PZLAQSY(l)

📦 version 1.5, in ubuntu jammy

PZLAQSY 💅✨

okay bestie, welcome to PZLAQSY, a LAPACK auxiliary routine (we're on version 1.5, she's vintage, she's iconic). This one equilibrates a symmetric distributed matrix. Yes, distributed. Yes, that means a whole grid of processes is involved. Sit down, get comfy, we're doing linear algebra with the girls 🫶

✨ NAME ✨

PZLAQSY — 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. That's it, that's the whole job, and she does it flawlessly 💅

✨ SYNOPSIS ✨

       SUBROUTINE PZLAQSY( 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 SC( * ), SR( * )

           COMPLEX*16      A( * )

🌸 PURPOSE 🌸

So like, PZLAQSY 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. That's the mission statement, and she is so committed to it 💖

Notes (iykyk, read these fr)

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".

Here's the whole notation situation, no cap:

  • 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. 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) COMPLEX*16 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. 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. 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.

💀 bestie reminder: those thresholds are decision-makers, not decoration. SCOND < THRESH → scaling happens. AMAX > LARGE or AMAX < SMALL → scaling happens. Periodt.

and that's the whole page, you ate that 🍓 this diva is brought to you by LAPACK version 1.5, dated 12 May 1997 — a literal queen from the archives, still serving accuracy 💖✨👑

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