PZLAHQR(3) ScaLAPACK routine of NEC Numeric Library Collection PZLAHQR(3)
NAME
PZLAHQR - 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 PZLAHQR( 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*16 A( * ), W( * ), WORK( * ), Z( * )
PURPOSE
PZLAHQR 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 Z'HZ,
with Z'Z=I, and H in Schur form.
Notes
=====
Each global data object is described by an associated description vec-
tor. This vector stores the information required to establish the map-
ping 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,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 col-
umn.
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
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). PZLAHQR 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*16 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*16 array, dimension (N)
The computed eigenvalues ILO to IHI are stored in the corre-
sponding 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*16 array.
If WANTZ is .TRUE., on entry Z must contain the current matrix
Z of transformations accumulated by PZHSEQR, 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 refer-
enced.
DESCZ (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix Z.
WORK (local output) COMPLEX*16 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 unrefer-
enced.
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: PZLAHQR 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 suc-
cessfully computed.
Logic:
======
This algorithm is very similar to DLAHQR. Unlike DLAHQR,
instead of sending one double shift through the largest unre-
duced 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 num-
ber of bulges started. K1(*),K2(*) = The current bulge loops
from K1(*) to K2(*).
Subroutines: From LAPACK, this routine calls: ZLAHQR ->
Serial QR used to determine shifts and eigenvalues ZLARFG
-> Determine the Householder transforms
This ScaLAPACK, this routine calls: PZLACONSB -> To determine
where to start each iteration ZLAMSH -> Sends multiple
shifts through a small submatrix to see how the consecutive
subdiagonals change (if PZLACONSB indicates we can start a run
in the middle) PZLAWIL -> Given the shift, get the transfor-
mation PZLACP3 -> Parallel array to local replicated array
copy & back. ZLAREF -> Row/column reflector applier. Core
routine here. PZLASMSUB -> Finds negligible subdiagonal ele-
ments.
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 sen-
sitive 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 rec-
ommended 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 ZLAHQR. 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
ZLAREF. 10.) This code has only been tested for WANTZ = .TRUE.
and may behave unpredictably for WANTZ set to .FALSE.
FURTHER DETAILS
Contributed by Mark Fahey, June, 2000.
ScaLAPACK routine 31 October 2017 PZLAHQR(3)