PDORGLQ(l)

📦 version 1.5, in ubuntu jammy

✨ PDORGLQ — bestie, it's giving orthonormal rows ✨

okay so real quick, this is PDORGLQ, a LAPACK routine (version 1.5, we're in the ScaLAPACK corner of the function library, iykyk). Strap in, because this one has layers. 🦋

💖 NAME 💖

PDORGLQ — bestie, it generates an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N, Q = H(k). Slay. 👑

🎀 SYNOPSIS 🎀

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

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

    INTEGER         DESCA( * )

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

the way that signature just sits there. gorgeous. no notes. 💅

🌸 PURPOSE 🌸

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

as returned by PDGELQF.

📓 Notes 📓

(original heading, but make it cute)

Every global data object out here has a description vector bestie — it's the little receipt that stores everything you need to map an object element to its matching process and memory location. 🧾

Let's call A a generic term for any 2D block cyclicly distributed array. A global array like that has an associated description vector DESCA. And in all the comments below, read the character _ as "of the global array". periodt.

  • 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 its process grid has dimension p x q.

LOCr( K ) = 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 ) = the number of elements of K that a process would receive if K were distributed over the q processes of its process row.

You can get the values of LOCr() and LOCc() by calling 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. 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) DOUBLE PRECISION 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 <= i <= IA+K-1, as returned by PDGELQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. ✨ glow up ✨
  • 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 LOCr(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. 🫶
  • 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 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.

heads up bestie: 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. no drama, no PXERBLA yelling at you. 💀
  • 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.

that's the whole page, no cap. 💜

sign-off bestie: this lil masterpiece is from LAPACK version 1.5, dated 12 May 1997 — yes she's vintage, yes she still serves. go forth and generate those orthonormal rows. mwah. 💅✨🎀

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