PCLAUUM(l)

📦 version 1.5, in ubuntu jammy

💅 PCLAUUM 💅

okay bestie, sit down, hydrate, because we're doing a LAPACK auxiliary routine and she is gorgeous. This is PCLAUUM from LAPACK version 1.5, and yes she's an (l) — the lowercase l, the auxiliary girlies, the ones who do the actual labor while everyone else takes credit. 🫶

✨ NAME ✨

PCLAUUM — she 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 PCLAUUM( UPLO, N, A, IA, JA, DESCA )

    CHARACTER       UPLO

    INTEGER         IA, JA, N

    INTEGER         DESCA( * )

    COMPLEX         A( * )

🎀 PURPOSE 🎀

So PCLAUUM 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. (Blocked = she's doing it in chunky, cache-friendly bites instead of one sad little element at a time. main character efficiency, no cap. 🔥)

🧚 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".

  • 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

👑 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. (Zero is allowed, she's not judging. 🙈)
  • 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 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 really said in place, bestie, no copies.
  • 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.
⚠️ heads up: A is input and output, and the result lands right on top of your triangular factor. The original U or L is gone-gone after this. Plan accordingly, iykyk. 💅

that's the whole page, bestie, and she ate. no crumbs. this was PCLAUUM(l) from LAPACK version 1.5, dated 12 May 1997 — the footer said so, so I'm saying so. go forth and multiply your triangles. 🦋✨💜

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