PSLAQGE(l)

📦 version 1.5, in ubuntu jammy

💖 PSLAQGE — the LAPACK auxiliary routine, bestie edition 💖

okay so. you opened the manpage for PSLAQGE from LAPACK auxiliary routine (version 1.5) and you're probably like "what is this" — don't worry, I got you. this is a real one, a distributed-matrix glow-up routine, and we're gonna walk through it together like it's a group chat. grab a strawberry 🍓 and let's go.

✨ NAME ✨

PSLAQGE — equilibrate a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) using the row and scaling factors in the vectors R and C

✨ SYNOPSIS ✨

SUBROUTINE PSLAQGE( M, N, A, IA, JA, DESCA, R, C, ROWCND, COLCND, AMAX,
                    EQUED )

    CHARACTER       EQUED

    INTEGER         IA, JA, M, N

    REAL            AMAX, COLCND, ROWCND

    INTEGER         DESCA( * )

    REAL            A( * ), C( * ), R( * )

✨ PURPOSE ✨

PSLAQGE equilibrates a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) using the row and scaling factors in the vectors R and C.

that's the whole vibe. it takes your matrix, it takes your row factors and your column factors, and it does the little scale-scale. 💅

Notes

okay so this is the part where we talk about description vectors and honestly it's giving "assigned reading" but it's SO important, so lock in 🫶

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 ✨

  • M (global input) INTEGER — the number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. no cap, if it's negative we're not doing this. 😤
  • N (global input) INTEGER — the number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0.
  • A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)) — containing on entry the M-by-N matrix sub( A ). On exit, the equilibrated distributed matrix. See EQUED for the form of the equilibrated distributed submatrix. she goes in normal and comes out ✨equilibrated✨
  • 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. the little ID card for your matrix, fr fr.
  • R (local input) REAL array, dimension LOCr(M_A) — the row scale factors for sub( A ). R is aligned with the distributed matrix A, and replicated across every process column. R is tied to the distributed matrix A.
  • C (local input) REAL array, dimension LOCc(N_A) — the column scale factors of sub( A ). C is aligned with the distributed matrix A, and replicated down every process row. C is tied to the distributed matrix A.
  • ROWCND (global input) REAL — the global ratio of the smallest R(i) to the largest R(i), IA <= i <= IA+M-1.
  • COLCND (global input) REAL — the global ratio of the smallest C(i) to the largest C(i), JA <= j <= JA+N-1.
  • AMAX (global input) REAL — absolute value of largest distributed submatrix entry.
  • EQUED (global output) CHARACTER — specifies the form of equilibration that was done.
    • = 'N': No equilibration
    • = 'R': Row equilibration, i.e., sub( A ) has been pre-multiplied by diag(R(IA:IA+M-1)),
    • = 'C': Column equilibration, i.e., sub( A ) has been post-multiplied by diag(C(JA:JA+N-1)),
    • = 'B': Both row and column equilibration, i.e., sub( A ) has been replaced by diag(R(IA:IA+M-1)) * sub( A ) * diag(C(JA:JA+N-1)).

✨ PARAMETERS ✨

these ones are the behind-the-scenes thresholds — they're not arguments, they're just vibes the routine lives by 🦋

  • THRESH is a threshold value used to decide if row or column scaling should be done based on the ratio of the row or column scaling factors. If ROWCND < THRESH, row scaling is done, and if COLCND < THRESH, column scaling is done.
  • LARGE and SMALL are threshold values used to decide if row scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, row scaling is done.
💀 heads up bestie: those are the only conditions written down here. don't go inventing extra ones, i'm literally just reading you the page. 😭

and that's the whole thing!! you understood it, you slayed it, main character energy confirmed. go equilibrate something. 👑

this little love letter to linear algebra came from LAPACK version 1.5, dated 12 May 1997 — the source and the date, preserved with love, because the lore matters. 🎀🫶✨

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