PDGEEQU(l)

📦 version 1.5, in ubuntu jammy

💖 PDGEEQU 💖

okay so bestie, welcome to PDGEEQU, a LAPACK routine (that's LAPACK version 1.5, she's vintage but she ATE) and she's here to do equilibration on your distributed matrix. sit down, grab a boba, we're doing linear algebra 💅

✨ NAME ✨

PDGEEQU — she 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. that's it, that's the girl 🎀

🦋 SYNOPSIS 🦋

SUBROUTINE PDGEEQU( 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 A( * ), C( * ), R( * )

look at that signature. iconic. no notes. ✨

🎀 PURPOSE 🎀

So PDGEEQU 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 returns the row scale factors and C the column scale factors, chosen to try to make the largest entry in each row and column of the distributed matrix B with elements B(i,j) = R(i) * A(i,j) * C(j) have absolute value 1. like, every entry gets its moment, everyone's a main character, but nobody's louder than 1.0 🫶

R(i) and C(j) are restricted to be between SMLNUM = smallest safe number and BIGNUM = largest safe number. Use of these scaling factors is not guaranteed to reduce the condition number of sub( A ) but works well in practice. ngl it's giving "trust me bro" but she's earned it 💀

📝 Notes

okay this part is the lore, bestie. 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. it's the group chat admin of data structures, fr fr.

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

math girlies stay winning 📐✨

👑 ARGUMENTS 👑

every single one of them, bestie, no skips:

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) DOUBLE PRECISION 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) 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).
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.
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, read the INFO thing twice. negative means somebody handed her a bad argument, positive means a whole row or column of your submatrix is literally all zeros. that's not a vibe check failure, that's a real math situation. handle it. 💀

and that's the whole page, no cap. go forth and equilibrate responsibly. periodt. 💅

this has been PDGEEQU(l), LAPACK routine, LAPACK version 1.5, dated 12 May 1997 — shoutout to the vintage era, she's been serving condition numbers since before you were born 🎀✨

📦 where this page comes from

ubuntu jammy: scalapack-doc 1.5-11
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-10-14: a language model's rewrite of the original, under its terms