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