PDGELQ2(l)

📦 version 1.5, in ubuntu jammy

✨ NAME ✨

PDGELQ2 — bestie, this one computes a LQ factorization of a real distributed M-by-N matrix, and I mean literally:

sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = L * Q

🎀 SYNOPSIS 🎀

SUBROUTINE PDGELQ2( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO )

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

    INTEGER         DESCA( * )

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

💖 PURPOSE 💖

okay so PDGELQ2 computes a LQ factorization of a real distributed M-by-N matrix, and here's the whole vibe:

sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = L * Q

📓 Notes 📓

Every single global data object out here is described by its own lil description vector, and that vector is holding all the info you need to map an element of the object to its process and its memory location. she's organized like that. 💅

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

Here's the whole rundown, no cap:

  • 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. And she has standards: LLD_A >= MAX(1,LOCr(M_A)).

So 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.

Same energy for 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 those LOCr() and LOCc() values by calling the ScaLAPACK tool function NUMROC, like:

LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).

And an upper bound for those quantities can be computed like this:

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 sub( A ). M >= 0. queen said no negatives. 🙈
  • N — (global input) INTEGER. The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 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 local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the elements on and below the diagonal of sub( A ) contain the M by min(M,N) lower trapezoidal matrix L (and L is lower triangular if M <= N, periodt); the elements above the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors (see Further Details, bestie, it's right there 👀).
  • 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 output) DOUBLE PRECISION, array, dimension LOCr(IA+MIN(M,N)-1). This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A — they're bonded fr fr. 🦋
  • WORK — (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK). On exit, WORK(1) returns the minimal and optimal LWORK. she gives you both, because she's generous like that. ✨
  • LWORK — (local or global input) INTEGER. The dimension of the array WORK. LWORK is local input and must be at least LWORK >= Nq0 + MAX( 1, Mp0 ), where:
IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW =
INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA,
NB_A, MYCOL, CSRC_A, NPCOL ), Mp0   = NUMROC( M+IROFF, MB_A,
MYROW, IAROW, NPROW ), Nq0   = NUMROC( N+ICOFF, NB_A, MYCOL,
IACOL, NPCOL ),

and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO.

And listen: 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. she's quiet in query mode. 💀
  • 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.

🔍 FURTHER DETAILS 🔍

The matrix Q is represented as a product of elementary reflectors:

Q = H(ia+k-1) H(ia+k-2) . . . H(ia), where k = min(m,n).

Each H(i) has the form:

H(i) = I - tau * v * v'

where tau is a real scalar, and v is a real vector with v(1:i-1)=0 and v(i) = 1; v(i+1:n) is stored on exit in A(ia+i-1,ja+i:ja+n-1), and tau in TAU(ia+i-1).

okay and one last lil sign-off bestie 💜 — this whole thing is LAPACK version 1.5, dated 12 May 1997, and yeah she's a classic for a reason. slay, ate, no crumbs. 👑🌈

📦 where this page comes from