PCGERQF(l)

📦 version 1.5, in ubuntu jammy

💖 PCGERQF — the RQ glow-up 💖

okay bestie, sit down, hydrate, because we're about to do a RQ factorization on a complex distributed M-by-N matrix and it's gonna be so good. This is a LAPACK routine, manual page PCGERQF(l), and she is giving ✨ parallel linear algebra royalty ✨.

🌸 NAME 🌸

PCGERQF — compute a RQ factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = R * Q. no cap, that's the whole job description.

🎀 SYNOPSIS 🎀

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

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

    INTEGER         DESCA( * )

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

literally the whole cast list right there. iconique.

🦋 PURPOSE 🦋

PCGERQF computes a RQ factorization of a complex distributed M-by-N matrix

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

she's not doing it alone though — this is a distributed situation, so let's talk vibes.

💜 Notes 💜

Every global data object has a lil bestie called a description vector. That vector holds all the info you need to map an object element to its process and memory location. it's giving "personal assistant for your array."

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, we're doing grammar now. multitasking queen.)

So here's the lore — notation, where it's stored, and what it means:

  • 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 fluid, we love that for her 💅
  • 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)).

Okay so let K be the number of rows or columns of a distributed matrix, and assume its process grid has dimension p x q. (main character grid energy fr fr.)

  • 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.
  • 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.

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 sub( A ). M >= 0. she can be zero, she can be huge, she contains multitudes.
  • 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) COMPLEX 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:
    • if M <= N, the upper triangle of A( IA:IA+M-1, JA+N-M:JA+N-1 ) contains the M by M upper triangular matrix R;
    • if M >= N, the elements on and above the (M-N)-th subdiagonal contain the M by N upper trapezoidal matrix R;
    • the remaining elements, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors (see Further Details).

🫶 glow-up arc: goes in as your matrix, comes out as R plus the receipts for Q.

  • 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. her little info card 🎀
  • TAU — (local output) COMPLEX, array, dimension LOCr(IA+M-1). This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. they're bonded, iykyk.
  • 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 >= MB_A * ( Mp0 + Nq0 + MB_A ),

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.

🚨 bestie 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 just window shopping, not buying. no drama, no error.
  • INFO — (global output) INTEGER.
    • = 0: successful exit 🎉 slay.
    • < 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)' H(ia+1)' . . . H(ia+k-1)', where k = min(m,n).

Each H(i) has the form

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

where tau is a complex scalar, and v is a complex vector with

v(n-k+i+1:n) = 0 and v(n-k+i) = 1;

conjg(v(1:n-k+i-1)) is stored on exit in A(ia+m-k+i-1,ja:ja+n-k+i-2), and tau in TAU(ia+m-k+i-1).

like. she left the reflectors behind so you can rebuild Q later. that's called leaving the recipe, and it's iconic behavior. 👑

⭐ SEE ALSO ⭐

no SEE ALSO section on this page, bestie — this one's a standalone queen. the tool functions she leans on (NUMROC, INDXG2P, BLACS_GRIDINFO) get their own shoutouts up in ARGUMENTS. 🦋

💀 one more time for the people in the back: IA, JA, DESCA, M, N, TAU, WORK, LWORK, INFO — every single one matters, and LWORK >= MB_A * ( Mp0 + Nq0 + MB_A ) is not a suggestion. respect the workspace or the routine will simply not vibe with you.

this has been PCGERQF(l), a LAPACK routine, version 1.5, dated 12 May 1997. yeah she's vintage, and vintage is just another word for timeless. mwah. 💋

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