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