PCUNG2R(l)

📦 version 1.5, in ubuntu jammy

💖 PCUNG2R — bestie, we're generating Q 💖

okay so this is a LAPACK routine, PCUNG2R(l), from LAPACK version 1.5, and it's genuinely iconic. it lives in the l manual (that's the letter ell, lowercase, she's fancy). let's get into it 💅

✨ NAME ✨

PCUNG2R — she generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M

Q = H(1) H(2)

like, that's the whole job description. no cap. 🦋

✨ SYNOPSIS ✨

SUBROUTINE PCUNG2R( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO )

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

    INTEGER         DESCA( * )

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

✨ PURPOSE ✨

PCUNG2R generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M

as returned by PCGEQRF. 🎀

📝 Notes

every 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. (translation: the descriptor is the map, the array is the territory, 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".

so here's the notation breakdown, and it's giving very much ✨organized✨:

  • 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 ✨

  • 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. M >= N >= 0.
  • K — (global input) INTEGER. the number of elementary reflectors whose product defines the matrix Q. N >= K >= 0.
  • A — (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). on entry, the j-th column must contain the vector which defines the elementary reflector H(j), JA <= j <= JA+K-1, as returned by PCGEQRF in the K columns of its array argument A(IA:*,JA:JA+K-1). on exit, this array contains the local pieces of the M-by-N distributed matrix Q. she goes in as reflectors and comes out as a whole entire Q, main character energy fr fr 👑
  • 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, array, dimension LOCc(JA+K-1). this array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PCGEQRF. TAU is tied to the distributed matrix A.
  • WORK — (local workspace/local output) COMPLEX 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 >= MpA0 + MAX( 1, NqA0 )

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. so like, she's quiet about it. no drama. 🤫
  • 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.

that's the whole vibe check bestie, every flag accounted for, nothing skipped 💖

this was LAPACK routine, version 1.5, dated 12 May 1997 — shoutout to the 90s for this absolute banger, she's been slaying since before some of us were born, periodt 🫶✨🌈

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