PZUNGRQ(l)

📦 version 1.5, in ubuntu jammy

💖 PZUNGRQ 💖

okay so bestie, gather round, because today we're talking about PZUNGRQ — a LAPACK routine, version 1.5, and she is that girl. 🫶

✨ NAME ✨

PZUNGRQ generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N

Q = H(1)' H(2)'

like. she builds Q. she gives it orthonormal rows. she's doing the most and we love her for it. 💅

✨ SYNOPSIS ✨

SUBROUTINE PZUNGRQ( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO )

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

    INTEGER         DESCA( * )

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

no cap, that's the whole calling card. eleven arguments, all iconic, all necessary. 🌸

✨ PURPOSE ✨

PZUNGRQ generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N

as returned by PZGERQF.

like yes she's the glow-up phase of PZGERQF's output. 🦋

📝 Notes 📝

so every global data object is described by an associated description vector. that vector stores the info you need to map between an object element and its corresponding process and memory location. it's giving ✨admin✨ but make it distributed.

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

fr fr, that's the whole vibe check on distribution. 🧚

✨ ARGUMENTS ✨

  • M — (global input) INTEGER. The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0.
  • N — (global input) INTEGER. The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. N >= M >= 0.
  • K — (global input) INTEGER. The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0.
  • A — (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA+M-K <= i <= IA+M-1, as returned by PZGERQF in the K rows of its distributed matrix argument A(IA+M-K:IA+M-1,JA:*). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. 💖 she comes in as reflectors and leaves as a whole entire Q, main character arc honestly.
  • 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 LOCr(IA+M-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PZGERQF. TAU is tied to the distributed matrix A. like they're linked. inseparable. the way that they're bonded. 😭
  • 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
LWORK >= MB_A * ( MpA0 + NqA0 + MB_A ), where
IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW =
INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA,
NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A,
MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL,
IACOL, 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. 👑 sneak peek mode, we love a preview.

  • 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.
okay so heads up bestie — that negative INFO thing is the routine's way of snitching on exactly which argument went feral. read it carefully, it's literally telling you who the problem is. 💀

and that's the whole page, bestie. this was LAPACK version 1.5, dated 12 May 1997 — yes she's vintage, yes she still ate, no cap. mwah 💅💖✨🎀🦋🌸🫶🔥🌈⭐💜

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