PDLAQSY(l)

📦 version 1.5, in ubuntu jammy

💖 NAME 💖

okay bestie, so PDLAQSY is the moment — it equilibrates a symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the scaling factors sitting in the vectors SR and SC. that's it, that's the whole job, and it does it gorgeously 💅

✨ SYNOPSIS ✨

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

    CHARACTER       EQUED, UPLO

    INTEGER         IA, JA, N

    DOUBLE          PRECISION AMAX, SCOND

    INTEGER         DESCA( * )

    DOUBLE          PRECISION A( * ), SC( * ), SR( * )

🎯 PURPOSE 🎯

PDLAQSY 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. like, it takes your matrix and rescales it so the numbers stop being feral. no cap.

📓 Notes

every global data object has a little bestie of its own called an associated description vector, and that vector is the one holding all the info you need to map an element of the object to its process and memory location. iconic behavior honestly.

so let A be a generic term for any 2D block cyclicly distributed array, and say that global array has an associated description vector DESCA. in the comments below, read the character _ 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)).

now let K be the number of rows or columns of a distributed matrix, and assume its process grid has dimension p x q.

LOCr( K ) is 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 ) is the number of elements of K that a process would receive if K were distributed over the q processes of its process row.

you can get the values of LOCr() and LOCc() by calling the ScaLAPACK tool function NUMROC, like so:

LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).

and an upper bound for these quantities can be computed with:

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. tells us whether the upper or lower triangular part of the symmetric distributed matrix sub( A ) is the one being referenced:
    • = 'U': upper triangular
    • = 'L': lower triangular
  • N — (global input) INTEGER. the number of rows and columns to be operated on, aka the order of the distributed submatrix sub( A ). N >= 0.
  • A — (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION. the 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) DOUBLE PRECISION. the absolute value of the largest distributed submatrix entry.
  • EQUED — (output) CHARACTER*1. says 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. slay.
  • 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.
🫶 that's the whole routine, bestie. read the descriptor notes twice, they're the iykyk part.

LAPACK version 1.5 ✨ 12 May 1997 — yeah she's an old icon but she still eats, mwah 💋👑

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