PZGEEQU - 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 PZGEEQU(

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 C( * ), R( * )

COMPLEX*16 A( * )

PZGEEQU 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*16 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 distri- buted 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.