PDORMLQ(l)

🎀 NAME 🎀

PDORMLQ — the routine that overwrites the general real M-by-N distributed matrix <code>sub( C ) = C(IC:IC+M-1,JC:JC+N-1)</code> with

SIDE = 'L'      SIDE = 'R'      TRANS = 'N'

no cap, that's the whole vibe. it's the PDORMLQ manual page, LAPACK routine, and she's a legend. 💅

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🦋 SYNOPSIS 🦋

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

    CHARACTER       SIDE, TRANS

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

    INTEGER         DESCA( * ), DESCC( * )

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

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✨ PURPOSE ✨

okay so PDORMLQ overwrites the general real M-by-N distributed matrix <code>sub( C ) = C(IC:IC+M-1,JC:JC+N-1)</code> with

TRANS = 'T':      Q**T * sub( C )      sub( C ) * Q**T

where Q is a real orthogonal distributed matrix defined as the product of K elementary reflectors

Q = H(k) . . . H(2) H(1)

as returned by PDGELQF. 🫶 Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'.

📓 Notes

bestie, this part is giving ✨descriptor energy✨ so listen up. 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".

NOTATION        STORED IN      EXPLANATION
--------------- -------------- --------------------------------------
DTYPE_A(global) DESCA( DTYPE_ )The descriptor type.  In this case,
                               DTYPE_A = 1.
CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
                               the BLACS process grid A is distribu-
                               ted over. The context itself is glo-
                               bal, but the handle (the integer
                               value) may vary.
M_A    (global) DESCA( M_ )    The number of rows in the global
                               array A.
N_A    (global) DESCA( N_ )    The number of columns in the global
                               array A.
MB_A   (global) DESCA( MB_ )   The blocking factor used to distribute
                               the rows of the array.
NB_A   (global) DESCA( NB_ )   The blocking factor used to distribute
                               the columns of the array.
RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
                               row of the array A is distributed.
CSRC_A (global) DESCA( CSRC_ ) The process column over which the
                               first column of the array A is
                               distributed.
LLD_A  (local)  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

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💅 ARGUMENTS 💅

here's where the real tea is. every single one, no skips, we don't gatekeep. 🍓

• SIDE (global input) CHARACTER
  • = '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.
• 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 number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0.
• A (local input) DOUBLE PRECISION pointer into the local memory
  • to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= max(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PDGELQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*).
  • A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. (she borrows it, she gives it back, she's polite like that 💜)
• 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.
• TAU (local input) DOUBLE PRECISION, array, dimension LOCc(IA+K-1).
  • This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PDGELQF. TAU is tied to the distributed matrix A.
• 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, 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/local output) DOUBLE PRECISION array, dimension (LWORK)
  • On exit, WORK(1) returns the minimal and optimal LWORK.
• LWORK (local or global input) INTEGER
  • The dimension of the array WORK. LWORK is local input and must be at least if SIDE = 'L',

LWORK >= MAX( (MB_A*(MB_A-1))/2, ( MpC0 + MAX( MqA0 + NUMROC( NUMROC( M+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

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

IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MqA0 = NUMROC( M+ICOFFA, NB_A, MYCOL, IACOL, 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.

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. ⭐

• INFO (global output) INTEGER
  • = 0: successful exit 🎉
  • < 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.

🧚 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 SIDE = 'L',

( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC )

If SIDE = 'R',

( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL )

bestie this is a MUST. not a maybe. the alignment check is not playing around. 🙈

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and that's the whole page, you ate it, you understood it, main character energy fr fr. 💖✨

this masterpiece is LAPACK version 1.5, dated 12 May 1997, and yeah the PDORMLQ(l) girlies have been thriving ever since. mwah. 💅🦋🌈