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
- 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( * )
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 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.
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.
Notes
=====
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".
NOTATION STORED IN EXPLANATION
--------------- -------------- --------------------------------------
DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case,
DTYPE_A = 1.
CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
the BLACS process grid A is distribu-
ted over. The context itself is glo-
bal, but the handle (the integer
value) may vary.
M_A (global) DESCA( M_ ) The number of rows in the global
array A.
N_A (global) DESCA( N_ ) The number of columns in the global
array A.
MB_A (global) DESCA( MB_ ) The blocking factor used to distribute
the rows of the array.
NB_A (global) DESCA( NB_ ) The blocking factor used to distribute
the columns of the array.
RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process
column over which the
first column of the array A is
distributed.
LLD_A (local) 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
- 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 distri- buted 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.