subroutine ssbevx_2stage (JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO)
SSBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
SSBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
Purpose:
SSBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A using the 2stage technique for the reduction to tridiagonal. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.
Parameters:
JOBZ is CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
Not available in this release.
RANGE
RANGE is CHARACTER*1
= 'A': all eigenvalues will be found;
= 'V': all eigenvalues in the half-open interval (VL,VU]
will be found;
= 'I': the IL-th through IU-th eigenvalues will be found.
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
KD
KD is INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KD >= 0.
AB
AB is REAL array, dimension (LDAB, N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
On exit, AB is overwritten by values generated during the
reduction to tridiagonal form. If UPLO = 'U', the first
superdiagonal and the diagonal of the tridiagonal matrix T
are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
the diagonal and first subdiagonal of T are returned in the
first two rows of AB.
LDAB
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD + 1.
Q
Q is REAL array, dimension (LDQ, N)
If JOBZ = 'V', the N-by-N orthogonal matrix used in the
reduction to tridiagonal form.
If JOBZ = 'N', the array Q is not referenced.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. If JOBZ = 'V', then
LDQ >= max(1,N).
VL
VL is REAL
If RANGE='V', the lower bound of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = 'A' or 'I'.
VU
VU is REAL
If RANGE='V', the upper bound of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = 'A' or 'I'.
IL
IL is INTEGER
If RANGE='I', the index of the
smallest eigenvalue to be returned.
1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
Not referenced if RANGE = 'A' or 'V'.
IU
IU is INTEGER
If RANGE='I', the index of the
largest eigenvalue to be returned.
1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
Not referenced if RANGE = 'A' or 'V'.
ABSTOL
ABSTOL is REAL
The absolute error tolerance for the eigenvalues.
An approximate 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*|T| will be used in its place,
where |T| is the 1-norm of the tridiagonal matrix obtained
by reducing AB to tridiagonal form.
Eigenvalues will be computed most accurately when ABSTOL is
set to twice the underflow threshold 2*SLAMCH('S'), not zero.
If this routine returns with INFO>0, indicating that some
eigenvectors did not converge, try setting ABSTOL to
2*SLAMCH('S').
See "Computing Small Singular Values of Bidiagonal Matrices
with Guaranteed High Relative Accuracy," by Demmel and
Kahan, LAPACK Working Note #3.
M
M is INTEGER
The total number of eigenvalues found. 0 <= M <= N.
If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
W
W is REAL array, dimension (N)
The first M elements contain the selected eigenvalues in
ascending order.
Z
Z is REAL array, dimension (LDZ, max(1,M))
If JOBZ = 'V', then if INFO = 0, the first M columns of Z
contain the orthonormal eigenvectors of the matrix A
corresponding to the selected eigenvalues, with the i-th
column of Z holding the eigenvector associated with W(i).
If an eigenvector fails to converge, then that column of Z
contains the latest approximation to the eigenvector, and the
index of the eigenvector is returned in IFAIL.
If JOBZ = 'N', then Z is not referenced.
Note: the user must ensure that at least max(1,M) columns are
supplied in the array Z; if RANGE = 'V', the exact value of M
is not known in advance and an upper bound must be used.
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1, and if
JOBZ = 'V', LDZ >= max(1,N).
WORK
WORK is REAL array, dimension (LWORK)
LWORK
LWORK is INTEGER
The length of the array WORK. LWORK >= 1, when N <= 1;
otherwise
If JOBZ = 'N' and N > 1, LWORK must be queried.
LWORK = MAX(1, 7*N, dimension) where
dimension = (2KD+1)*N + KD*NTHREADS + 2*N
where KD is the size of the band.
NTHREADS is the number of threads used when
openMP compilation is enabled, otherwise =1.
If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
IWORK
IWORK is INTEGER array, dimension (5*N)
IFAIL
IFAIL is INTEGER array, dimension (N)
If JOBZ = 'V', then if INFO = 0, the first M elements of
IFAIL are zero. If INFO > 0, then IFAIL contains the
indices of the eigenvectors that failed to converge.
If JOBZ = 'N', then IFAIL is not referenced.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, then i eigenvectors failed to converge.
Their indices are stored in array IFAIL.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Further Details:
All details about the 2stage techniques are available in: Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394 A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292 A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196
Definition at line 324 of file ssbevx_2stage.f.
Generated automatically by Doxygen for LAPACK from the source code.