PDSYEVX(l) ) PDSYEVX(l)NAME
PDSYEVX - compute selected eigenvalues and, optionally, eigenvectors of
a real symmetric matrix A by calling the recommended sequence of ScaLA‐
PACK routines
SYNOPSIS
SUBROUTINE PDSYEVX( JOBZ, RANGE, UPLO, N, A, IA, JA, DESCA, VL, VU, IL,
IU, ABSTOL, M, NZ, W, ORFAC, Z, IZ, JZ, DESCZ,
WORK, LWORK, IWORK, LIWORK, IFAIL, ICLUSTR, GAP,
INFO )
CHARACTER JOBZ, RANGE, UPLO
INTEGER IA, IL, INFO, IU, IZ, JA, JZ, LIWORK, LWORK, M, N,
NZ
DOUBLE PRECISION ABSTOL, ORFAC, VL, VU
INTEGER DESCA( * ), DESCZ( * ), ICLUSTR( * ), IFAIL( * ),
IWORK( * )
DOUBLE PRECISION A( * ), GAP( * ), W( * ), WORK( * ), Z( *
)
PURPOSE
PDSYEVX computes selected eigenvalues and, optionally, eigenvectors of
a real symmetric matrix A by calling the recommended sequence of ScaLA‐
PACK routines. Eigenvalues/vectors can be selected by specifying a
range of values or a range of indices for the desired eigenvalues.
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
PDSYEVX assumes IEEE 754 standard compliant arithmetic. To port to a
system which does not have IEEE 754 arithmetic, modify the appropriate
SLmake.inc file to include the compiler switch -DNO_IEEE. This switch
only affects the compilation of pdlaiect.c.
ARGUMENTS
NP = the number of rows local to a given process. NQ = the number of
columns local to a given process.
JOBZ (global input) CHARACTER*1
Specifies whether or not to compute the eigenvectors:
= 'N': Compute eigenvalues only.
= 'V': Compute eigenvalues and eigenvectors.
RANGE (global input) CHARACTER*1
= 'A': all eigenvalues will be found.
= 'V': all eigenvalues in the interval [VL,VU] will be found.
= 'I': the IL-th through IU-th eigenvalues will be found.
UPLO (global input) CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N (global input) INTEGER
The number of rows and columns of the matrix A. N >= 0.
A (local input/workspace) block cyclic DOUBLE PRECISION array,
global dimension (N, N), local dimension ( LLD_A, LOCc(JA+N-1)
)
On entry, the symmetric matrix A. If UPLO = 'U', only the
upper triangular part of A is used to define the elements of
the symmetric matrix. If UPLO = 'L', only the lower triangular
part of A is used to define the elements of the symmetric
matrix.
On exit, the lower triangle (if UPLO='L') or the upper triangle
(if UPLO='U') of A, including the diagonal, is destroyed.
IA (global input) INTEGER
A's global row index, which points to the beginning of the sub‐
matrix which is to be operated on.
JA (global input) INTEGER
A's global column index, which points to the beginning of the
submatrix which is to be operated on.
DESCA (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix A. If DESCA(
CTXT_ ) is incorrect, PDSYEVX cannot guarantee correct error
reporting.
VL (global input) DOUBLE PRECISION
If RANGE='V', the lower bound of the interval to be searched
for eigenvalues. Not referenced if RANGE = 'A' or 'I'.
VU (global input) DOUBLE PRECISION
If RANGE='V', the upper bound of the interval to be searched
for eigenvalues. Not referenced if RANGE = 'A' or 'I'.
IL (global input) INTEGER
If RANGE='I', the index (from smallest to largest) of the
smallest eigenvalue to be returned. IL >= 1. Not referenced
if RANGE = 'A' or 'V'.
IU (global input) INTEGER
If RANGE='I', the index (from smallest to largest) of the
largest eigenvalue to be returned. min(IL,N) <= IU <= N. Not
referenced if RANGE = 'A' or 'V'.
ABSTOL (global input) DOUBLE PRECISION
If JOBZ='V', setting ABSTOL to PDLAMCH( CONTEXT, 'U') yields
the most orthogonal eigenvectors.
The absolute error tolerance for the eigenvalues. An approxi‐
mate eigenvalue is accepted as converged when it is determined
to lie in an interval [a,b] of width less than or equal to
ABSTOL + EPS * max( |a|,|b| ) ,
where EPS is the machine precision. If ABSTOL is less than or
equal to zero, then EPS*norm(T) will be used in its place,
where norm(T) is the 1-norm of the tridiagonal matrix obtained
by reducing A to tridiagonal form.
Eigenvalues will be computed most accurately when ABSTOL is set
to twice the underflow threshold 2*PDLAMCH('S') not zero. If
this routine returns with ((MOD(INFO,2).NE.0) .OR.
(MOD(INFO/8,2).NE.0)), indicating that some eigenvalues or
eigenvectors did not converge, try setting ABSTOL to
2*PDLAMCH('S').
See "Computing Small Singular Values of Bidiagonal Matrices
with Guaranteed High Relative Accuracy," by Demmel and Kahan,
LAPACK Working Note #3.
See "On the correctness of Parallel Bisection in Floating
Point" by Demmel, Dhillon and Ren, LAPACK Working Note #70
M (global output) INTEGER
Total number of eigenvalues found. 0 <= M <= N.
NZ (global output) INTEGER
Total number of eigenvectors computed. 0 <= NZ <= M. The num‐
ber of columns of Z that are filled. If JOBZ .NE. 'V', NZ is
not referenced. If JOBZ .EQ. 'V', NZ = M unless the user sup‐
plies insufficient space and PDSYEVX is not able to detect this
before beginning computation. To get all the eigenvectors
requested, the user must supply both sufficient space to hold
the eigenvectors in Z (M .LE. DESCZ(N_)) and sufficient
workspace to compute them. (See LWORK below.) PDSYEVX is
always able to detect insufficient space without computation
unless RANGE .EQ. 'V'.
W (global output) DOUBLE PRECISION array, dimension (N)
On normal exit, the first M entries contain the selected eigen‐
values in ascending order.
ORFAC (global input) DOUBLE PRECISION
Specifies which eigenvectors should be reorthogonalized.
Eigenvectors that correspond to eigenvalues which are within
tol=ORFAC*norm(A) of each other are to be reorthogonalized.
However, if the workspace is insufficient (see LWORK), tol may
be decreased until all eigenvectors to be reorthogonalized can
be stored in one process. No reorthogonalization 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) DOUBLE PRECISION array,
global dimension (N, N), local dimension ( LLD_Z, LOCc(JZ+N-1)
) If JOBZ = 'V', then on normal exit the first M columns of Z
contain the orthonormal eigenvectors of the matrix correspond‐
ing to the selected eigenvalues. If an eigenvector fails to
converge, then that column of Z contains the latest approxima‐
tion to the eigenvector, and the index of the eigenvector is
returned in IFAIL. If JOBZ = 'N', then Z is not referenced.
IZ (global input) INTEGER
Z's global row index, which points to the beginning of the sub‐
matrix 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. DESCZ(
CTXT_ ) must equal DESCA( CTXT_ )
WORK (local workspace/output) DOUBLE PRECISION array,
dimension (LWORK) On return, WROK(1) contains the optimal
amount of workspace required for efficient execution. if
JOBZ='N' WORK(1) = optimal amount of workspace required to com‐
pute eigenvalues efficiently if JOBZ='V' WORK(1) = optimal
amount of workspace required to compute eigenvalues and eigen‐
vectors efficiently with no guarantee on orthogonality. If
RANGE='V', it is assumed that all eigenvectors may be required.
LWORK (local input) INTEGER
Size of WORK See below for definitions of variables used to
define LWORK. If no eigenvectors are requested (JOBZ = 'N')
then LWORK >= 5 * N + MAX( 5 * NN, NB * ( NP0 + 1 ) ) If eigen‐
vectors are requested (JOBZ = 'V' ) then the amount of
workspace required to guarantee that all eigenvectors are com‐
puted is: LWORK >= 5*N + MAX( 5*NN, NP0 * MQ0 + 2 * NB * NB ) +
ICEIL( NEIG, NPROW*NPCOL)*NN
The computed eigenvectors may not be orthogonal if the minimal
workspace is supplied and ORFAC is too small. If you want to
guarantee orthogonality (at the cost of potentially poor per‐
formance) you should add the following to LWORK: (CLUSTER‐
SIZE-1)*N where CLUSTERSIZE is the number of eigenvalues in the
largest cluster, where a cluster is defined as a set of close
eigenvalues: { W(K),...,W(K+CLUSTERSIZE-1) | W(J+1) <= W(J) +
ORFAC*2*norm(A) } Variable definitions: NEIG = number of eigen‐
vectors requested NB = DESCA( MB_ ) = DESCA( NB_ ) = DESCZ( MB_
) = DESCZ( NB_ ) NN = MAX( N, NB, 2 ) DESCA( RSRC_ ) = DESCA(
NB_ ) = DESCZ( RSRC_ ) = DESCZ( CSRC_ ) = 0 NP0 = NUMROC( NN,
NB, 0, 0, NPROW ) MQ0 = NUMROC( MAX( NEIG, NB, 2 ), NB, 0, 0,
NPCOL ) ICEIL( X, Y ) is a ScaLAPACK function returning ceil‐
ing(X/Y)
When LWORK is too small: If LWORK is too small to guarantee
orthogonality, PDSYEVX attempts to maintain orthogonality in
the clusters with the smallest spacing between the eigenvalues.
If LWORK is too small to compute all the eigenvectors
requested, no computation is performed and INFO=-23 is
returned. Note that when RANGE='V', PDSYEVX does not know how
many eigenvectors are requested until the eigenvalues are com‐
puted. Therefore, when RANGE='V' and as long as LWORK is large
enough to allow PDSYEVX to compute the eigenvalues, PDSYEVX
will compute the eigenvalues and as many eigenvectors as it
can.
Relationship between workspace, orthogonality & performance:
Greater performance can be achieved if adequate workspace is
provided. On the other hand, in some situations, performance
can decrease as the workspace provided increases above the
workspace amount shown below:
For optimal performance, greater workspace may be needed, i.e.
LWORK >= MAX( LWORK, 5*N + NSYTRD_LWOPT ) Where: LWORK, as
defined previously, depends upon the number of eigenvectors
requested, and NSYTRD_LWOPT = N + 2*( ANB+1 )*( 4*NPS+2 ) + (
NPS + 3 ) * NPS
ANB = PJLAENV( DESCA( CTXT_), 3, 'PDSYTTRD', 'L', 0, 0, 0, 0)
SQNPC = INT( SQRT( DBLE( NPROW * NPCOL ) ) ) NPS = MAX( NUMROC(
N, 1, 0, 0, SQNPC ), 2*ANB )
NUMROC is a ScaLAPACK tool functions; PJLAENV is a ScaLAPACK
envionmental inquiry function MYROW, MYCOL, NPROW and NPCOL can
be determined by calling the subroutine BLACS_GRIDINFO.
For large N, no extra workspace is needed, however the biggest
boost in performance comes for small N, so it is wise to pro‐
vide the extra workspace (typically less than a Megabyte per
process).
If CLUSTERSIZE >= N/SQRT(NPROW*NPCOL), then providing enough
space to compute all the eigenvectors orthogonally will cause
serious degradation in performance. In the limit (i.e. CLUSTER‐
SIZE = N-1) PDSTEIN will perform no better than DSTEIN on 1
processor. For CLUSTERSIZE = N/SQRT(NPROW*NPCOL) reorthogonal‐
izing all eigenvectors will increase the total execution time
by a factor of 2 or more. For CLUSTERSIZE >
N/SQRT(NPROW*NPCOL) execution time will grow as the square of
the cluster size, all other factors remaining equal and assum‐
ing enough workspace. Less workspace means less reorthogonal‐
ization but faster execution.
If LWORK = -1, then LWORK is global input and a workspace query
is assumed; the routine only calculates the size required for
optimal performance for all work arrays. Each of these values
is returned in the first entry of the corresponding work
arrays, and no error message is issued by PXERBLA.
IWORK (local workspace) INTEGER array
On return, IWORK(1) contains the amount of integer workspace
required.
LIWORK (local input) INTEGER
size of IWORK LIWORK >= 6 * NNP Where: NNP = MAX( N,
NPROW*NPCOL + 1, 4 ) If LIWORK = -1, then LIWORK is global
input and a workspace query is assumed; the routine only calcu‐
lates the minimum and optimal size for all work arrays. Each of
these values is returned in the first entry of the correspond‐
ing work array, and no error message is issued by PXERBLA.
IFAIL (global output) INTEGER array, dimension (N)
If JOBZ = 'V', then on normal exit, the first M elements of
IFAIL are zero. If (MOD(INFO,2).NE.0) on exit, then IFAIL con‐
tains the indices of the eigenvectors that failed to converge.
If JOBZ = 'N', then IFAIL is not referenced.
ICLUSTR (global output) integer array, dimension
(2*NPROW*NPCOL) This array contains indices of eigenvectors
corresponding to a cluster of eigenvalues that could not be
reorthogonalized due to insufficient workspace (see LWORK,
ORFAC and INFO). Eigenvectors corresponding to clusters of ei‐
genvalues indexed ICLUSTR(2*I-1) to ICLUSTR(2*I), could not be
reorthogonalized due to lack of workspace. Hence the eigenvec‐
tors 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
ICLUSTR is not referenced if JOBZ = 'N'
GAP (global output) DOUBLE PRECISION array,
dimension (NPROW*NPCOL) This array contains the gap between ei‐
genvalues whose eigenvectors could not be reorthogonalized. The
output values in this array correspond to the clusters indi‐
cated by the array ICLUSTR. As a result, the dot product
between eigenvectors correspoding to the I^th cluster may be as
high as ( C * n ) / GAP(I) where C is a small constant.
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
(MOD(INFO,2).NE.0), then one or more eigenvectors failed to
converge. Their indices are stored in IFAIL. Ensure
ABSTOL=2.0*PDLAMCH( 'U' ) Send e-mail to scalapack@cs.utk.edu
if (MOD(INFO/2,2).NE.0),then eigenvectors corresponding to one
or more clusters of eigenvalues could not be reorthogonalized
because of insufficient workspace. The indices of the clusters
are stored in the array ICLUSTR. if (MOD(INFO/4,2).NE.0), then
space limit prevented PDSYEVX from computing all of the eigen‐
vectors between VL and VU. The number of eigenvectors computed
is returned in NZ. if (MOD(INFO/8,2).NE.0), then PDSTEBZ
failed to compute eigenvalues. Ensure ABSTOL=2.0*PDLAMCH( 'U'
) Send e-mail to scalapack@cs.utk.edu
Alignment requirements ======================
The distributed submatrices A(IA:*, JA:*) and
C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties,
namely the following expressions should be true:
( MB_A.EQ.NB_A.EQ.MB_Z .AND. IROFFA.EQ.IROFFZ .AND. IROFFA.EQ.0
.AND. IAROW.EQ.IZROW ) where IROFFA = MOD( IA-1, MB_A ) and
ICOFFA = MOD( JA-1, NB_A ).
ScaLAPACK version 1.7 13 August 2001 PDSYEVX(l)
