Provided by: liblapack-doc_3.11.0-2_all 
NAME
doubleSYeigen - double
SYNOPSIS
Functions
subroutine dsyev (JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, INFO)
DSYEV computes the eigenvalues and, optionally, the left and/or right eigenvectors
for SY matrices
subroutine dsyev_2stage (JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, INFO)
DSYEV_2STAGE computes the eigenvalues and, optionally, the left and/or right
eigenvectors for SY matrices
subroutine dsyevd (JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK, LIWORK, INFO)
DSYEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors
for SY matrices
subroutine dsyevd_2stage (JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK, LIWORK, INFO)
DSYEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right
eigenvectors for SY matrices
subroutine dsyevr (JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ,
ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)
DSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors
for SY matrices
subroutine dsyevr_2stage (JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z,
LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)
DSYEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right
eigenvectors for SY matrices
subroutine dsyevx (JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ,
WORK, LWORK, IWORK, IFAIL, INFO)
DSYEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors
for SY matrices
subroutine dsyevx_2stage (JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z,
LDZ, WORK, LWORK, IWORK, IFAIL, INFO)
DSYEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right
eigenvectors for SY matrices
subroutine dsygv (ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK, LWORK, INFO)
DSYGV
subroutine dsygv_2stage (ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK, LWORK, INFO)
DSYGV_2STAGE
subroutine dsygvd (ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK, LWORK, IWORK, LIWORK,
INFO)
DSYGVD
subroutine dsygvx (ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, VL, VU, IL, IU, ABSTOL, M,
W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO)
DSYGVX
Detailed Description
This is the group of double eigenvalue driver functions for SY matrices
Function Documentation
subroutine dsyev (character JOBZ, character UPLO, integer N, double precision, dimension( lda,
* ) A, integer LDA, double precision, dimension( * ) W, double precision, dimension( * )
WORK, integer LWORK, integer INFO)
DSYEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY
matrices
Purpose:
DSYEV computes all eigenvalues and, optionally, eigenvectors of a
real symmetric matrix A.
Parameters
JOBZ
JOBZ is CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
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.
A
A is DOUBLE PRECISION array, dimension (LDA, N)
On entry, the symmetric matrix A. If UPLO = 'U', the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If UPLO = 'L',
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, if JOBZ = 'V', then if INFO = 0, A contains the
orthonormal eigenvectors of the matrix A.
If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')
or the upper triangle (if UPLO='U') of A, including the
diagonal, is destroyed.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
W
W is DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The length of the array WORK. LWORK >= max(1,3*N-1).
For optimal efficiency, LWORK >= (NB+2)*N,
where NB is the blocksize for DSYTRD returned by ILAENV.
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.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the algorithm failed to converge; i
off-diagonal elements of an intermediate tridiagonal
form did not converge to zero.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine dsyev_2stage (character JOBZ, character UPLO, integer N, double precision,
dimension( lda, * ) A, integer LDA, double precision, dimension( * ) W, double precision,
dimension( * ) WORK, integer LWORK, integer INFO)
DSYEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors
for SY matrices
Purpose:
DSYEV_2STAGE computes all eigenvalues and, optionally, eigenvectors of a
real symmetric matrix A using the 2stage technique for
the reduction to tridiagonal.
Parameters
JOBZ
JOBZ is CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
Not available in this release.
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.
A
A is DOUBLE PRECISION array, dimension (LDA, N)
On entry, the symmetric matrix A. If UPLO = 'U', the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If UPLO = 'L',
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, if JOBZ = 'V', then if INFO = 0, A contains the
orthonormal eigenvectors of the matrix A.
If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')
or the upper triangle (if UPLO='U') of A, including the
diagonal, is destroyed.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
W
W is DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
WORK
WORK is DOUBLE PRECISION array, dimension LWORK
On exit, if INFO = 0, WORK(1) returns the optimal 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, dimension) where
dimension = max(stage1,stage2) + (KD+1)*N + 2*N
= N*KD + N*max(KD+1,FACTOPTNB)
+ max(2*KD*KD, KD*NTHREADS)
+ (KD+1)*N + 2*N
where KD is the blocking size of the reduction,
FACTOPTNB is the blocking used by the QR or LQ
algorithm, usually FACTOPTNB=128 is a good choice
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.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the algorithm failed to converge; i
off-diagonal elements of an intermediate tridiagonal
form did not converge to zero.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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
subroutine dsyevd (character JOBZ, character UPLO, integer N, double precision, dimension(
lda, * ) A, integer LDA, double precision, dimension( * ) W, double precision, dimension(
* ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
DSYEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for
SY matrices
Purpose:
DSYEVD computes all eigenvalues and, optionally, eigenvectors of a
real symmetric matrix A. If eigenvectors are desired, it uses a
divide and conquer algorithm.
The divide and conquer algorithm makes very mild assumptions about
floating point arithmetic. It will work on machines with a guard
digit in add/subtract, or on those binary machines without guard
digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
Cray-2. It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none.
Because of large use of BLAS of level 3, DSYEVD needs N**2 more
workspace than DSYEVX.
Parameters
JOBZ
JOBZ is CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
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.
A
A is DOUBLE PRECISION array, dimension (LDA, N)
On entry, the symmetric matrix A. If UPLO = 'U', the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If UPLO = 'L',
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, if JOBZ = 'V', then if INFO = 0, A contains the
orthonormal eigenvectors of the matrix A.
If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')
or the upper triangle (if UPLO='U') of A, including the
diagonal, is destroyed.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
W
W is DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
WORK
WORK is DOUBLE PRECISION array,
dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK.
If N <= 1, LWORK must be at least 1.
If JOBZ = 'N' and N > 1, LWORK must be at least 2*N+1.
If JOBZ = 'V' and N > 1, LWORK must be at least
1 + 6*N + 2*N**2.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal sizes of the WORK and IWORK
arrays, returns these values as the first entries of the WORK
and IWORK arrays, and no error message related to LWORK or
LIWORK is issued by XERBLA.
IWORK
IWORK is INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK
LIWORK is INTEGER
The dimension of the array IWORK.
If N <= 1, LIWORK must be at least 1.
If JOBZ = 'N' and N > 1, LIWORK must be at least 1.
If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N.
If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal sizes of the WORK and
IWORK arrays, returns these values as the first entries of
the WORK and IWORK arrays, and no error message related to
LWORK or LIWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i and JOBZ = 'N', then the algorithm failed
to converge; i off-diagonal elements of an intermediate
tridiagonal form did not converge to zero;
if INFO = i and JOBZ = 'V', then the algorithm failed
to compute an eigenvalue while working on the submatrix
lying in rows and columns INFO/(N+1) through
mod(INFO,N+1).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee
Modified description of INFO. Sven, 16 Feb 05.
subroutine dsyevd_2stage (character JOBZ, character UPLO, integer N, double precision,
dimension( lda, * ) A, integer LDA, double precision, dimension( * ) W, double precision,
dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer
INFO)
DSYEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right
eigenvectors for SY matrices
Purpose:
DSYEVD_2STAGE computes all eigenvalues and, optionally, eigenvectors of a
real symmetric matrix A using the 2stage technique for
the reduction to tridiagonal. If eigenvectors are desired, it uses a
divide and conquer algorithm.
The divide and conquer algorithm makes very mild assumptions about
floating point arithmetic. It will work on machines with a guard
digit in add/subtract, or on those binary machines without guard
digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
Cray-2. It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none.
Parameters
JOBZ
JOBZ is CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
Not available in this release.
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.
A
A is DOUBLE PRECISION array, dimension (LDA, N)
On entry, the symmetric matrix A. If UPLO = 'U', the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If UPLO = 'L',
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, if JOBZ = 'V', then if INFO = 0, A contains the
orthonormal eigenvectors of the matrix A.
If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')
or the upper triangle (if UPLO='U') of A, including the
diagonal, is destroyed.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
W
W is DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
WORK
WORK is DOUBLE PRECISION array,
dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK.
If N <= 1, LWORK must be at least 1.
If JOBZ = 'N' and N > 1, LWORK must be queried.
LWORK = MAX(1, dimension) where
dimension = max(stage1,stage2) + (KD+1)*N + 2*N+1
= N*KD + N*max(KD+1,FACTOPTNB)
+ max(2*KD*KD, KD*NTHREADS)
+ (KD+1)*N + 2*N+1
where KD is the blocking size of the reduction,
FACTOPTNB is the blocking used by the QR or LQ
algorithm, usually FACTOPTNB=128 is a good choice
NTHREADS is the number of threads used when
openMP compilation is enabled, otherwise =1.
If JOBZ = 'V' and N > 1, LWORK must be at least
1 + 6*N + 2*N**2.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal sizes of the WORK and IWORK
arrays, returns these values as the first entries of the WORK
and IWORK arrays, and no error message related to LWORK or
LIWORK is issued by XERBLA.
IWORK
IWORK is INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK
LIWORK is INTEGER
The dimension of the array IWORK.
If N <= 1, LIWORK must be at least 1.
If JOBZ = 'N' and N > 1, LIWORK must be at least 1.
If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N.
If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal sizes of the WORK and
IWORK arrays, returns these values as the first entries of
the WORK and IWORK arrays, and no error message related to
LWORK or LIWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i and JOBZ = 'N', then the algorithm failed
to converge; i off-diagonal elements of an intermediate
tridiagonal form did not converge to zero;
if INFO = i and JOBZ = 'V', then the algorithm failed
to compute an eigenvalue while working on the submatrix
lying in rows and columns INFO/(N+1) through
mod(INFO,N+1).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee
Modified description of INFO. Sven, 16 Feb 05.
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
subroutine dsyevr (character JOBZ, character RANGE, character UPLO, integer N, double
precision, dimension( lda, * ) A, integer LDA, double precision VL, double precision VU,
integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( *
) W, double precision, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ,
double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK,
integer LIWORK, integer INFO)
DSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for
SY matrices
Purpose:
DSYEVR computes selected eigenvalues and, optionally, eigenvectors
of a real symmetric matrix A. Eigenvalues and eigenvectors can be
selected by specifying either a range of values or a range of
indices for the desired eigenvalues.
DSYEVR first reduces the matrix A to tridiagonal form T with a call
to DSYTRD. Then, whenever possible, DSYEVR calls DSTEMR to compute
the eigenspectrum using Relatively Robust Representations. DSTEMR
computes eigenvalues by the dqds algorithm, while orthogonal
eigenvectors are computed from various "good" L D L^T representations
(also known as Relatively Robust Representations). Gram-Schmidt
orthogonalization is avoided as far as possible. More specifically,
the various steps of the algorithm are as follows.
For each unreduced block (submatrix) of T,
(a) Compute T - sigma I = L D L^T, so that L and D
define all the wanted eigenvalues to high relative accuracy.
This means that small relative changes in the entries of D and L
cause only small relative changes in the eigenvalues and
eigenvectors. The standard (unfactored) representation of the
tridiagonal matrix T does not have this property in general.
(b) Compute the eigenvalues to suitable accuracy.
If the eigenvectors are desired, the algorithm attains full
accuracy of the computed eigenvalues only right before
the corresponding vectors have to be computed, see steps c) and d).
(c) For each cluster of close eigenvalues, select a new
shift close to the cluster, find a new factorization, and refine
the shifted eigenvalues to suitable accuracy.
(d) For each eigenvalue with a large enough relative separation compute
the corresponding eigenvector by forming a rank revealing twisted
factorization. Go back to (c) for any clusters that remain.
The desired accuracy of the output can be specified by the input
parameter ABSTOL.
For more details, see DSTEMR's documentation and:
- Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
- Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
2004. Also LAPACK Working Note 154.
- Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
tridiagonal eigenvalue/eigenvector problem",
Computer Science Division Technical Report No. UCB/CSD-97-971,
UC Berkeley, May 1997.
Note 1 : DSYEVR calls DSTEMR when the full spectrum is requested
on machines which conform to the ieee-754 floating point standard.
DSYEVR calls DSTEBZ and DSTEIN on non-ieee machines and
when partial spectrum requests are made.
Normal execution of DSTEMR may create NaNs and infinities and
hence may abort due to a floating point exception in environments
which do not handle NaNs and infinities in the ieee standard default
manner.
Parameters
JOBZ
JOBZ is CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
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.
For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
DSTEIN are called
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.
A
A is DOUBLE PRECISION array, dimension (LDA, N)
On entry, the symmetric matrix A. If UPLO = 'U', the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If UPLO = 'L',
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, the lower triangle (if UPLO='L') or the upper
triangle (if UPLO='U') of A, including the diagonal, is
destroyed.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
VL
VL is DOUBLE PRECISION
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 DOUBLE PRECISION
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 DOUBLE PRECISION
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 A to tridiagonal form.
See "Computing Small Singular Values of Bidiagonal Matrices
with Guaranteed High Relative Accuracy," by Demmel and
Kahan, LAPACK Working Note #3.
If high relative accuracy is important, set ABSTOL to
DLAMCH( 'Safe minimum' ). Doing so will guarantee that
eigenvalues are computed to high relative accuracy when
possible in future releases. The current code does not
make any guarantees about high relative accuracy, but
future releases will. See J. Barlow and J. Demmel,
"Computing Accurate Eigensystems of Scaled Diagonally
Dominant Matrices", LAPACK Working Note #7, for a discussion
of which matrices define their eigenvalues to high relative
accuracy.
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 DOUBLE PRECISION array, dimension (N)
The first M elements contain the selected eigenvalues in
ascending order.
Z
Z is DOUBLE PRECISION 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 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.
Supplying N columns is always safe.
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1, and if
JOBZ = 'V', LDZ >= max(1,N).
ISUPPZ
ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
The support of the eigenvectors in Z, i.e., the indices
indicating the nonzero elements in Z. The i-th eigenvector
is nonzero only in elements ISUPPZ( 2*i-1 ) through
ISUPPZ( 2*i ). This is an output of DSTEMR (tridiagonal
matrix). The support of the eigenvectors of A is typically
1:N because of the orthogonal transformations applied by DORMTR.
Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,26*N).
For optimal efficiency, LWORK >= (NB+6)*N,
where NB is the max of the blocksize for DSYTRD and DORMTR
returned by ILAENV.
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 (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LWORK.
LIWORK
LIWORK is INTEGER
The dimension of the array IWORK. LIWORK >= max(1,10*N).
If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the IWORK array,
returns this value as the first entry of the IWORK array, and
no error message related to LIWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: Internal error
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Ken Stanley, Computer Science Division, University of California at Berkeley, USA
Jason Riedy, Computer Science Division, University of California at Berkeley, USA
subroutine dsyevr_2stage (character JOBZ, character RANGE, character UPLO, integer N, double
precision, dimension( lda, * ) A, integer LDA, double precision VL, double precision VU,
integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( *
) W, double precision, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ,
double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK,
integer LIWORK, integer INFO)
DSYEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right
eigenvectors for SY matrices
Purpose:
DSYEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors
of a real symmetric 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.
DSYEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call
to DSYTRD. Then, whenever possible, DSYEVR_2STAGE calls DSTEMR to compute
the eigenspectrum using Relatively Robust Representations. DSTEMR
computes eigenvalues by the dqds algorithm, while orthogonal
eigenvectors are computed from various "good" L D L^T representations
(also known as Relatively Robust Representations). Gram-Schmidt
orthogonalization is avoided as far as possible. More specifically,
the various steps of the algorithm are as follows.
For each unreduced block (submatrix) of T,
(a) Compute T - sigma I = L D L^T, so that L and D
define all the wanted eigenvalues to high relative accuracy.
This means that small relative changes in the entries of D and L
cause only small relative changes in the eigenvalues and
eigenvectors. The standard (unfactored) representation of the
tridiagonal matrix T does not have this property in general.
(b) Compute the eigenvalues to suitable accuracy.
If the eigenvectors are desired, the algorithm attains full
accuracy of the computed eigenvalues only right before
the corresponding vectors have to be computed, see steps c) and d).
(c) For each cluster of close eigenvalues, select a new
shift close to the cluster, find a new factorization, and refine
the shifted eigenvalues to suitable accuracy.
(d) For each eigenvalue with a large enough relative separation compute
the corresponding eigenvector by forming a rank revealing twisted
factorization. Go back to (c) for any clusters that remain.
The desired accuracy of the output can be specified by the input
parameter ABSTOL.
For more details, see DSTEMR's documentation and:
- Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
- Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
2004. Also LAPACK Working Note 154.
- Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
tridiagonal eigenvalue/eigenvector problem",
Computer Science Division Technical Report No. UCB/CSD-97-971,
UC Berkeley, May 1997.
Note 1 : DSYEVR_2STAGE calls DSTEMR when the full spectrum is requested
on machines which conform to the ieee-754 floating point standard.
DSYEVR_2STAGE calls DSTEBZ and SSTEIN on non-ieee machines and
when partial spectrum requests are made.
Normal execution of DSTEMR may create NaNs and infinities and
hence may abort due to a floating point exception in environments
which do not handle NaNs and infinities in the ieee standard default
manner.
Parameters
JOBZ
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.
For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
DSTEIN are called
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.
A
A is DOUBLE PRECISION array, dimension (LDA, N)
On entry, the symmetric matrix A. If UPLO = 'U', the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If UPLO = 'L',
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, the lower triangle (if UPLO='L') or the upper
triangle (if UPLO='U') of A, including the diagonal, is
destroyed.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
VL
VL is DOUBLE PRECISION
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 DOUBLE PRECISION
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 DOUBLE PRECISION
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 A to tridiagonal form.
See "Computing Small Singular Values of Bidiagonal Matrices
with Guaranteed High Relative Accuracy," by Demmel and
Kahan, LAPACK Working Note #3.
If high relative accuracy is important, set ABSTOL to
DLAMCH( 'Safe minimum' ). Doing so will guarantee that
eigenvalues are computed to high relative accuracy when
possible in future releases. The current code does not
make any guarantees about high relative accuracy, but
future releases will. See J. Barlow and J. Demmel,
"Computing Accurate Eigensystems of Scaled Diagonally
Dominant Matrices", LAPACK Working Note #7, for a discussion
of which matrices define their eigenvalues to high relative
accuracy.
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 DOUBLE PRECISION array, dimension (N)
The first M elements contain the selected eigenvalues in
ascending order.
Z
Z is DOUBLE PRECISION 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 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.
Supplying N columns is always safe.
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1, and if
JOBZ = 'V', LDZ >= max(1,N).
ISUPPZ
ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
The support of the eigenvectors in Z, i.e., the indices
indicating the nonzero elements in Z. The i-th eigenvector
is nonzero only in elements ISUPPZ( 2*i-1 ) through
ISUPPZ( 2*i ). This is an output of DSTEMR (tridiagonal
matrix). The support of the eigenvectors of A is typically
1:N because of the orthogonal transformations applied by DORMTR.
Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK.
If JOBZ = 'N' and N > 1, LWORK must be queried.
LWORK = MAX(1, 26*N, dimension) where
dimension = max(stage1,stage2) + (KD+1)*N + 5*N
= N*KD + N*max(KD+1,FACTOPTNB)
+ max(2*KD*KD, KD*NTHREADS)
+ (KD+1)*N + 5*N
where KD is the blocking size of the reduction,
FACTOPTNB is the blocking used by the QR or LQ
algorithm, usually FACTOPTNB=128 is a good choice
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 (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LWORK.
LIWORK
LIWORK is INTEGER
The dimension of the array IWORK. LIWORK >= max(1,10*N).
If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the IWORK array,
returns this value as the first entry of the IWORK array, and
no error message related to LIWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: Internal error
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Ken Stanley, Computer Science Division, University of
California at Berkeley, USA
Jason Riedy, Computer Science Division, University of
California at Berkeley, USA
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
subroutine dsyevx (character JOBZ, character RANGE, character UPLO, integer N, double
precision, dimension( lda, * ) A, integer LDA, double precision VL, double precision VU,
integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( *
) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( *
) WORK, integer LWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL,
integer INFO)
DSYEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for
SY matrices
Purpose:
DSYEVX computes selected eigenvalues and, optionally, eigenvectors
of a real symmetric matrix A. Eigenvalues and eigenvectors can be
selected by specifying either a range of values or a range of indices
for the desired eigenvalues.
Parameters
JOBZ
JOBZ is CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
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.
A
A is DOUBLE PRECISION array, dimension (LDA, N)
On entry, the symmetric matrix A. If UPLO = 'U', the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If UPLO = 'L',
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, the lower triangle (if UPLO='L') or the upper
triangle (if UPLO='U') of A, including the diagonal, is
destroyed.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
VL
VL is DOUBLE PRECISION
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 DOUBLE PRECISION
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 DOUBLE PRECISION
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 A to tridiagonal form.
Eigenvalues will be computed most accurately when ABSTOL is
set to twice the underflow threshold 2*DLAMCH('S'), not zero.
If this routine returns with INFO>0, indicating that some
eigenvectors did not converge, try setting ABSTOL to
2*DLAMCH('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 DOUBLE PRECISION array, dimension (N)
On normal exit, the first M elements contain the selected
eigenvalues in ascending order.
Z
Z is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The length of the array WORK. LWORK >= 1, when N <= 1;
otherwise 8*N.
For optimal efficiency, LWORK >= (NB+3)*N,
where NB is the max of the blocksize for DSYTRD and DORMTR
returned by ILAENV.
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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine dsyevx_2stage (character JOBZ, character RANGE, character UPLO, integer N, double
precision, dimension( lda, * ) A, integer LDA, double precision VL, double precision VU,
integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( *
) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( *
) WORK, integer LWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL,
integer INFO)
DSYEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right
eigenvectors for SY matrices
Purpose:
DSYEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors
of a real symmetric 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
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.
A
A is DOUBLE PRECISION array, dimension (LDA, N)
On entry, the symmetric matrix A. If UPLO = 'U', the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If UPLO = 'L',
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, the lower triangle (if UPLO='L') or the upper
triangle (if UPLO='U') of A, including the diagonal, is
destroyed.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
VL
VL is DOUBLE PRECISION
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 DOUBLE PRECISION
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 DOUBLE PRECISION
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 A to tridiagonal form.
Eigenvalues will be computed most accurately when ABSTOL is
set to twice the underflow threshold 2*DLAMCH('S'), not zero.
If this routine returns with INFO>0, indicating that some
eigenvectors did not converge, try setting ABSTOL to
2*DLAMCH('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 DOUBLE PRECISION array, dimension (N)
On normal exit, the first M elements contain the selected
eigenvalues in ascending order.
Z
Z is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal 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, 8*N, dimension) where
dimension = max(stage1,stage2) + (KD+1)*N + 3*N
= N*KD + N*max(KD+1,FACTOPTNB)
+ max(2*KD*KD, KD*NTHREADS)
+ (KD+1)*N + 3*N
where KD is the blocking size of the reduction,
FACTOPTNB is the blocking used by the QR or LQ
algorithm, usually FACTOPTNB=128 is a good choice
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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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
subroutine dsygv (integer ITYPE, character JOBZ, character UPLO, integer N, double precision,
dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB,
double precision, dimension( * ) W, double precision, dimension( * ) WORK, integer LWORK,
integer INFO)
DSYGV
Purpose:
DSYGV computes all the eigenvalues, and optionally, the eigenvectors
of a real generalized symmetric-definite eigenproblem, of the form
A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x.
Here A and B are assumed to be symmetric and B is also
positive definite.
Parameters
ITYPE
ITYPE is INTEGER
Specifies the problem type to be solved:
= 1: A*x = (lambda)*B*x
= 2: A*B*x = (lambda)*x
= 3: B*A*x = (lambda)*x
JOBZ
JOBZ is CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangles of A and B are stored;
= 'L': Lower triangles of A and B are stored.
N
N is INTEGER
The order of the matrices A and B. N >= 0.
A
A is DOUBLE PRECISION array, dimension (LDA, N)
On entry, the symmetric matrix A. If UPLO = 'U', the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If UPLO = 'L',
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, if JOBZ = 'V', then if INFO = 0, A contains the
matrix Z of eigenvectors. The eigenvectors are normalized
as follows:
if ITYPE = 1 or 2, Z**T*B*Z = I;
if ITYPE = 3, Z**T*inv(B)*Z = I.
If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
or the lower triangle (if UPLO='L') of A, including the
diagonal, is destroyed.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B
B is DOUBLE PRECISION array, dimension (LDB, N)
On entry, the symmetric positive definite matrix B.
If UPLO = 'U', the leading N-by-N upper triangular part of B
contains the upper triangular part of the matrix B.
If UPLO = 'L', the leading N-by-N lower triangular part of B
contains the lower triangular part of the matrix B.
On exit, if INFO <= N, the part of B containing the matrix is
overwritten by the triangular factor U or L from the Cholesky
factorization B = U**T*U or B = L*L**T.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
W
W is DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The length of the array WORK. LWORK >= max(1,3*N-1).
For optimal efficiency, LWORK >= (NB+2)*N,
where NB is the blocksize for DSYTRD returned by ILAENV.
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.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: DPOTRF or DSYEV returned an error code:
<= N: if INFO = i, DSYEV failed to converge;
i off-diagonal elements of an intermediate
tridiagonal form did not converge to zero;
> N: if INFO = N + i, for 1 <= i <= N, then the leading
minor of order i of B is not positive definite.
The factorization of B could not be completed and
no eigenvalues or eigenvectors were computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine dsygv_2stage (integer ITYPE, character JOBZ, character UPLO, integer N, double
precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B,
integer LDB, double precision, dimension( * ) W, double precision, dimension( * ) WORK,
integer LWORK, integer INFO)
DSYGV_2STAGE
Purpose:
DSYGV_2STAGE computes all the eigenvalues, and optionally, the eigenvectors
of a real generalized symmetric-definite eigenproblem, of the form
A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x.
Here A and B are assumed to be symmetric and B is also
positive definite.
This routine use the 2stage technique for the reduction to tridiagonal
which showed higher performance on recent architecture and for large
sizes N>2000.
Parameters
ITYPE
ITYPE is INTEGER
Specifies the problem type to be solved:
= 1: A*x = (lambda)*B*x
= 2: A*B*x = (lambda)*x
= 3: B*A*x = (lambda)*x
JOBZ
JOBZ is CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
Not available in this release.
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangles of A and B are stored;
= 'L': Lower triangles of A and B are stored.
N
N is INTEGER
The order of the matrices A and B. N >= 0.
A
A is DOUBLE PRECISION array, dimension (LDA, N)
On entry, the symmetric matrix A. If UPLO = 'U', the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If UPLO = 'L',
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, if JOBZ = 'V', then if INFO = 0, A contains the
matrix Z of eigenvectors. The eigenvectors are normalized
as follows:
if ITYPE = 1 or 2, Z**T*B*Z = I;
if ITYPE = 3, Z**T*inv(B)*Z = I.
If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
or the lower triangle (if UPLO='L') of A, including the
diagonal, is destroyed.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B
B is DOUBLE PRECISION array, dimension (LDB, N)
On entry, the symmetric positive definite matrix B.
If UPLO = 'U', the leading N-by-N upper triangular part of B
contains the upper triangular part of the matrix B.
If UPLO = 'L', the leading N-by-N lower triangular part of B
contains the lower triangular part of the matrix B.
On exit, if INFO <= N, the part of B containing the matrix is
overwritten by the triangular factor U or L from the Cholesky
factorization B = U**T*U or B = L*L**T.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
W
W is DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal 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, dimension) where
dimension = max(stage1,stage2) + (KD+1)*N + 2*N
= N*KD + N*max(KD+1,FACTOPTNB)
+ max(2*KD*KD, KD*NTHREADS)
+ (KD+1)*N + 2*N
where KD is the blocking size of the reduction,
FACTOPTNB is the blocking used by the QR or LQ
algorithm, usually FACTOPTNB=128 is a good choice
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.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: DPOTRF or DSYEV returned an error code:
<= N: if INFO = i, DSYEV failed to converge;
i off-diagonal elements of an intermediate
tridiagonal form did not converge to zero;
> N: if INFO = N + i, for 1 <= i <= N, then the leading
minor of order i of B is not positive definite.
The factorization of B could not be completed and
no eigenvalues or eigenvectors were computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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
subroutine dsygvd (integer ITYPE, character JOBZ, character UPLO, integer N, double precision,
dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB,
double precision, dimension( * ) W, double precision, dimension( * ) WORK, integer LWORK,
integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
DSYGVD
Purpose:
DSYGVD computes all the eigenvalues, and optionally, the eigenvectors
of a real generalized symmetric-definite eigenproblem, of the form
A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and
B are assumed to be symmetric and B is also positive definite.
If eigenvectors are desired, it uses a divide and conquer algorithm.
The divide and conquer algorithm makes very mild assumptions about
floating point arithmetic. It will work on machines with a guard
digit in add/subtract, or on those binary machines without guard
digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
Cray-2. It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none.
Parameters
ITYPE
ITYPE is INTEGER
Specifies the problem type to be solved:
= 1: A*x = (lambda)*B*x
= 2: A*B*x = (lambda)*x
= 3: B*A*x = (lambda)*x
JOBZ
JOBZ is CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangles of A and B are stored;
= 'L': Lower triangles of A and B are stored.
N
N is INTEGER
The order of the matrices A and B. N >= 0.
A
A is DOUBLE PRECISION array, dimension (LDA, N)
On entry, the symmetric matrix A. If UPLO = 'U', the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If UPLO = 'L',
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, if JOBZ = 'V', then if INFO = 0, A contains the
matrix Z of eigenvectors. The eigenvectors are normalized
as follows:
if ITYPE = 1 or 2, Z**T*B*Z = I;
if ITYPE = 3, Z**T*inv(B)*Z = I.
If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
or the lower triangle (if UPLO='L') of A, including the
diagonal, is destroyed.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B
B is DOUBLE PRECISION array, dimension (LDB, N)
On entry, the symmetric matrix B. If UPLO = 'U', the
leading N-by-N upper triangular part of B contains the
upper triangular part of the matrix B. If UPLO = 'L',
the leading N-by-N lower triangular part of B contains
the lower triangular part of the matrix B.
On exit, if INFO <= N, the part of B containing the matrix is
overwritten by the triangular factor U or L from the Cholesky
factorization B = U**T*U or B = L*L**T.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
W
W is DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK.
If N <= 1, LWORK >= 1.
If JOBZ = 'N' and N > 1, LWORK >= 2*N+1.
If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal sizes of the WORK and IWORK
arrays, returns these values as the first entries of the WORK
and IWORK arrays, and no error message related to LWORK or
LIWORK is issued by XERBLA.
IWORK
IWORK is INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK
LIWORK is INTEGER
The dimension of the array IWORK.
If N <= 1, LIWORK >= 1.
If JOBZ = 'N' and N > 1, LIWORK >= 1.
If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.
If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal sizes of the WORK and
IWORK arrays, returns these values as the first entries of
the WORK and IWORK arrays, and no error message related to
LWORK or LIWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: DPOTRF or DSYEVD returned an error code:
<= N: if INFO = i and JOBZ = 'N', then the algorithm
failed to converge; i off-diagonal elements of an
intermediate tridiagonal form did not converge to
zero;
if INFO = i and JOBZ = 'V', then the algorithm
failed to compute an eigenvalue while working on
the submatrix lying in rows and columns INFO/(N+1)
through mod(INFO,N+1);
> N: if INFO = N + i, for 1 <= i <= N, then the leading
minor of order i of B is not positive definite.
The factorization of B could not be completed and
no eigenvalues or eigenvectors were computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
Modified so that no backsubstitution is performed if DSYEVD fails to
converge (NEIG in old code could be greater than N causing out of
bounds reference to A - reported by Ralf Meyer). Also corrected the
description of INFO and the test on ITYPE. Sven, 16 Feb 05.
Contributors:
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
subroutine dsygvx (integer ITYPE, character JOBZ, character RANGE, character UPLO, integer N,
double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, *
) B, integer LDB, double precision VL, double precision VU, integer IL, integer IU, double
precision ABSTOL, integer M, double precision, dimension( * ) W, double precision,
dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK,
integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)
DSYGVX
Purpose:
DSYGVX computes selected eigenvalues, and optionally, eigenvectors
of a real generalized symmetric-definite eigenproblem, of the form
A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A
and B are assumed to be symmetric and B is also positive definite.
Eigenvalues and eigenvectors can be selected by specifying either a
range of values or a range of indices for the desired eigenvalues.
Parameters
ITYPE
ITYPE is INTEGER
Specifies the problem type to be solved:
= 1: A*x = (lambda)*B*x
= 2: A*B*x = (lambda)*x
= 3: B*A*x = (lambda)*x
JOBZ
JOBZ is CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
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 and B are stored;
= 'L': Lower triangle of A and B are stored.
N
N is INTEGER
The order of the matrix pencil (A,B). N >= 0.
A
A is DOUBLE PRECISION array, dimension (LDA, N)
On entry, the symmetric matrix A. If UPLO = 'U', the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If UPLO = 'L',
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, the lower triangle (if UPLO='L') or the upper
triangle (if UPLO='U') of A, including the diagonal, is
destroyed.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B
B is DOUBLE PRECISION array, dimension (LDB, N)
On entry, the symmetric matrix B. If UPLO = 'U', the
leading N-by-N upper triangular part of B contains the
upper triangular part of the matrix B. If UPLO = 'L',
the leading N-by-N lower triangular part of B contains
the lower triangular part of the matrix B.
On exit, if INFO <= N, the part of B containing the matrix is
overwritten by the triangular factor U or L from the Cholesky
factorization B = U**T*U or B = L*L**T.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
VL
VL is DOUBLE PRECISION
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 DOUBLE PRECISION
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 DOUBLE PRECISION
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 C to tridiagonal form, where C is the symmetric
matrix of the standard symmetric problem to which the
generalized problem is transformed.
Eigenvalues will be computed most accurately when ABSTOL is
set to twice the underflow threshold 2*DLAMCH('S'), not zero.
If this routine returns with INFO>0, indicating that some
eigenvectors did not converge, try setting ABSTOL to
2*DLAMCH('S').
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 DOUBLE PRECISION array, dimension (N)
On normal exit, the first M elements contain the selected
eigenvalues in ascending order.
Z
Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
If JOBZ = 'N', then Z is not referenced.
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).
The eigenvectors are normalized as follows:
if ITYPE = 1 or 2, Z**T*B*Z = I;
if ITYPE = 3, Z**T*inv(B)*Z = 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.
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 DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The length of the array WORK. LWORK >= max(1,8*N).
For optimal efficiency, LWORK >= (NB+3)*N,
where NB is the blocksize for DSYTRD returned by ILAENV.
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: DPOTRF or DSYEVX returned an error code:
<= N: if INFO = i, DSYEVX failed to converge;
i eigenvectors failed to converge. Their indices
are stored in array IFAIL.
> N: if INFO = N + i, for 1 <= i <= N, then the leading
minor of order i of B is not positive definite.
The factorization of B could not be completed and
no eigenvalues or eigenvectors were computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
Author
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