PDLAUU2(l)

📦 version 1.5, in ubuntu jammy

💖 PDLAUU2 — the LAPACK auxiliary routine, bestie edition ✨

okay so real talk, this is PDLAUU2 from LAPACK, and it's an auxiliary routine, version 1.5, living its best life in the (l) manual. If you're here you're probably doing distributed linear algebra and you need to know exactly what this thing does, so let's get into it, no cap. 🫶

🌸 NAME 🌸

PDLAUU2 — 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 matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). iconic behaviour, honestly. 👑

✨ SYNOPSIS ✨

SUBROUTINE PDLAUU2( UPLO, N, A, IA, JA, DESCA )

    CHARACTER       UPLO

    INTEGER         IA, JA, N

    INTEGER         DESCA( * )

    DOUBLE          PRECISION A( * )

💅 PURPOSE 💅

So PDLAUU2 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 matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). that's the whole assignment, periodt. 💖

  • 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 unblocked form of the algorithm, calling Level 2 BLAS. 🏃♀️ And listen — no communication is performed by this routine, the matrix to operate on should be strictly local to one process. she's not networked, she's not phoning anyone, she's local-only and that's her whole vibe. 🦋

📝 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 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 order of the triangular factor U or L. N >= 0.
  • 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.
  • 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 bestie: A gets overwritten on exit. whatever triangular factor you had in there? gone, replaced by the product. so don't come crying later. 😭💀

and that's the whole page! this was LAPACK version 1.5, dated 12 May 1997 — yeah she's vintage, she's a classic, she's been serving since the 90s and she still ate. 🌈⭐💜 mwah, go call your NUMROC and stay local. 🧚🍓

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