PCGEEQU(3) ScaLAPACK routine of NEC Numeric Library Collection PCGEEQU(3)
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
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
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 num-
ber 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 vec-
tor. This vector stores the information required to establish the map-
ping 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 col-
umn.
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 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.
ScaLAPACK routine 31 October 2017 PCGEEQU(3)