PZUNMTR(l)

📦 version 1.5, in ubuntu jammy

💖 PZUNMTR 💖

okay bestie, welcome to PZUNMTR, a LAPACK routine (that's LAPACK version 1.5, vintage but iconic) and it's here to do one very specific, very sparkly thing to your distributed complex matrices. pull up a chair. 🫶

✨ NAME ✨

PZUNMTR — overwrite the general complex 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 job, she's a specialist 💅

🌸 SYNOPSIS 🌸

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

    CHARACTER       SIDE, TRANS, UPLO

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

    INTEGER         DESCA( * ), DESCC( * )

    COMPLEX*16      A( * ), C( * ), TAU( * ), WORK( * )

🎀 PURPOSE 🎀

PZUNMTR overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H

where Q is a complex unitary distributed matrix of order nq, with nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of nq-1 elementary reflectors, as returned by PZHETRD:

  • if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1);
  • if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1).

📝 Notes 📝

every global data object is described by an associated description vector. this vector stores the info you need to establish the mapping between an object element and its corresponding process and memory location. yes bestie, it's the ✨map✨.

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 whole notation situation, laid out all cute:

  • 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**H from the Left;
    • = 'R': apply Q or Q**H from the Right.
  • UPLO — (global input) CHARACTER
    • = 'U': Upper triangle of A(IA:*,JA:*) contains elementary reflectors from PZHETRD;
    • = 'L': Lower triangle of A(IA:*,JA:*) contains elementary reflectors from PZHETRD.
  • TRANS — (global input) CHARACTER
    • = 'N': No transpose, apply Q;
    • = 'C': Conjugate transpose, apply 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.
  • A — (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', or (LLD_A,LOCc(JA+N-1)) if SIDE = 'R'. the vectors which define the elementary reflectors, as returned by PZHETRD. if SIDE = 'L', LLD_A >= max(1,LOCr(IA+M-1)); if SIDE = 'R', LLD_A >= max(1,LOCr(IA+N-1)).
  • 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) COMPLEX*16 array, dimension LTAU, where if SIDE = 'L' and UPLO = 'U', LTAU = LOCc(M_A), if SIDE = 'L' and UPLO = 'L', LTAU = LOCc(JA+M-2), if SIDE = 'R' and UPLO = 'U', LTAU = LOCc(N_A), if SIDE = 'R' and UPLO = 'L', LTAU = LOCc(JA+N-2). TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by PZHETRD. TAU is tied to the distributed matrix A. 🦋
  • 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 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) COMPLEX*16 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 UPLO = 'U', IAA = IA, JAA = JA+1, ICC = IC, JCC = JC; else
UPLO = 'L', IAA = IA+1, JAA = JA; if SIDE = 'L', ICC = IC+1; JCC
= JC; else ICC = IC; JCC = JC+1; end if end if

If SIDE = 'L', MI = M-1; NI = N; LWORK >= MAX(
(NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if
SIDE = 'R', MI = M; MI = N-1; 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

where 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 ), 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 ),

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. so if you were waiting for a lil scream from PXERBLA, bestie, it's not coming. 😭
  • 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', ( 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 )

and that's the whole page, babe. this was PZUNMTR(l) from the LAPACK routine pile — LAPACK version 1.5, dated 12 May 1997 — so go forth, align your descriptors, and slay 💅👑🌈💖

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