PCGEEQU(l)

📦 version 1.5, in ubuntu jammy

💖 PCGEEQU(l) 💖

okay bestie, listen up — this is a LAPACK routine, and it's lowkey one of the good ones. Her name is PCGEEQU, and she's here to fix your matrix's whole situation. 🎀

✨ NAME ✨

PCGEEQU — 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 vibe. she balances. she scales. she serves. 💅

🦋 SYNOPSIS 🦋

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

    INTEGER         IA, INFO, JA, M, N

    REAL            AMAX, COLCND, ROWCND

    INTEGER         DESCA( * )

    REAL            C( * ), R( * )

    COMPLEX         A( * )

🌸 PURPOSE 🌸

okay so PCGEEQU 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 the row scale factors, C hands you the column scale factors — and they're picked specifically to try to make the largest entry in each row and column of the distributed matrix B (whose elements are B(i,j) = R(i) * A(i,j) * C(j)) have absolute value 1. iconic behavior honestly. ✨

R(i) and C(j) are restricted to be between SMLNUM = smallest safe number and BIGNUM = largest safe number. and like, real talk? using these scaling factors is not guaranteed to reduce the condition number of sub( A ) — but it works well in practice. no cap. 🫶

💜 Notes 💜

every global data object is described by an associated description vector. this vector stores the info 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". (iykyk 🧚)

  • 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. 💖
  • 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 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. ✨
  • 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 output) REAL 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) REAL 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) REAL. 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). 💅
  • COLCND — (global output) REAL. 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) REAL. absolute value of largest distributed matrix element. if AMAX is very close to overflow or very close to underflow, the matrix should be scaled. 🚨
  • INFO — (global output) INTEGER.
    • = 0: successful exit 🎉
    • < 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. 😭
⚠️ bestie be aware: INFO < 0 means somebody handed over an illegal value, and INFO > 0 means you've got a row or column of sub( A ) that is exactly zero. that's not a vibe. check your inputs. 💀

and that's the whole page, bestie!! 🎀 this one's from LAPACK version 1.5, dated 12 May 1997 — she's vintage but she's still eating. go equilibrate something. 💅✨

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