PCLARFC(l)

📦 version 1.5, in ubuntu jammy

💖 PCLARFC 💖

okay bestie, so this is a LAPACK auxiliary routine — LAPACK version 1.5 energy — and it lives in the (l) manual, lowercase l, very demure. Pull up a chair, we're doing distributed linear algebra and we're doing it cutely.

✨ NAME ✨

PCLARFC — she applies a complex elementary reflector Q**H to a complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1). That's it. That's the job. 💅

✨ SYNOPSIS ✨

SUBROUTINE PCLARFC( SIDE, M, N, V, IV, JV, DESCV, INCV, TAU, C, IC, JC,
                    DESCC, WORK )

    CHARACTER       SIDE

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

    INTEGER         DESCC( * ), DESCV( * )

    COMPLEX         C( * ), TAU( * ), V( * ), WORK( * )

🦋 PURPOSE 🦋

So PCLARFC applies a complex elementary reflector Q**H to a complex M-by-N distributed matrix

sub( C ) = C(IC:IC+M-1,JC:JC+N-1)

…either from the left or from the right. Choose your fighter. Q itself is represented in the form

Q = I - tau * v * v'

where tau is a complex scalar and v is a complex vector. Slay.

And listen — if tau = 0, then Q is taken to be the unit matrix. Nothing happens. She's off duty. 😌

📝 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.

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". iykyk.

Here's the notation table, but make it a list bestie, we're in a group chat:

  • 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

Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. She contains multitudes. 🧚

🚧 Restrictions 🚧

Read these, bestie. Fr fr. These are not optional vibes.

If SIDE = 'Left' and INCV = 1, then the row process having the first entry V(IV,JV) must also have the first row of sub( C ). Moreover, MOD(IV-1,MB_V) must be equal to MOD(IC-1,MB_C), if INCV=M_V, only the last equality must be satisfied.

If SIDE = 'Right' and INCV = M_V then the column process having the first entry V(IV,JV) must also have the first column of sub( C ) and MOD(JV-1,NB_V) must be equal to MOD(JC-1,NB_C), if INCV = 1 only the last equality must be satisfied.

👑 ARGUMENTS 👑

  • SIDE (global input) CHARACTER — the direction, the moment, the whole plot:
    • = 'L': form Q**H * sub( C ),
    • = 'R': form sub( C ) * Q**H.
  • 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.
  • V (local input) COMPLEX pointer into the local memory — to an array of dimension (LLD_V,*) containing the local pieces of the distributed vectors V representing the Householder transformation Q:
    • V(IV:IV+M-1,JV) if SIDE = 'L' and INCV = 1,
    • V(IV,JV:JV+M-1) if SIDE = 'L' and INCV = M_V,
    • V(IV:IV+N-1,JV) if SIDE = 'R' and INCV = 1,
    • V(IV,JV:JV+N-1) if SIDE = 'R' and INCV = M_V,

The vector v in the representation of Q. V is not used if TAU = 0. (She's literally not invited to that party.)

  • 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.
  • INCV (global input) INTEGER — the global increment for the elements of V. Only two values of INCV are supported in this version, namely 1 and M_V. INCV must not be zero. Zero is a crime. 🚫
  • TAU (local input) COMPLEX, array, dimension LOCc(JV) if INCV = 1, and LOCr(IV) otherwise — this array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V. They're bonded. 🫶
  • C (local input/local output) COMPLEX pointer into the local memory — to an array of dimension (LLD_C, LOCc(JC+N-1) ), containing the local pieces of sub( C ). On exit, sub( C ) is overwritten by the Q**H * sub( C ) if SIDE = 'L', or sub( C ) * Q**H if SIDE = 'R'. Not me, she's getting rewritten. 😭
  • 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 array, dimension (LWORK) — okay bestie this one's a whole nested-if situation, so here it is in its full glory:
If INCV = 1,
   if SIDE = 'L',
      if IVCOL = ICCOL, LWORK >= NqC0
      else LWORK >= MpC0 + MAX( 1, NqC0 )
      end if
   else if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC(
        N+ICOFFC,NB_V,0,0,NPCOL ),NB_V,0,0,LCMQ ) )
   end if
else if INCV = M_V,
   if SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC(
        M+IROFFC,MB_V,0,0,NPROW ),MB_V,0,0,LCMP ) )
   else if SIDE = 'R',
      if IVROW = ICROW, LWORK >= MpC0
      else LWORK >= NqC0 + MAX( 1, MpC0 )
      end if
   end if
end if

where LCM is the least common multiple of NPROW and NPCOL and

LCM = ILCM( NPROW, NPCOL ),
LCMP = LCM / NPROW,
LCMQ = LCM / NPCOL,
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 ),
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:

MB_V = NB_V,

If INCV = 1,
   If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND.
        IVROW.EQ.ICROW )
   If SIDE = 'Right', ( MB_V.EQ.NB_A .AND. MB_V.EQ.NB_C .AND.
        IROFFV.EQ.ICOFFC )
else if INCV = M_V,
   If SIDE = 'Left', ( MB_V.EQ.NB_V .AND. MB_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 those aren't satisfied it's giving chaos. Check them. 💀

🌈 Sign-off 🌈

LAPACK version 1.5 🌸 12 May 1997 🌸 and that's the whole page, PCLARFC(l), archived and iconic. Go forth and apply your reflectors, bestie. You got this. Periodt. 👑✨💖

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