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
N_A (global) DESCA( N_ ) The number of columns in the global
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
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
Logic: This algorithm is very similar to _LAHQR. Unlike _LAHQR, instead of sending one double shift through the largest unreduced submatrix, this algorithm sends multiple double shifts and spaces them apart so that there can be parallelism across several processor row/columns. Another critical difference is that this algorithm aggregrates multiple transforms together in order to apply them in a block fashion.
Important Local Variables: IBLK = The maximum number of bulges that can be computed. Currently fixed. Future releases this won't be fixed. HBL = The square block size (HBL=DESCA(MB_)=DESCA(NB_)) ROTN = The number of transforms to block together NBULGE = The number of bulges that will be attempted on the current submatrix. IBULGE = The current number of bulges started. K1(*),K2(*) = The current bulge loops from K1(*) to K2(*).
Subroutines: From LAPACK, this routine calls: SLAHQR -> Serial QR used to determine shifts and eigenvalues SLARFG -> Determine the Householder transforms
This ScaLAPACK, this routine calls: PSLACONSB -> To determine where to start each iteration SLAMSH -> Sends multiple shifts through a small submatrix to see how the consecutive subdiagonals change (if PSLACONSB indicates we can start a run in the middle) PSLAWIL -> Given the shift, get the transformation SLASORTE -> Pair up eigenvalues so that reals are paired. PSLACP3 -> Parallel array to local replicated array copy & back. SLAREF -> Row/column reflector applier. Core routine here. PSLASMSUB -> Finds negligible subdiagonal elements.
Current Notes and/or Restrictions: 1.) This code requires the distributed block size to be square and at least six (6); unlike simpler codes like LU, this algorithm is extremely sensitive to block size. Unwise choices of too small a block size can lead to bad performance. 2.) This code requires A and Z to be distributed identically and have identical contxts. A future version may allow Z to have a different contxt to 1D row map it to all nodes (so no communication on Z is necessary.) 3.) This release currently does not have a routine for resolving the Schur blocks into regular 2x2 form after this code is completed. Because of this, a significant performance impact is required while the deflation is done by sometimes a single column of processors. 4.) This code does not currently block the initial transforms so that none of the rows or columns for any bulge are completed until all are started. To offset pipeline start-up it is recommended that at least 2*LCM(NPROW,NPCOL) bulges are used (if possible) 5.) The maximum number of bulges currently supported is fixed at 32. In future versions this will be limited only by the incoming WORK and IWORK array. 6.) The matrix A must be in upper Hessenberg form. If elements below the subdiagonal are nonzero, the resulting transforms may be nonsimilar. This is also true with the LAPACK routine SLAHQR. 7.) For this release, this code has only been tested for RSRC_=CSRC_=0, but it has been written for the general case. 8.) Currently, all the eigenvalues are distributed to all the nodes. Future releases will probably distribute the eigenvalues by the column partitioning. 9.) The internals of this routine are subject to change. 10.) To optimize this for your architecture, try tuning SLAREF. 11.) This code has only been tested for WANTZ = .TRUE. and may behave unpredictably for WANTZ set to .FALSE.
Implemented by: G. Henry, May 1, 1997