PDGEQR2(l)

📦 version 1.5, in ubuntu jammy

✨ PDGEQR2 — the LAPACK girlie who does QR factorization 💅

okay so bestie, this is PDGEQR2 from the LAPACK routine crew (version 1.5 era, we're talking vintage ✨). It's a ScaLAPACK parallel routine, it's lowkey iconic, and if you're doing distributed linear algebra you're gonna want to be on a first-name basis with it. Sit down, hydrate, let's go. 🫶

🌸 NAME 🌸

PDGEQR2 — she computes a QR factorization of a real distributed M-by-N matrix, specifically:

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

No cap. That's the whole job. 👑

✨ SYNOPSIS ✨

SUBROUTINE PDGEQR2( 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 💖

PDGEQR2 computes a QR factorization of a real distributed M-by-N matrix, and the vibe is:

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

🦋 Notes 🦋

Okay so before we get into it — every global data object out here 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. Basically it's the group chat where the array tells you which process it's living with. 💀

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". (iykyk)

Here's the whole notation breakdown, bestie — every single one:

  • 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. She's global but her number is giving local. 🙈
  • 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
Quick reminder bestie: those ceils are doing a LOT of heavy lifting. Don't skip the upper bound check, she's there for a reason. 🌈

👑 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. Zero is fine, negative is not, periodt.
  • 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. Same energy as M. 💜
  • 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 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). She comes in one way and leaves as two matrices, that's range bestie. 🦋
  • 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. The whole lore drop from the Notes section lives in here. 👑
  • TAU — (local output) DOUBLE PRECISION, 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, no separating them. 🎀
  • 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 >= 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. So the -1 move is the "just tell me the size, don't actually do anything" move. Very demure, very mindful. 💅

  • INFO — (local output) INTEGER
    • = 0: successful exit. We ate. ✨
    • < 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. So if INFO comes back negative, that's not a vibe, that's a bug report. 💀

🔥 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).

That's it, that's the whole reflector lore. The zeros at the front are intentional, don't come for her. 🧚

okay and that's the whole page!! This was LAPACK version 1.5, dated 12 May 1997 — yes bestie, she's an elder, she predates your entire childhood, and she still runs. Respect the classics. 👑💖

go forth and factorize, my love — see also whatever your local LAPACK index throws at you, because that's how the family tree goes 🫶✨

📦 where this page comes from