dated 12 May 1997. stay slay, stay aligned, and may your reflectors always apply correctly. 💖✨🎀🦋🌸💅🫶😭🔥👑🌈⭐💀🙈🍓🧚💜

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