PSGEQR2(l)

📦 version 1.5, in ubuntu jammy

✨ PSGEQR2(l) — the QR factorization glow-up, distributed edition ✨

okay bestie, gather round 💖 this is PSGEQR2, a LAPACK routine (the .l suffix is the ScaLAPACK/parallel-flavored one, iykyk), and she is that girl when it comes to QR factorizations of real distributed matrices. no cap.

🌸 NAME 🌸

PSGEQR2 — compute a QR factorization of a real distributed M-by-N matrix

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

periodt. 💅

🎀 SYNOPSIS 🎀

SUBROUTINE PSGEQR2( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO )

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

    INTEGER         DESCA( * )

    REAL            A( * ), TAU( * ), WORK( * )

🦋 PURPOSE 🦋

PSGEQR2 computes a QR factorization of a real distributed M-by-N matrix

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

like, that's the whole job. she does it, she does it well, she leaves. 👑

💜 Notes 💜

ok so before we get into it, you have to understand descriptors, because every global data object here rolls with an associated description vector. this vector is basically the group chat where all the info about how an object element maps to its process + memory location lives. no descriptor, no mapping, no slay.

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". (yes literally. she's saying "the M of the global array A". iconic shorthand. 💅)

here's the whole roster, and bestie every single one of these matters:

  • DTYPE_A (global) — stored in DESCA( DTYPE_ ). this is 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)).

now 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

write that down. fr fr. 📝

💖 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.
  • 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) REAL 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 above the diagonal of sub( A ) contain the min(M,N) by N upper trapezoidal matrix R (R is upper triangular if M >= N); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors (see Further Details). 💅
  • 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) REAL, array, dimension LOCc(JA+MIN(M,N)-1). this array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. (they're a package deal, don't separate them 🫶)
  • WORK — (local workspace/local output) REAL 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 >= Mp0 + MAX( 1, Nq0 ), 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.

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.

🫶 lowkey the most useful trick here: LWORK = -1 is the "just tell me how much room I need, bestie, I'm not committing yet" query. use it.
  • 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(ja) H(ja+1) . . . H(ja+k-1), where k = min(m,n).

each H(i) has the form

H(j) = 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:m) is stored on exit in A(ia+i:ia+m-1,ja+i-1), and tau in TAU(ja+i-1).

like. she builds Q out of little reflector pieces and tucks the extra bits into A and TAU. iconic behavior, honestly. 🦋✨

and that's the whole page, bestie — LAPACK version 1.5, dated 12 May 1997. yes she's vintage, yes she still ate, no you may not skip the descriptor section. go forth and factor 💖👑💅

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