PZGEEQU(l)

📦 version 1.5, in ubuntu jammy

💖 PZGEEQU 💖

ok so bestie, gather round, because we're about to talk about PZGEEQU — a LAPACK routine, version 1.5, from the l manual section, and she is lowkey a whole vibe. she computes row and column scalings that try to equilibrate a distributed matrix and shrink its condition number. no cap. let's get into it 🦋

✨ NAME ✨

PZGEEQU — compute row and column scalings intended to equilibrate an M-by-N distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA:JA+N-1) and reduce its condition number. that's the whole job description, she's focused, she's employed, she's that girl 💅

🌸 SYNOPSIS 🌸

SUBROUTINE PZGEEQU( M, N, A, IA, JA, DESCA, R, C, ROWCND, COLCND, AMAX,
                    INFO )

    INTEGER         IA, INFO, JA, M, N

    DOUBLE          PRECISION AMAX, COLCND, ROWCND

    INTEGER         DESCA( * )

    DOUBLE          PRECISION C( * ), R( * )

    COMPLEX*16      A( * )

👑 PURPOSE 👑

ok so PZGEEQU computes row and column scalings intended to equilibrate an M-by-N distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA:JA+N-1) and reduce its condition number. R hands you back the row scale factors and C hands you back the column scale factors, and the whole point of them is to try to make the largest entry in each row and column of the distributed matrix B — where B(i,j) = R(i) * A(i,j) * C(j) — have absolute value 1. it's giving ✨main character normalization✨

R(i) and C(j) are restricted to be between SMLNUM = smallest safe number and BIGNUM = largest safe number. and listen, use of these scaling factors is not guaranteed to reduce the condition number of sub( A ), but it works well in practice. fr fr, she tries her best and usually she eats 🔥

📝 Notes 📝

every 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".

  • 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 🎀

  • 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. she does not do negative rows, bestie 🚫
  • 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) COMPLEX*16 pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ), the local pieces of the M-by-N distributed matrix whose equilibration factors are to be computed. this is the star of the show ⭐
  • 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. she's holding all the lore 🫶
  • R — (local output) DOUBLE PRECISION array, dimension LOCr(M_A). if INFO = 0 or INFO > IA+M-1, R(IA:IA+M-1) contains 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 output) DOUBLE PRECISION array, dimension LOCc(N_A). if INFO = 0, C(JA:JA+N-1) contains the column scale factors for 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 output) DOUBLE PRECISION. if INFO = 0 or INFO > IA+M-1, ROWCND contains the ratio of the smallest R(i) to the largest R(i) (IA <= i <= IA+M-1). if ROWCND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by R(IA:IA+M-1). save yourself the effort bestie 💀
  • COLCND — (global output) DOUBLE PRECISION. if INFO = 0, COLCND contains the ratio of the smallest C(j) to the largest C(j) (JA <= j <= JA+N-1). if COLCND >= 0.1, it is not worth scaling by C(JA:JA+N-1).
  • AMAX — (global output) DOUBLE PRECISION. absolute value of largest distributed matrix element. if AMAX is very close to overflow or very close to underflow, the matrix should be scaled. ⚠️ like, actually should.
  • INFO — (global output) INTEGER.
    • = 0: successful exit 🎉 slay
    • < 0: if the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j); if the i-th argument is a scalar and had an illegal value, then INFO = -i.
    • > 0: if INFO = i, and i is <= M: the i-th row of the distributed matrix sub( A ) is exactly zero; > M: the (i-M)-th column of the distributed matrix sub( A ) is exactly zero.
🚨 heads up bestie: that > 0 case means an entire row or column is exactly zero, which is not a cute look for your matrix. also the < 0 business with -(i*100+j) is the classic "which array, which entry" detective game. iykyk 🔍

and that's the whole page, no notes, every flag and every formula accounted for. she's iconic, she's precise, she's PZGEEQU 💜

LAPACK version 1.5 · 12 May 1997 · bye bestie, go equilibrate something 🌈✨

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