|SUBROUTINE PCLAHQR(||WANTT, WANTZ, N, ILO, IHI, A, DESCA, W, ILOZ, IHIZ, Z, DESCZ, WORK, LWORK, IWORK, ILWORK, INFO )|
|LOGICAL WANTT, WANTZ|
|INTEGER IHI, IHIZ, ILO, ILOZ, ILWORK, INFO, LWORK, N|
|INTEGER DESCA( * ), DESCZ( * ), IWORK( * )|
|COMPLEX A( * ), W( * ), WORK( * ), Z( * )|
PCLAHQR is an auxiliary routine used to find the Schur decomposition and or eigenvalues of a matrix already in Hessenberg form from cols ILO to IHI. If Z = I, and WANTT=WANTZ=.TRUE., H gets replaced with ZHZ,
with ZZ=I, and H in Schur form.
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,LOCp(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.
LOCp( 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, LOCq( 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 LOCp() and LOCq() may be determined via a call to the ScaLAPACK tool function, NUMROC:
LOCp( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
LOCq( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by:
LOCp( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
LOCq( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A
WANTT (global input) LOGICAL = .TRUE. : the full Schur form T is required;
= .FALSE.: only eigenvalues are required.
WANTZ (global input) LOGICAL = .TRUE. : the matrix of Schur vectors Z is required;
= .FALSE.: Schur vectors are not required.
N (global input) INTEGER The order of the Hessenberg matrix A (and Z if WANTZ). N >= 0. ILO (global input) INTEGER IHI (global input) INTEGER It is assumed that A is already upper quasi-triangular in rows and columns IHI+1:N, and that A(ILO,ILO-1) = 0 (unless ILO = 1). PCLAHQR works primarily with the Hessenberg submatrix in rows and columns ILO to IHI, but applies transformations to all of H if WANTT is .TRUE.. 1 <= ILO <= max(1,IHI); IHI <= N. A (global input/output) COMPLEX array, dimension (DESCA(LLD_),*) On entry, the upper Hessenberg matrix A. On exit, if WANTT is .TRUE., A is upper triangular in rows and columns ILO:IHI. If WANTT is .FALSE., the contents of A are unspecified on exit. DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A.
W (global replicated output) COMPLEX array, dimension (N) The computed eigenvalues ILO to IHI are stored in the corresponding elements of W. If WANTT is .TRUE., the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in A. A may be returned with larger diagonal blocks until the next release.
ILOZ (global input) INTEGER IHIZ (global input) INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= ILO; IHI <= IHIZ <= N. Z (global input/output) COMPLEX array. If WANTZ is .TRUE., on entry Z must contain the current matrix Z of transformations accumulated by PCHSEQR, and on exit Z has been updated; transformations are applied only to the submatrix Z(ILOZ:IHIZ,ILO:IHI). If WANTZ is .FALSE., Z is not referenced. DESCZ (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Z. WORK (local output) COMPLEX array of size LWORK (Unless LWORK=-1, in which case WORK must be at least size 1) LWORK (local input) INTEGER WORK(LWORK) is a local array and LWORK is assumed big enough so that LWORK >= 3*N + MAX( 2*MAX(DESCZ(LLD_),DESCA(LLD_)) + 2*LOCq(N), 7*Ceil(N/HBL)/LCM(NPROW,NPCOL)) + MAX( 2*N, (8*LCM(NPROW,NPCOL)+2)**2 ) If LWORK=-1, then WORK(1) gets set to the above number and the code returns immediately. IWORK (global and local input) INTEGER array of size ILWORK This will hold some of the IBLK integer arrays. This is held as a place holder for a future release. Currently unreferenced. ILWORK (local input) INTEGER This will hold the size of the IWORK array. This is held as a place holder for a future release. Currently unreferenced. INFO (global output) INTEGER < 0: parameter number -INFO incorrect or inconsistent
= 0: successful exit
> 0: PCLAHQR failed to compute all the eigenvalues ILO to IHI in a total of 30*(IHI-ILO+1) iterations; if INFO = i, elements i+1:ihi of W contains those eigenvalues which have been successfully computed.
Logic: This algorithm is very similar to SLAHQR. Unlike SLAHQR, 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 wont 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: CLAHQR -> Serial QR used to determine shifts and eigenvalues CLARFG -> Determine the Householder transforms
This ScaLAPACK, this routine calls: PCLACONSB -> To determine where to start each iteration CLAMSH -> Sends multiple shifts through a small submatrix to see how the consecutive subdiagonals change (if PCLACONSB indicates we can start a run in the middle) PCLAWIL -> Given the shift, get the transformation PCLACP3 -> Parallel array to local replicated array copy & back. CLAREF -> Row/column reflector applier. Core routine here. PCLASMSUB -> 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 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) 4.) 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. 5.) 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 CLAHQR. 6.) For this release, this code has only been tested for RSRC_=CSRC_=0, but it has been written for the general case. 7.) Currently, all the eigenvalues are distributed to all the nodes. Future releases will probably distribute the eigenvalues by the column partitioning. 8.) The internals of this routine are subject to change. 9.) To optimize this for your architecture, try tuning CLAREF. 10.) This code has only been tested for WANTZ = .TRUE. and may behave unpredictably for WANTZ set to .FALSE.
Contributed by Mark Fahey, June, 2000.
|ScaLAPACK version 1.7||PCLAHQR (l)||13 August 2001|