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PCLAQGE - equilibrate a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) using the row and scaling factors in the vectors R and C

SUBROUTINE PCLAQGE( M, N, A, IA, JA, DESCA, R, C, ROWCND, COLCND, AMAX, EQUED ) CHARACTER EQUED INTEGER IA, JA, M, N REAL AMAX, COLCND, ROWCND INTEGER DESCA( * ) REAL C( * ), R( * ) COMPLEX A( * )

PCLAQGE equilibrates a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) using the row and scaling factors in the vectors R and C. 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/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)) containing on entry the M-by-N matrix sub( A ). On exit, the equilibrated distributed matrix. See EQUED for the form of the equilibrated distributed submatrix. 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 input) REAL array, dimension LOCr(M_A) 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 input) REAL array, dimension LOCc(N_A) The column scale factors of 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 input) REAL The global ratio of the smallest R(i) to the largest R(i), IA <= i <= IA+M-1. COLCND (global input) REAL The global ratio of the smallest C(i) to the largest C(i), JA <= j <= JA+N-1. AMAX (global input) REAL Absolute value of largest distributed submatrix entry. EQUED (global output) CHARACTER Specifies the form of equilibration that was done. = ’N’: No equilibration = ’R’: Row equilibration, i.e., sub( A ) has been pre- multiplied by diag(R(IA:IA+M-1)), = ’C’: Column equilibration, i.e., sub( A ) has been post- multiplied by diag(C(JA:JA+N-1)), = ’B’: Both row and column equilibration, i.e., sub( A ) has been replaced by diag(R(IA:IA+M-1)) * sub( A ) * diag(C(JA:JA+N-1)).

THRESH is a threshold value used to decide if row or column scaling should be done based on the ratio of the row or column scaling factors. If ROWCND < THRESH, row scaling is done, and if COLCND < THRESH, column scaling is done. LARGE and SMALL are threshold values used to decide if row scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, row scaling is done.