NAME
PZSTEIN - compute the eigenvectors of a symmetric tridiagonal matrix in
parallel, using inverse iteration
SYNOPSIS
SUBROUTINE PZSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, ORFAC, Z, IZ, JZ,
DESCZ, WORK, LWORK, IWORK, LIWORK, IFAIL, ICLUSTR,
GAP, INFO )
INTEGER INFO, IZ, JZ, LIWORK, LWORK, M, N
DOUBLE PRECISION ORFAC
INTEGER DESCZ( * ), IBLOCK( * ), ICLUSTR( * ), IFAIL( * ),
ISPLIT( * ), IWORK( * )
DOUBLE PRECISION D( * ), E( * ), GAP( * ), W( * ), WORK( *
)
COMPLEX*16 Z( * )
PURPOSE
PZSTEIN computes the eigenvectors of a symmetric tridiagonal matrix in
parallel, using inverse iteration. The eigenvectors found correspond to
user specified eigenvalues. PZSTEIN does not orthogonalize vectors that
are on different processes. The extent of orthogonalization is
controlled by the input parameter LWORK. Eigenvectors that are to be
orthogonalized are computed by the same process. PZSTEIN decides on the
allocation of work among the processes and then calls DSTEIN2 (modified
LAPACK routine) on each individual process. If insufficient workspace
is allocated, the expected orthogonalization may not be done.
Note : If the eigenvectors obtained are not orthogonal, increase
LWORK and run the code again.
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
P = NPROW * NPCOL is the total number of processes
N (global input) INTEGER
The order of the tridiagonal matrix T. N >= 0.
D (global input) DOUBLE PRECISION array, dimension (N)
The n diagonal elements of the tridiagonal matrix T.
E (global input) DOUBLE PRECISION array, dimension (N-1)
The (n-1) off-diagonal elements of the tridiagonal matrix T.
M (global input) INTEGER
The total number of eigenvectors to be found. 0 <= M <= N.
W (global input/global output) DOUBLE PRECISION array, dim (M)
On input, the first M elements of W contain all the eigenvalues
for which eigenvectors are to be computed. The eigenvalues
should be grouped by split-off block and ordered from smallest
to largest within the block (The output array W from PDSTEBZ
with ORDER=’b’ is expected here). This array should be
replicated on all processes. On output, the first M elements
contain the input eigenvalues in ascending order.
Note : To obtain orthogonal vectors, it is best if eigenvalues
are computed to highest accuracy ( this can be done by setting
ABSTOL to the underflow threshold = DLAMCH(’U’) --- ABSTOL is
an input parameter to PDSTEBZ )
IBLOCK (global input) INTEGER array, dimension (N)
The submatrix indices associated with the corresponding
eigenvalues in W -- 1 for eigenvalues belonging to the first
submatrix from the top, 2 for those belonging to the second
submatrix, etc. (The output array IBLOCK from PDSTEBZ is
expected here).
ISPLIT (global input) INTEGER array, dimension (N)
The splitting points, at which T breaks up into submatrices.
The first submatrix consists of rows/columns 1 to ISPLIT(1),
the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc.,
and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1
through ISPLIT(NSPLIT)=N (The output array ISPLIT from PDSTEBZ
is expected here.)
ORFAC (global input) DOUBLE PRECISION
ORFAC specifies which eigenvectors should be orthogonalized.
Eigenvectors that correspond to eigenvalues which are within
ORFAC*||T|| of each other are to be orthogonalized. However,
if the workspace is insufficient (see LWORK), this tolerance
may be decreased until all eigenvectors to be orthogonalized
can be stored in one process. No orthogonalization will be
done if ORFAC equals zero. A default value of 10^-3 is used if
ORFAC is negative. ORFAC should be identical on all processes.
Z (local output) COMPLEX*16 array,
dimension (DESCZ(DLEN_), N/npcol + NB) Z contains the computed
eigenvectors associated with the specified eigenvalues. Any
vector which fails to converge is set to its current iterate
after MAXITS iterations ( See DSTEIN2 ). On output, Z is
distributed across the P processes in block cyclic format.
IZ (global input) INTEGER
Z’s global row index, which points to the beginning of the
submatrix which is to be operated on.
JZ (global input) INTEGER
Z’s global column index, which points to the beginning of the
submatrix which is to be operated on.
DESCZ (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix Z.
WORK (local workspace/global output) DOUBLE PRECISION array,
dimension ( LWORK ) On output, WORK(1) gives a lower bound on
the workspace ( LWORK ) that guarantees the user desired
orthogonalization (see ORFAC). Note that this may overestimate
the minimum workspace needed.
LWORK (local input) integer
LWORK controls the extent of orthogonalization which can be
done. The number of eigenvectors for which storage is allocated
on each process is NVEC = floor(( LWORK- max(5*N,NP00*MQ00)
)/N). Eigenvectors corresponding to eigenvalue clusters of
size NVEC - ceil(M/P) + 1 are guaranteed to be orthogonal ( the
orthogonality is similar to that obtained from ZSTEIN2). Note
: LWORK must be no smaller than: max(5*N,NP00*MQ00) +
ceil(M/P)*N, and should have the same input value on all
processes. It is the minimum value of LWORK input on different
processes that is significant.
If LWORK = -1, then LWORK is global input and a workspace query
is assumed; the routine only calculates the minimum and optimal
size for all work arrays. Each of these values is returned in
the first entry of the corresponding work array, and no error
message is issued by PXERBLA.
IWORK (local workspace/global output) INTEGER array,
dimension ( 3*N+P+1 ) On return, IWORK(1) contains the amount
of integer workspace required. On return, the IWORK(2) through
IWORK(P+2) indicate the eigenvectors computed by each process.
Process I computes eigenvectors indexed IWORK(I+2)+1 thru’
IWORK(I+3).
LIWORK (local input) INTEGER
Size of array IWORK. Must be >= 3*N + P + 1
If LIWORK = -1, then LIWORK is global input and a workspace
query is assumed; the routine only calculates the minimum and
optimal size for all work arrays. Each of these values is
returned in the first entry of the corresponding work array,
and no error message is issued by PXERBLA.
IFAIL (global output) integer array, dimension (M)
On normal exit, all elements of IFAIL are zero. If one or more
eigenvectors fail to converge after MAXITS iterations (as in
ZSTEIN), then INFO > 0 is returned. If mod(INFO,M+1)>0, then
for I=1 to mod(INFO,M+1), the eigenvector corresponding to the
eigenvalue W(IFAIL(I)) failed to converge ( W refers to the
array of eigenvalues on output ).
ICLUSTR (global output) integer array, dimension (2*P) This
output array contains indices of eigenvectors corresponding to
a cluster of eigenvalues that could not be orthogonalized due
to insufficient workspace (see LWORK, ORFAC and INFO).
Eigenvectors corresponding to clusters of eigenvalues indexed
ICLUSTR(2*I-1) to ICLUSTR(2*I), I = 1 to INFO/(M+1), could not
be orthogonalized due to lack of workspace. Hence the
eigenvectors corresponding to these clusters may not be
orthogonal. ICLUSTR is a zero terminated array --- (
ICLUSTR(2*K).NE.0 .AND. ICLUSTR(2*K+1).EQ.0 ) if and only if K
is the number of clusters.
GAP (global output) DOUBLE PRECISION array, dimension (P)
This output array contains the gap between eigenvalues whose
eigenvectors could not be orthogonalized. The INFO/M output
values in this array correspond to the INFO/(M+1) clusters
indicated by the array ICLUSTR. As a result, the dot product
between eigenvectors corresponding to the I^th cluster may be
as high as ( O(n)*macheps ) / GAP(I).
INFO (global output) INTEGER
= 0: successful exit
< 0: If the i-th argument is an array and the j-entry had an
illegal value, then INFO = -(i*100+j), if the i-th argument is
a scalar and had an illegal value, then INFO = -i. < 0 : if
INFO = -I, the I-th argument had an illegal value
> 0 : if mod(INFO,M+1) = I, then I eigenvectors failed to
converge in MAXITS iterations. Their indices are stored in the
array IFAIL. if INFO/(M+1) = I, then eigenvectors
corresponding to I clusters of eigenvalues could not be
orthogonalized due to insufficient workspace. The indices of
the clusters are stored in the array ICLUSTR.