complexHEeigen

NAME

complexHEeigen - complex

SYNOPSIS

Functions

subroutine cheev (JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, RWORK, INFO)
CHEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices
subroutine cheev_2stage (JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, RWORK, INFO)
CHEEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices
subroutine cheevd (JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
CHEEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices
subroutine cheevd_2stage (JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
CHEEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices
subroutine cheevr (JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
CHEEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices
subroutine cheevr_2stage (JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
CHEEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices
subroutine cheevx (JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK, IWORK, IFAIL, INFO)
CHEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices
subroutine cheevx_2stage (JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK, IWORK, IFAIL, INFO)
CHEEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices
subroutine chegv (ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK, LWORK, RWORK, INFO)
CHEGV
subroutine chegv_2stage (ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK, LWORK, RWORK, INFO)
CHEGV_2STAGE
subroutine chegvd (ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
CHEGVD
subroutine chegvx (ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK, IWORK, IFAIL, INFO)
CHEGVX

Detailed Description

This is the group of complex eigenvalue driver functions for HE matrices

Function Documentation

subroutine cheev (character JOBZ, character UPLO, integer N, complex,dimension( lda, * ) A, integer LDA, real, dimension( * ) W, complex,dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integerINFO)

CHEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Purpose:

CHEEV computes all eigenvalues and, optionally, eigenvectors of a
complex Hermitian 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 COMPLEX array, dimension (LDA, N)
On entry, the Hermitian 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 REAL array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.

WORK

WORK is COMPLEX 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(1,2*N-1).
For optimal efficiency, LWORK >= (NB+1)*N,
where NB is the blocksize for CHETRD 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.

RWORK

RWORK is REAL array, dimension (max(1, 3*N-2))

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 cheev_2stage (character JOBZ, character UPLO, integer N, complex,dimension( lda, * ) A, integer LDA, real, dimension( * ) W, complex,dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integerINFO)

CHEEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Purpose:

CHEEV_2STAGE computes all eigenvalues and, optionally, eigenvectors of a
complex Hermitian 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 COMPLEX array, dimension (LDA, N)
On entry, the Hermitian 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 REAL array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.

WORK

WORK is COMPLEX 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 + N
= N*KD + N*max(KD+1,FACTOPTNB)
+ max(2*KD*KD, KD*NTHREADS)
+ (KD+1)*N + 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.

RWORK

RWORK is REAL array, dimension (max(1, 3*N-2))

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 cheevd (character JOBZ, character UPLO, integer N, complex,dimension( lda, * ) A, integer LDA, real, dimension( * ) W, complex,dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integerLRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

CHEEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Purpose:

CHEEVD computes all eigenvalues and, optionally, eigenvectors of a
complex Hermitian 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.

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 COMPLEX array, dimension (LDA, N)
On entry, the Hermitian 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 REAL array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.

WORK

WORK is COMPLEX 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.
If N <= 1, LWORK must be at least 1.
If JOBZ = ’N’ and N > 1, LWORK must be at least N + 1.
If JOBZ = ’V’ and N > 1, LWORK must be at least 2*N + N**2.

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal sizes of the WORK, RWORK and
IWORK arrays, returns these values as the first entries of
the WORK, RWORK and IWORK arrays, and no error message
related to LWORK or LRWORK or LIWORK is issued by XERBLA.

RWORK

RWORK is REAL array,
dimension (LRWORK)
On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.

LRWORK

LRWORK is INTEGER
The dimension of the array RWORK.
If N <= 1, LRWORK must be at least 1.
If JOBZ = ’N’ and N > 1, LRWORK must be at least N.
If JOBZ = ’V’ and N > 1, LRWORK must be at least
1 + 5*N + 2*N**2.

If LRWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal sizes of the WORK, RWORK
and IWORK arrays, returns these values as the first entries
of the WORK, RWORK and IWORK arrays, and no error message
related to LWORK or LRWORK 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, RWORK
and IWORK arrays, returns these values as the first entries
of the WORK, RWORK and IWORK arrays, and no error message
related to LWORK or LRWORK 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.

Further Details:

Modified description of INFO. Sven, 16 Feb 05.

Contributors:

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

subroutine cheevd_2stage (character JOBZ, character UPLO, integer N, complex,dimension( lda, * ) A, integer LDA, real, dimension( * ) W, complex,dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integerLRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

CHEEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Purpose:

CHEEVD_2STAGE computes all eigenvalues and, optionally, eigenvectors of a
complex Hermitian 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 COMPLEX array, dimension (LDA, N)
On entry, the Hermitian 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 REAL array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.

WORK

WORK is COMPLEX 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 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 + N+1
= N*KD + N*max(KD+1,FACTOPTNB)
+ max(2*KD*KD, KD*NTHREADS)
+ (KD+1)*N + 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 2*N + N**2

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal sizes of the WORK, RWORK and
IWORK arrays, returns these values as the first entries of
the WORK, RWORK and IWORK arrays, and no error message
related to LWORK or LRWORK or LIWORK is issued by XERBLA.

RWORK

RWORK is REAL array,
dimension (LRWORK)
On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.

LRWORK

LRWORK is INTEGER
The dimension of the array RWORK.
If N <= 1, LRWORK must be at least 1.
If JOBZ = ’N’ and N > 1, LRWORK must be at least N.
If JOBZ = ’V’ and N > 1, LRWORK must be at least
1 + 5*N + 2*N**2.

If LRWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal sizes of the WORK, RWORK
and IWORK arrays, returns these values as the first entries
of the WORK, RWORK and IWORK arrays, and no error message
related to LWORK or LRWORK 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, RWORK
and IWORK arrays, returns these values as the first entries
of the WORK, RWORK and IWORK arrays, and no error message
related to LWORK or LRWORK 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.

Further Details:

Modified description of INFO. Sven, 16 Feb 05.

Contributors:

Jeff Rutter, 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 cheevr (character JOBZ, character RANGE, character UPLO, integerN, complex, dimension( lda, * ) A, integer LDA, real VL, real VU, integerIL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, complex,dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ,complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK,integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, integerINFO)

CHEEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Purpose:

CHEEVR computes selected eigenvalues and, optionally, eigenvectors
of a complex Hermitian matrix A. Eigenvalues and eigenvectors can
be selected by specifying either a range of values or a range of
indices for the desired eigenvalues.

CHEEVR first reduces the matrix A to tridiagonal form T with a call
to CHETRD. Then, whenever possible, CHEEVR calls CSTEMR to compute
the eigenspectrum using Relatively Robust Representations. CSTEMR
computes eigenvalues by the dqds algorithm, while orthogonal
eigenvectors are computed from various ’good’ L D LˆT 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 LˆT, 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 CSTEMR’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(nˆ2) algorithm for the symmetric
tridiagonal eigenvalue/eigenvector problem’,
Computer Science Division Technical Report No. UCB/CSD-97-971,
UC Berkeley, May 1997.

Note 1 : CHEEVR calls CSTEMR when the full spectrum is requested
on machines which conform to the ieee-754 floating point standard.
CHEEVR calls SSTEBZ and CSTEIN on non-ieee machines and
when partial spectrum requests are made.

Normal execution of CSTEMR 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, SSTEBZ and
CSTEIN 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 COMPLEX array, dimension (LDA, N)
On entry, the Hermitian 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 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 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
SLAMCH( ’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 REAL array, dimension (N)
The first M elements contain the selected eigenvalues in
ascending order.

Z

Z is COMPLEX 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.

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 CSTEMR (tridiagonal
matrix). The support of the eigenvectors of A is typically
1:N because of the unitary transformations applied by CUNMTR.
Implemented only for RANGE = ’A’ or ’I’ and IU - IL = N - 1

WORK

WORK is COMPLEX 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(1,2*N).
For optimal efficiency, LWORK >= (NB+1)*N,
where NB is the max of the blocksize for CHETRD and for
CUNMTR as returned by ILAENV.

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal sizes of the WORK, RWORK and
IWORK arrays, returns these values as the first entries of
the WORK, RWORK and IWORK arrays, and no error message
related to LWORK or LRWORK or LIWORK is issued by XERBLA.

RWORK

RWORK is REAL array, dimension (MAX(1,LRWORK))
On exit, if INFO = 0, RWORK(1) returns the optimal
(and minimal) LRWORK.

LRWORK

LRWORK is INTEGER
The length of the array RWORK. LRWORK >= max(1,24*N).

If LRWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal sizes of the WORK, RWORK
and IWORK arrays, returns these values as the first entries
of the WORK, RWORK and IWORK arrays, and no error message
related to LWORK or LRWORK 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
(and minimal) LIWORK.

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 sizes of the WORK, RWORK
and IWORK arrays, returns these values as the first entries
of the WORK, RWORK and IWORK arrays, and no error message
related to LWORK or LRWORK 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: 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 cheevr_2stage (character JOBZ, character RANGE, character UPLO,integer N, complex, dimension( lda, * ) A, integer LDA, real VL, real VU,integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W,complex, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * )ISUPPZ, complex, dimension( * ) WORK, integer LWORK, real, dimension( * )RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK,integer INFO)

CHEEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Purpose:

CHEEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors
of a complex Hermitian 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.

CHEEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call
to CHETRD. Then, whenever possible, CHEEVR_2STAGE calls CSTEMR to compute
eigenspectrum using Relatively Robust Representations. CSTEMR
computes eigenvalues by the dqds algorithm, while orthogonal
eigenvectors are computed from various ’good’ L D LˆT 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 LˆT, 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 CSTEMR’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(nˆ2) algorithm for the symmetric
tridiagonal eigenvalue/eigenvector problem’,
Computer Science Division Technical Report No. UCB/CSD-97-971,
UC Berkeley, May 1997.

Note 1 : CHEEVR_2STAGE calls CSTEMR when the full spectrum is requested
on machines which conform to the ieee-754 floating point standard.
CHEEVR_2STAGE calls SSTEBZ and CSTEIN on non-ieee machines and
when partial spectrum requests are made.

Normal execution of CSTEMR 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, SSTEBZ and
CSTEIN 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 COMPLEX array, dimension (LDA, N)
On entry, the Hermitian 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 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 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
SLAMCH( ’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 REAL array, dimension (N)
The first M elements contain the selected eigenvalues in
ascending order.

Z

Z is COMPLEX 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.

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 CSTEMR (tridiagonal
matrix). The support of the eigenvectors of A is typically
1:N because of the unitary transformations applied by CUNMTR.
Implemented only for RANGE = ’A’ or ’I’ and IU - IL = N - 1

WORK

WORK is COMPLEX 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 + N
= N*KD + N*max(KD+1,FACTOPTNB)
+ max(2*KD*KD, KD*NTHREADS)
+ (KD+1)*N + 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 sizes of the WORK, RWORK and
IWORK arrays, returns these values as the first entries of
the WORK, RWORK and IWORK arrays, and no error message
related to LWORK or LRWORK or LIWORK is issued by XERBLA.

RWORK

RWORK is REAL array, dimension (MAX(1,LRWORK))
On exit, if INFO = 0, RWORK(1) returns the optimal
(and minimal) LRWORK.

LRWORK

LRWORK is INTEGER
The length of the array RWORK. LRWORK >= max(1,24*N).

If LRWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal sizes of the WORK, RWORK
and IWORK arrays, returns these values as the first entries
of the WORK, RWORK and IWORK arrays, and no error message
related to LWORK or LRWORK 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
(and minimal) LIWORK.

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 sizes of the WORK, RWORK
and IWORK arrays, returns these values as the first entries
of the WORK, RWORK and IWORK arrays, and no error message
related to LWORK or LRWORK 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: Internal error

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Inderjit Dhillon, IBM Almaden, USA \n
Osni Marques, LBNL/NERSC, USA \n
Ken Stanley, Computer Science Division, University of
California at Berkeley, USA \n
Jason Riedy, Computer Science Division, University of
California at Berkeley, USA \n

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 cheevx (character JOBZ, character RANGE, character UPLO, integerN, complex, dimension( lda, * ) A, integer LDA, real VL, real VU, integerIL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, complex,dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integerLWORK, real, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer,dimension( * ) IFAIL, integer INFO)

CHEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Purpose:

CHEEVX computes selected eigenvalues and, optionally, eigenvectors
of a complex Hermitian 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 COMPLEX array, dimension (LDA, N)
On entry, the Hermitian 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 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 A 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)
On normal exit, the first M elements contain the selected
eigenvalues in ascending order.

Z

Z is COMPLEX 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 COMPLEX 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 2*N.
For optimal efficiency, LWORK >= (NB+1)*N,
where NB is the max of the blocksize for CHETRD and for
CUNMTR as 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.

RWORK

RWORK is REAL array, dimension (7*N)

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 cheevx_2stage (character JOBZ, character RANGE, character UPLO,integer N, complex, dimension( lda, * ) A, integer LDA, real VL, real VU,integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W,complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK,integer LWORK, real, dimension( * ) RWORK, integer, dimension( * ) IWORK,integer, dimension( * ) IFAIL, integer INFO)

CHEEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Purpose:

CHEEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors
of a complex Hermitian 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 COMPLEX array, dimension (LDA, N)
On entry, the Hermitian 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 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 A 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)
On normal exit, the first M elements contain the selected
eigenvalues in ascending order.

Z

Z is COMPLEX 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 COMPLEX 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 + N
= N*KD + N*max(KD+1,FACTOPTNB)
+ max(2*KD*KD, KD*NTHREADS)
+ (KD+1)*N + 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.

RWORK

RWORK is REAL array, dimension (7*N)

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 chegv (integer ITYPE, character JOBZ, character UPLO, integer N,complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * )B, integer LDB, real, dimension( * ) W, complex, dimension( * ) WORK,integer LWORK, real, dimension( * ) RWORK, integer INFO)

CHEGV

Purpose:

CHEGV computes all the eigenvalues, and optionally, the eigenvectors
of a complex generalized Hermitian-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 Hermitian 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 COMPLEX array, dimension (LDA, N)
On entry, the Hermitian 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**H*B*Z = I;
if ITYPE = 3, Z**H*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 COMPLEX array, dimension (LDB, N)
On entry, the Hermitian 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**H*U or B = L*L**H.

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

W

W is REAL array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.

WORK

WORK is COMPLEX 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(1,2*N-1).
For optimal efficiency, LWORK >= (NB+1)*N,
where NB is the blocksize for CHETRD 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.

RWORK

RWORK is REAL array, dimension (max(1, 3*N-2))

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: CPOTRF or CHEEV returned an error code:
<= N: if INFO = i, CHEEV 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 chegv_2stage (integer ITYPE, character JOBZ, character UPLO,integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension(ldb, * ) B, integer LDB, real, dimension( * ) W, complex, dimension( * )WORK, integer LWORK, real, dimension( * ) RWORK, integer INFO)

CHEGV_2STAGE

Purpose:

CHEGV_2STAGE computes all the eigenvalues, and optionally, the eigenvectors
of a complex generalized Hermitian-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 Hermitian 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 COMPLEX array, dimension (LDA, N)
On entry, the Hermitian 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**H*B*Z = I;
if ITYPE = 3, Z**H*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 COMPLEX array, dimension (LDB, N)
On entry, the Hermitian 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**H*U or B = L*L**H.

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

W

W is REAL array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.

WORK

WORK is COMPLEX 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 + N
= N*KD + N*max(KD+1,FACTOPTNB)
+ max(2*KD*KD, KD*NTHREADS)
+ (KD+1)*N + 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.

RWORK

RWORK is REAL array, dimension (max(1, 3*N-2))

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: CPOTRF or CHEEV returned an error code:
<= N: if INFO = i, CHEEV 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 chegvd (integer ITYPE, character JOBZ, character UPLO, integer N,complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * )B, integer LDB, real, dimension( * ) W, complex, dimension( * ) WORK,integer LWORK, real, dimension( * ) RWORK, integer LRWORK, integer,dimension( * ) IWORK, integer LIWORK, integer INFO)

CHEGVD

Purpose:

CHEGVD computes all the eigenvalues, and optionally, the eigenvectors
of a complex generalized Hermitian-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 Hermitian 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 COMPLEX array, dimension (LDA, N)
On entry, the Hermitian 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**H*B*Z = I;
if ITYPE = 3, Z**H*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 COMPLEX array, dimension (LDB, N)
On entry, the Hermitian 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**H*U or B = L*L**H.

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

W

W is REAL array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.

WORK

WORK is COMPLEX 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.
If N <= 1, LWORK >= 1.
If JOBZ = ’N’ and N > 1, LWORK >= N + 1.
If JOBZ = ’V’ and N > 1, LWORK >= 2*N + N**2.

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal sizes of the WORK, RWORK and
IWORK arrays, returns these values as the first entries of
the WORK, RWORK and IWORK arrays, and no error message
related to LWORK or LRWORK or LIWORK is issued by XERBLA.

RWORK

RWORK is REAL array, dimension (MAX(1,LRWORK))
On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.

LRWORK

LRWORK is INTEGER
The dimension of the array RWORK.
If N <= 1, LRWORK >= 1.
If JOBZ = ’N’ and N > 1, LRWORK >= N.
If JOBZ = ’V’ and N > 1, LRWORK >= 1 + 5*N + 2*N**2.

If LRWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal sizes of the WORK, RWORK
and IWORK arrays, returns these values as the first entries
of the WORK, RWORK and IWORK arrays, and no error message
related to LWORK or LRWORK 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, RWORK
and IWORK arrays, returns these values as the first entries
of the WORK, RWORK and IWORK arrays, and no error message
related to LWORK or LRWORK 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: CPOTRF or CHEEVD 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 CHEEVD 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 chegvx (integer ITYPE, character JOBZ, character RANGE, characterUPLO, integer N, complex, dimension( lda, * ) A, integer LDA, complex,dimension( ldb, * ) B, integer LDB, real VL, real VU, integer IL, integerIU, real ABSTOL, integer M, real, dimension( * ) W, complex, dimension(ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, real,dimension( * ) RWORK, integer, dimension( * ) IWORK, integer, dimension( *) IFAIL, integer INFO)

CHEGVX

Purpose:

CHEGVX computes selected eigenvalues, and optionally, eigenvectors
of a complex generalized Hermitian-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 Hermitian 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 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 COMPLEX array, dimension (LDA, N)
On entry, the Hermitian 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 COMPLEX array, dimension (LDB, N)
On entry, the Hermitian 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**H*U or B = L*L**H.

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= 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 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*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’).

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 COMPLEX 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 COMPLEX 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(1,2*N).
For optimal efficiency, LWORK >= (NB+1)*N,
where NB is the blocksize for CHETRD 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.

RWORK

RWORK is REAL array, dimension (7*N)

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: CPOTRF or CHEEVX returned an error code:
<= N: if INFO = i, CHEEVX 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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