PSLAUUM(l)

📦 version 1.5, in ubuntu jammy

💅 PSLAUUM 💅

okay bestie, gather round, because we're about to talk about a LAPACK auxiliary routine and honestly? she's iconic. this is PSLAUUM, and she lives in the LAPACK family (version 1.5, if you're keeping score) — the parallel, distributed-memory, ScaLAPACK-flavored corner of the family. 🌈

✨ NAME ✨

PSLAUUM — compute the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1)

like. that's the whole assignment. triangular factor goes in, gram-matrix-coded product comes out. periodt. 👑

✨ SYNOPSIS ✨

SUBROUTINE PSLAUUM( UPLO, N, A, IA, JA, DESCA )

    CHARACTER       UPLO

    INTEGER         IA, JA, N

    INTEGER         DESCA( * )

    REAL            A( * )

🎀 PURPOSE 🎀

PSLAUUM computes the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1).

so here's the tea, split by which flag you pick:

  • if UPLO = 'U' or 'u' then the upper triangle of the result is stored, overwriting the factor U in sub( A ).
  • if UPLO = 'L' or 'l' then the lower triangle of the result is stored, overwriting the factor L in sub( A ).

and listen — this is the blocked form of the algorithm, calling Level 3 PBLAS. she's not doing this one element at a time like some kind of amateur. no cap, blocked means efficiency and efficiency is a love language. 💖

📝 Notes

Every global data object 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.

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".

ok so here's the notation breakdown, because the original had a whole table and we're doing it as a list instead — same facts, cuter energy, you're welcome:

  • 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

LOCr and LOCc, the way that they just tell you how much of the array your little process is holding... it's giving "who's carrying the group project" and the answer is everyone, fairly, block-cyclically. 🫶

🌸 ARGUMENTS 🌸

  • UPLO — (global input) CHARACTER*1 — specifies whether the triangular factor stored in the distributed matrix sub( A ) is upper or lower triangular:
    • = 'U': upper triangular
    • = 'L': lower triangular
  • N — (global input) INTEGER — the number of rows and columns to be operated on, i.e. the order of the triangular factor U or L. 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 triangular factor L or U. On exit, if UPLO = 'U', the upper triangle of the distributed matrix sub( A ) is overwritten with the upper triangle of the product U * U'; if UPLO = 'L', the lower triangle of sub( A ) is overwritten with the lower triangle of the product L' * L.
bestie, she overwrites. in place. no copy, no backup, no take-backs. if you needed that triangular factor for something else, that's a you problem and it's already gone. 💀
  • 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.
and that's the whole page, no filler, no skipped options, every single one accounted for. we ate and left absolutely zero crumbs. 🍓

okay and for the cute sign-off: this whole vibe check comes to you courtesy of LAPACK version 1.5, dated 12 May 1997, and she's the PSLAUUM(l) page in the LAPACK auxiliary routine lineup. yes she's older than most of the internet and yes she still slays. 👑💖

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