Provided by: scalapack-doc_1.5-10_all 
NAME
PDLAHQR - i an auxiliary routine used to find the Schur decomposition and or eigenvalues of a matrix
already in Hessenberg form from cols ILO to IHI
SYNOPSIS
SUBROUTINE PDLAHQR( WANTT, WANTZ, N, ILO, IHI, A, DESCA, WR, WI, ILOZ, IHIZ, Z, DESCZ, WORK, LWORK,
IWORK, ILWORK, INFO )
LOGICAL WANTT, WANTZ
INTEGER IHI, IHIZ, ILO, ILOZ, ILWORK, INFO, LWORK, N, ROTN
INTEGER DESCA( * ), DESCZ( * ), IWORK( * )
DOUBLE PRECISION A( * ), WI( * ), WORK( * ), WR( * ), Z( * )
PURPOSE
PDLAHQR 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.
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
ARGUMENTS
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). PDLAHQR 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) DOUBLE PRECISION array, dimension
(DESCA(LLD_),*) On entry, the upper Hessenberg matrix A. On exit, if WANTT is .TRUE., A is upper
quasi-triangular in rows and columns ILO:IHI, with any 2-by-2 or larger diagonal blocks not yet
in standard form. 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.
WR (global replicated output) DOUBLE PRECISION array,
dimension (N) WI (global replicated output) DOUBLE PRECISION array, dimension (N) The real
and imaginary parts, respectively, of the computed eigenvalues ILO to IHI are stored in the
corresponding elements of WR and WI. If two eigenvalues are computed as a complex conjugate pair,
they are stored in consecutive elements of WR and WI, say the i-th and (i+1)th, with WI(i) > 0
and WI(i+1) < 0. 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) DOUBLE PRECISION array.
If WANTZ is .TRUE., on entry Z must contain the current matrix Z of transformations accumulated
by PDHSEQR, 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) DOUBLE PRECISION 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*LOCc(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: PDLAHQR 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 WR and WI contain those eigenvalues which have been
successfully computed.
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: DLAHQR -> Serial QR used to determine shifts
and eigenvalues DLARFG -> Determine the Householder transforms
This ScaLAPACK, this routine calls: PDLACONSB -> To determine where to start each iteration
DLAMSH -> Sends multiple shifts through a small submatrix to see how the consecutive
subdiagonals change (if PDLACONSB indicates we can start a run in the middle) PDLAWIL -> Given
the shift, get the transformation DLASORTE -> Pair up eigenvalues so that reals are paired.
PDLACP3 -> Parallel array to local replicated array copy & back. DLAREF -> Row/column
reflector applier. Core routine here. PDLASMSUB -> 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 DLAHQR. 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 DLAREF. 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
LAPACK version 1.5 12 May 1997 PDLAHQR(l)