PCLAUU2(l)

📦 version 1.5, in ubuntu jammy

💖 PCLAUU2 💖

okay bestie, buckle up, because we're doing LAPACK auxiliary routine realness today — this one lives in the PCLAUU2(l) girlies' section of the manual, version 1.5 energy, and she is so much chiller than she looks. no cap.

✨ NAME ✨

PCLAUU2 — 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 matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1)

🎀 SYNOPSIS 🎀

SUBROUTINE PCLAUU2( UPLO, N, A, IA, JA, DESCA )

    CHARACTER       UPLO

    INTEGER         IA, JA, N

    INTEGER         DESCA( * )

    COMPLEX         A( * )

like. look at her. five arguments and a dream 💅

🌸 PURPOSE 🌸

PCLAUU2 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).

  • 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. no communication is performed by this routine — the matrix to operate on should be strictly local to one process. fr fr, she does not talk to the other processes. she is on her own journey. 🦋

📝 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".

so like, the whole notation table but make it a vibe check:

  • 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. (yes it literally says "the order of the order", she said it twice because she meant it 😭)
  • 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.
  • 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.
⚠️ friendly reminder bestie: this routine does no communication. if your matrix isn't strictly local to one process you are going to have a bad time. keep it local, keep it cute. 🫶

that's the whole page, no notes, she ate and left no crumbs. this icon is from LAPACK version 1.5, dated 12 May 1997 — yeah she's vintage, she's archival, she's still serving. 💜✨🎀

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