PCLAQGE(l)

📦 version 1.5, in ubuntu jammy

💖 PCLAQGE(l) 💖

okay bestie, sit DOWN, because we're about to talk about PCLAQGE — a LAPACK auxiliary routine, version 1.5, and she is so underrated. this is the girl who balances your distributed matrix so your math doesn't go feral. 🎀

✨ NAME ✨

PCLAQGE — 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. periodt. 💅

🦋 SYNOPSIS 🦋

SUBROUTINE PCLAQGE( 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            C( * ), R( * )

    COMPLEX         A( * )

like. look at her. gorgeous. no cap. ✨

🌸 PURPOSE 🌸

PCLAQGE 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 it, that's the job, and she does it flawlessly. 💖

🫶 Notes 🫶

okay so before we go ANY further, you need to understand descriptors, because this is the lore and it's lowkey essential, iykyk.

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. she's basically the group chat admin of memory. 👑

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".

here's the whole cast, bestie:

  • 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

memorize it or don't, but it's RIGHT there for you. 🧚

🌈 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.
  • 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) COMPLEX 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 enters a mess and leaves balanced, main character energy fr. ✨
  • 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.
  • 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. this is the receipts, bestie, she tells you exactly what she did:
    • = '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 💅

  • 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.
🫶 friendly reminder bestie: EQUED is a global output and she is the vibe check for whether anything got scaled at all. 'N' means no notes, nothing changed. don't come for her. 💀

and that's the whole page!! you ate that. go equilibrate something. 🔥👑

LAPACK version 1.5 — dated 12 May 1997. yes she's vintage, yes she still slays, we love a legacy icon. 🎀✨

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