PSLAQSY(l)

📦 version 1.5, in ubuntu jammy

✨ PSLAQSY ✨

okay bestie, gather round 💖 this one is PSLAQSY, a LAPACK auxiliary routine (manual page l, lowercase, we love her), and she is literally here to equilibrate a symmetric distributed matrix. no cap. let's get into it 🫶

💫 NAME 💫

PSLAQSY — equilibrate a symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the scaling factors in the vectors SR and SC 🌸

like. that's the whole vibe. scaling. balancing. glow-up for your matrix 💅

🎀 SYNOPSIS 🎀

SUBROUTINE PSLAQSY( UPLO, N, A, IA, JA, DESCA, SR, SC, SCOND, AMAX,
                    EQUED )

    CHARACTER       EQUED, UPLO

    INTEGER         IA, JA, N

    REAL            AMAX, SCOND

    INTEGER         DESCA( * )

    REAL            A( * ), SC( * ), SR( * )

🦋 PURPOSE 🦋

PSLAQSY equilibrates a symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the scaling factors in the vectors SR and SC. periodt ✨

🌈 Notes 🌈

okay so this is the part everyone skips and then everyone regrets skipping. read it. i'm serious. 🙈

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". iykyk 💫

  • 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 distributed over. The context itself is global, 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 — Specifies whether the upper or lower triangular part of the symmetric distributed matrix sub( A ) is to be referenced:
    • = '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 distributed submatrix sub( A ). N >= 0.
  • A — (input/output) REAL pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1)). On entry, the local pieces of the distributed symmetric matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and the strictly lower triangular part of sub( A ) is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and the strictly upper triangular part of sub( A ) is not referenced. On exit, if EQUED = 'Y', the equilibrated matrix:
diag(SR(IA:IA+N-1)) * sub( A ) * diag(SC(JA:JA+N-1)).
  • 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.
  • SR — (local input) REAL array, dimension LOCr(M_A) — The scale factors for A(IA:IA+M-1,JA:JA+N-1). SR is aligned with the distributed matrix A, and replicated across every process column. SR is tied to the distributed matrix A.
  • SC — (local input) REAL array, dimension LOCc(N_A) — The scale factors for sub( A ). SC is aligned with the distributed matrix A, and replicated down every process row. SC is tied to the distributed matrix A.
  • SCOND — (global input) REAL — Ratio of the smallest SR(i) (respectively SC(j)) to the largest SR(i) (respectively SC(j)), with IA <= i <= IA+N-1 and JA <= j <= JA+N-1.
  • AMAX — (global input) REAL — Absolute value of the largest distributed submatrix entry.
  • EQUED — (output) CHARACTER*1 — Specifies whether or not equilibration was done.
    • = 'N': No equilibration.
    • = 'Y': Equilibration was done, i.e., sub( A ) has been replaced by:
diag(SR(IA:IA+N-1)) * sub( A ) * diag(SC(JA:JA+N-1)).

🍓 PARAMETERS 🍓

THRESH is a threshold value used to decide if scaling should be done based on the ratio of the scaling factors. If SCOND < THRESH, scaling is done. 👀

LARGE and SMALL are threshold values used to decide if scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, scaling is done. 🔥

and that's the whole page bestie, you ate that 💖✨

this icon was brought to you by LAPACK version 1.5, dated 12 May 1997, from the manual page PSLAQSY(l) 🌸 mwah, go equilibrate something gorgeous 🎀💅

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