PDORML2(l)

📦 version 1.5, in ubuntu jammy

💖 PDORML2 💖

okay bestie, welcome to PDORML2 — a real LAPACK routine, version 1.5 energy, and she is that girl. sit down, grab a strawberry, we're doing distributed linear algebra 🍓✨

✨ NAME ✨

PDORML2 — she overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L', SIDE = 'R', TRANS = 'N'. that's the whole vibe. no cap. 💅

✨ SYNOPSIS ✨

SUBROUTINE PDORML2( 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( * )

✨ PURPOSE ✨

so PDORML2 overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) 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'. iconic behaviour honestly 🎀

🌈 Notes 🌈

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. iykyk.

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
    • = '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 borrowed it and gave it back, so polite 🫶)
  • 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. the glow-up is real 💅
  • 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 >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_A,0,0,NPROW ),MB_A,0,0,LCMP ) ); if SIDE = 'R', LWORK >= NqC0 + MAX( 1, MpC0 )

where

LCMP = LCM / NPROW with LCM = ICLM( NPROW, 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. the ✨ just-asking-for-a-friend ✨ mode, we love her.

  • INFO — (local 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 )
not me warning you: if those alignment expressions are not true, bestie, that is a you problem, not a PDORML2 problem. check your descriptors before you ship it. 💀

and that's the whole page, no notes left behind, every option accounted for 👑

this has been PDORML2 from LAPACK version 1.5, dated 12 May 1997 — she's vintage, she's distributed, she's doing great. mwah 💖🎀🦋✨

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