fix reduction scope for column scale bounds in p?geequ

[kyungminlee]

Fix reduction scope for column scale bounds in p?geequ

Summary

p{s,d,c,z}geequ compute COLCND = min(C(j)) / max(C(j)) over the global column-scale vector. The min/max reduction is currently issued along the column axis of the BLACS process grid, but at that point C has already been replicated along that same axis — so the reduction is a no-op and each process column ends up with the min/max of only its own subset of global columns. The returned COLCND (and INFO, on the singular-column path) is therefore wrong whenever NPCOL ≥ 2 and the per-process-column scale ranges differ.

The fix switches three reductions in each of p{s,d,c,z}geequ.f from 'Columnwise'/COLCTOP to 'Rowwise'/ROWCTOP so the bounds are combined across process columns. R, ROWCND, AMAX, and the per-column entries of C are unaffected.

Root cause

In SRC/pdgeequ.f (other precisions identical), after the per-column max is computed and replicated along each process column:

  321  CALL DGAMX2D( ICTXT, 'Columnwise', COLCTOP, 1, NQ, C(JJA), 1, ...)   ! replicates C down each proc-column
       ...
       DO 100 J = JJA, JJA+NQ-1
          RCMIN = MIN( RCMIN, C(J) )      ! local min over THIS proc-column's NQ cols
          RCMAX = MAX( RCMAX, C(J) )
  100  CONTINUE
- 332  CALL DGAMX2D( ICTXT, 'Columnwise', COLCTOP, 1, 1, RCMAX, 1, ...)     ! BUG: along axis already replicated → no-op
- 334  CALL DGAMN2D( ICTXT, 'Columnwise', COLCTOP, 1, 1, RCMIN, 1, ...)     ! BUG
+ 332  CALL DGAMX2D( ICTXT, 'Rowwise',    ROWCTOP, 1, 1, RCMAX, 1, ...)     ! combine across proc-columns
+ 334  CALL DGAMN2D( ICTXT, 'Rowwise',    ROWCTOP, 1, 1, RCMIN, 1, ...)
       ...
- 346  CALL IGAMX2D( ICTXT, 'Columnwise', COLCTOP, 1, 1, INFO, 1, ...)      ! same axis bug on the singular-column path
+ 346  CALL IGAMX2D( ICTXT, 'Rowwise',    ROWCTOP, 1, 1, INFO, 1, ...)

After the patch, RCMIN/RCMAX are the true global min/max of C, so COLCND = max(RCMIN, SMLNUM) / min(RCMAX, BIGNUM) matches the serial LAPACK ?GEEQU result. The singular-column report (INFO = M + j_first_zero) is fixed the same way.

Reproducer

Build a matrix whose column-scale spread is asymmetric across process columns, e.g. A(i,j) = 10^(j-1). With M = N = 12, NB = 8, on NPROW × NPCOL = 1 × 2:

• proc col 0 owns global cols 1..8 → C(j) ∈ {10^11, …, 10^4} → local COLCND = 10^-7
• proc col 1 owns global cols 9..12 → C(j) ∈ {10^3, …, 10^0} → local COLCND = 10^-3
• true global → C ∈ {10^11, …, 10^0} → COLCND = 10^-11

Serial DGEEQU on the same matrix returns COLCND = 1.0e-11. Before the patch, PDGEEQU returns the rank-0 process column's local value 1.0e-7 — a factor-10^4 error. After the patch it returns 1.0e-11.

A minimal Fortran example

NPROW = 1 ; NPCOL = 2 ; M = 12 ; N = 12 ; NB = 8
...
DO J = 1, NQ
   GJ = INDXL2G( J, NB, MYCOL, 0, NPCOL )
   DO I = 1, NP
      A( I + (J-1)*LLDA ) = 10.0D0**(GJ-1)   ! constant within each global column
   END DO
END DO
CALL PDGEEQU( M, N, A, 1, 1, DESCA, R, C, ROWCND, COLCND, AMAX, INFO )
! Compare COLCND against serial DGEEQU on rank 0.

Observed (this matrix, several process grids):

grid	netlib master	this PR	serial DGEEQU
1×1	1.0e-11	1.0e-11	1.0e-11
1×2	1.0e-07	1.0e-11	1.0e-11
2×2	1.0e-07	1.0e-11	1.0e-11
1×4	1.0e-07	1.0e-11	1.0e-11
2×3	1.0e-07	1.0e-11	1.0e-11

R, ROWCND, and AMAX are correct on all grids and all builds; only COLCND (and the singular-column INFO report on the same code path) is affected. MKL inherits the bug from netlib; this PR is the only build that matches serial LAPACK on NPCOL ≥ 2.