PDLAQR3(3) ScaLAPACK routine of NEC Numeric Library Collection PDLAQR3(3)
NAME
PDLAQR3 - accepts as input an upper Hessenberg matrix H and performs an
orthogonal similarity transformation designed to detect and deflate
fully converged eigenvalues from a trailing principal submatrix
SYNOPSIS
RECURSIVE SUBROUTINE PDLAQR3(
WANTT, WANTZ, N, KTOP, KBOT, NW, H, DESCH, ILOZ,
IHIZ, Z, DESCZ, NS, ND, SR, SI, V, DESCV, NH, T,
DESCT, NV, WV, DESCW, WORK, LWORK, IWORK, LIWORK,
RECLEVEL )
INTEGER IHIZ, ILOZ, KBOT, KTOP, LWORK, N, ND, NH, NS, NV,
NW, LIWORK, RECLEVEL
LOGICAL WANTT, WANTZ
NTEGER DESCH( * ), DESCZ( * ), DESCT( * ), DESCV( * ),
DESCW( * ), IWORK( * )
DOUBLE PRECISION H( * ), SI( KBOT ), SR( KBOT ), T( * ),
V( * ), WORK( * ), WV( * ), Z( * )
PURPOSE
Aggressive early deflation:
This subroutine accepts as input an upper Hessenberg matrix H and per-
forms an orthogonal similarity transformation designed to detect and
deflate fully converged eigenvalues from a trailing principal subma-
trix. On output H has been overwritten by a new Hessenberg matrix that
is a perturbation of an orthogonal similarity transformation of H. It
is to be hoped that the final version of H has many zero subdiagonal
entries.
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
If .TRUE., then the Hessenberg matrix H is fully updated so
that the quasi-triangular Schur factor may be computed (in
cooperation with the calling subroutine).
If .FALSE., then only enough of H is updated to preserve the
eigenvalues.
WANTZ (global input) LOGICAL
If .TRUE., then the orthogonal matrix Z is updated so so that
the orthogonal Schur factor may be computed (in cooperation
with the calling subroutine).
If .FALSE., then Z is not referenced.
N (global input) INTEGER
The order of the matrix H and (if WANTZ is .TRUE.) the order of
the orthogonal matrix Z.
KTOP (global input) INTEGER
It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
KBOT and KTOP together determine an isolated block along the
diagonal of the Hessenberg matrix.
KBOT (global input) INTEGER
It is assumed without a check that either KBOT = N or
H(KBOT+1,KBOT)=0. KBOT and KTOP together determine an isolated
block along the diagonal of the Hessenberg matrix.
NW (global input) INTEGER
Deflation window size. 1 .LE. NW .LE. (KBOT-KTOP+1).
H (local input/output) DOUBLE PRECISION array, dimension
(DESCH(LLD_),*)
On input the initial N-by-N section of H stores the Hessenberg
matrix undergoing aggressive early deflation.
On output H has been transformed by an orthogonal similarity
transformation, perturbed, and the returned to Hessenberg form
that (it is to be hoped) has some zero subdiagonal entries.
DESCH (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix H.
ILOZ (global input) INTEGER
IHIZ (global input) INTEGER
Specify the rows of Z to which transformations must be applied
if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.
Z (input/output) DOUBLE PRECISION array, dimension
(DESCH(LLD_),*) IF WANTZ is .TRUE., then on output, the
orthogonal similarity transformation mentioned above has been
accumulated into Z(ILOZ:IHIZ,ILO:IHI) from the right.
If WANTZ is .FALSE., then Z is unreferenced.
DESCZ (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix Z.
NS (global output) INTEGER
The number of unconverged (ie approximate) eigenvalues returned
in SR and SI that may be used as shifts by the calling subrou-
tine.
ND (global output) INTEGER
The number of converged eigenvalues uncovered by this subrou-
tine.
SR (global output) DOUBLE PRECISION array, dimension KBOT
SI (global output) DOUBLE PRECISION array, dimension KBOT
On output, the real and imaginary parts of approximate eigen-
values that may be used for shifts are stored in SR(KBOT-ND-
NS+1) through SR(KBOT-ND) and SI(KBOT-ND-NS+1) through SI(KBOT-
ND), respectively.
The real and imaginary parts of converged eigenvalues are
stored in SR(KBOT-ND+1) through SR(KBOT) and SI(KBOT-ND+1)
through SI(KBOT), respectively.
V (global workspace) DOUBLE PRECISION array, dimension
(DESCV(LLD_),*)
An NW-by-NW distributed work array.
DESCV (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix V.
NH (input) INTEGER scalar
The number of columns of T. NH.GE.NW.
T (global workspace) DOUBLE PRECISION array, dimension
(DESCV(LLD_),*)
DESCT (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix T.
NV (global input) INTEGER
The number of rows of work array WV available for workspace.
NV.GE.NW.
WV (global workspace) DOUBLE PRECISION array, dimension
(DESCW(LLD_),*)
DESCW (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix WV.
WORK (local workspace) DOUBLE PRECISION array, dimension LWORK.
On exit, WORK(1) is set to an estimate of the optimal value of
LWORK for the given values of N, NW, KTOP and KBOT.
LWORK (local input) INTEGER
The dimension of the work array WORK. LWORK = 2*NW suffices,
but greater efficiency may result from larger values of LWORK.
If LWORK = -1, then a workspace query is assumed; PDLAQR3 only
estimates the optimal workspace size for the given values of N,
NW, KTOP and KBOT. The estimate is returned in WORK(1). No
error message related to LWORK is issued by XERBLA. Neither H
nor Z are accessed.
IWORK (local workspace) INTEGER array, dimension (LIWORK)
LIWORK (local input) INTEGER
The length of the workspace array IWORK
ScaLAPACK routine 31 October 2017 PDLAQR3(3)