PDLAUUM(l)

📦 version 1.5, in ubuntu jammy

💖 PDLAUUM 💖

okay so bestie, welcome to PDLAUUM(l) — it's a LAPACK auxiliary routine, version 1.5, and it's genuinely so good at what it does. like, she's a whole distributed-matrix girlie and she ate. let's get into it 🎀

✨ NAME ✨

PDLAUUM — 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). 💅

🦋 SYNOPSIS 🦋

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

    CHARACTER       UPLO

    INTEGER         IA, JA, N

    INTEGER         DESCA( * )

    DOUBLE          PRECISION A( * )

🌸 PURPOSE 🌸

PDLAUUM 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). 💖

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 ).

This is the blocked form of the algorithm, calling Level 3 PBLAS. no cap, she's calling the big-girl routines 👑

💜 Notes 💜

Each 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".

NOTATION        STORED IN      EXPLANATION
--------------- -------------- --------------------------------------
DTYPE_A(global) DESCA( DTYPE_ )The descriptor type.  In this case,
                               DTYPE_A = 1.
CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
                               the BLACS process grid A is distribu-
                               ted over. The context itself is glo-
                               bal, but the handle (the integer
                               value) may vary.
M_A    (global) DESCA( M_ )    The number of rows in the global
                               array A.
N_A    (global) DESCA( N_ )    The number of columns in the global
                               array A.
MB_A   (global) DESCA( MB_ )   The blocking factor used to distribute
                               the rows of the array.
NB_A   (global) DESCA( NB_ )   The blocking factor used to distribute
                               the columns of the array.
RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
                               row of the array A is distributed.
CSRC_A (global) DESCA( CSRC_ ) The process column over which the
                               first column of the array A is
                               distributed.
LLD_A  (local)  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

🎀 ARGUMENTS 🎀

  • UPLO (global input) CHARACTER*1 — tells you whether the triangular factor chilling 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. bestie, keep it non-negative, it's the vibe check of this whole routine 🫶
  • 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 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. she's IN-PLACE, she does not ask permission, that's the whole point 💀
  • 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. literally the blueprint, the map, the whole entire lore 🌈
heads up: A is local input and local output, and the result gets written right back over the triangular factor. no copy, no backup, no take-backs. 🔥

and that's the whole page, bestie — slay 💖✨

signing off from LAPACK version 1.5, dated 12 May 1997 — iconic behavior honestly, she's been serving since the nineties 👑🦋

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