PSLAQR1(3) ScaLAPACK routine of NEC Numeric Library Collection PSLAQR1(3)
NAME
PSLAQR1 - 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
SYNOPSIS
RECURSIVE SUBROUTINE PSLAQR1(
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
INTEGER DESCA( * ), DESCZ( * ), IWORK( * )
REAL A( * ), WI( * ), WORK( * ), WR( * ), Z( * )
PURPOSE
PSLAQR1 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.
This is a modified version of PSLAHQR from ScaLAPACK version 1.7.3.
The following modifications were made:
o Recently removed workspace query functionality was added.
o Aggressive early deflation is implemented.
o Aggressive deflation (looking for two consecutive small
subdiagonal elements by PSLACONSB) is abandoned.
o The returned Schur form is now in canonical form, i.e., the
returned 2-by-2 blocks really correspond to complex conjugate
pairs of eigenvalues.
o For some reason, the original version of PSLAHQR sometimes did
not read out the converged eigenvalues correclty. This is now
fixed.
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,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 col-
umn.
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). PSLAQR1 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) REAL 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 con-
tents 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) REAL array, dimension (N)
WI (global replicated output) REAL 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 con-
jugate 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) REAL array.
If WANTZ is .TRUE., on entry Z must contain the current matrix
Z of transformations accumulated by PSHSEQR, 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) REAL array of size LWORK
LWORK (local input) INTEGER WORK(LWORK) is a local array and
LWORK is assumed big enough so that LWORK >= 6*N + 6*385*385 +
MAX( 2*MAX(DESCZ(LLD_),DESCA(LLD_)) + 2*LOCc(N),
7*Ceil(N/HBL)/LCM(NPROW,NPCOL)) )
IWORK (global and local input) INTEGER array of size ILWORK
ILWORK (local input) INTEGER
This holds the some of the IBLK integer arrays. This is held
as a place holder for the next release.
INFO (global output) INTEGER
< 0: parameter number -INFO incorrect or inconsistent
= 0: successful exit
> 0: PSLAQR1 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:
============
This routine calls:
PSLAWIL -> Given the shift, get the transformation
SLASORTE -> Pair up eigenvalues so that reals are paired.
PSLACP3 -> Parallel array to local replicated array copy &
back.
SLAREF -> Row/column reflector applier. Core routine here.
PSLASMSUB -> 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.
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 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.
7.) For this release, it is assumed RSRC_=CSRC_=0
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.
ScaLAPACK routine 31 October 2017 PSLAQR1(3)