PCGEQLF(l)

📦 version 1.5, in ubuntu jammy

✨ PCGEQLF(l) ✨

okay so bestie, you know that routine PCGEQLF(l) from the LAPACK crew? yeah her. she's a LAPACK routine, version 1.5, and she is so much cooler than her manpage makes her look. let's vibe check the whole thing together 💅

🌸 NAME 🌸

PCGEQLF — she computes a QL factorization of a complex distributed M-by-N matrix

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

periodt. that's the whole assignment 💖

🦋 SYNOPSIS 🦋

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

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

    INTEGER         DESCA( * )

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

look at her. ten arguments. not one of them optional. main character energy fr fr 👑

🎀 PURPOSE 🎀

PCGEQLF computes a QL factorization of a complex distributed M-by-N matrix

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

that's it. that's the job. she factors, she slays, she hands you back a Q and an L. 💅

Notes

so before we even get to the arguments, we gotta talk about descriptor vectors, because this is the part that makes everybody's brain short-circuit and honestly it's not even hard, it's just a lot of letters. 😭

each global data object is described by an associated description vector. this vector stores the info you need to establish the mapping between an object element and its corresponding process and memory location. (translation: it's the address book that tells you which process is holding which lil piece of your matrix 🫶)

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.

  • 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 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

see? not scary. just a lot of capital letters bestie 🌈

💜 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) 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 lower triangle of the distributed submatrix A( IA+M-N:IA+M-1, JA:JA+N-1 ) contains the N-by-N lower triangular matrix L; if M <= N, the elements on and below the (N-M)-th superdiagonal contain the M by N lower trapezoidal matrix L; the remaining elements, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors (see Further Details). ✨
not the manpage literally gluing IA's whole description onto the tail end of A's paragraph 💀 we love a formatting moment. here it is anyway so nothing gets lost:
  • 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) COMPLEX, array, dimension LOCc(JA+N-1) — this array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. 🔗
  • 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 >= NB_A * ( Mp0 + Nq0 + NB_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.

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. (yes bestie, the -1 trick. she's showing off and she's earned it 💅)

  • 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.
ok but the fact that the array-error code is -(i*100+j) and the scalar one is just -i?? it's giving "i ran out of budget for a nicer encoding" 🙈 remember both, fr fr.

🔥 FURTHER DETAILS 🔥

the matrix Q is represented as a product of elementary reflectors

Q = H(ja+k-1) . . . H(ja+1) H(ja), 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(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in A(ia:ia+m-k+i-2,ja+n-k+i-1), and tau in TAU(ja+n-k+i-1).

and that's the whole reflector recipe, bestie. she's not hiding anything from you 💖

and that's the page, no cap 🌈 this little love letter was brought to you by LAPACK version 1.5, dated 12 May 1997 — yes, she's an icon from the 90s and she's still eating 💅🫶 go factor something, you got this. 🎀🦋✨

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