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Manual

hbevx_2stage

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine chbevx_2stage (character jobz, character range, character uplo,integer n, integer kd, complex, dimension( ldab, * ) ab, integer ldab,complex, dimension( ldq, * ) q, integer ldq, real vl, real vu, integeril, 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)
subroutine dsbevx_2stage (character jobz, character range, character uplo,integer n, integer kd, double precision, dimension( ldab, * ) ab,integer ldab, double precision, dimension( ldq, * ) q, integer ldq,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)
subroutine ssbevx_2stage (character jobz, character range, character uplo,integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab,real, dimension( ldq, * ) q, integer ldq, real vl, real vu, integer il,integer iu, real abstol, integer m, real, dimension( * ) w, real,dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integerlwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail,integer info)
subroutine zhbevx_2stage (character jobz, character range, character uplo,integer n, integer kd, complex*16, dimension( ldab, * ) ab, integerldab, complex*16, dimension( ldq, * ) q, integer ldq, double precisionvl, double precision vu, integer il, integer iu, double precisionabstol, integer m, double precision, dimension( * ) w, complex*16,dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work,integer lwork, double precision, dimension( * ) rwork, integer,dimension( * ) iwork, integer, dimension( * ) ifail, integer info)
Author

NAME

hbevx_2stage - {hb,sb}evx_2stage: eig, bisection

SYNOPSIS

Functions

subroutine chbevx_2stage (jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, rwork, iwork, ifail, info)
CHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine dsbevx_2stage (jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, iwork, ifail, info)
DSBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine ssbevx_2stage (jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, iwork, ifail, info)
SSBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine zhbevx_2stage (jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, rwork, iwork, ifail, info)
ZHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Detailed Description

Function Documentation

subroutine chbevx_2stage (character jobz, character range, character uplo,integer n, integer kd, complex, dimension( ldab, * ) ab, integer ldab,complex, dimension( ldq, * ) q, integer ldq, real vl, real vu, integeril, 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)

CHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

CHBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors
of a complex Hermitian band matrix A using the 2stage technique for
the reduction to tridiagonal. Eigenvalues and eigenvectors
can be selected by specifying either a range of values or a range of
indices for the desired eigenvalues.

Parameters

JOBZ

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 is INTEGER
The order of the matrix A. N >= 0.

KD is INTEGER
The number of superdiagonals of the matrix A if UPLO = ’U’,
or the number of subdiagonals if UPLO = ’L’. KD >= 0.

AB is COMPLEX array, dimension (LDAB, N)
On entry, the upper or lower triangle of the Hermitian band
matrix A, stored in the first KD+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = ’U’, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = ’L’, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

On exit, AB is overwritten by values generated during the
reduction to tridiagonal form.

LDAB

LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD + 1.

Q is COMPLEX array, dimension (LDQ, N)
If JOBZ = ’V’, the N-by-N unitary matrix used in the
reduction to tridiagonal form.
If JOBZ = ’N’, the array Q is not referenced.

LDQ

LDQ is INTEGER
The leading dimension of the array Q. If JOBZ = ’V’, then
LDQ >= max(1,N).

VL 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 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 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 is INTEGER
If RANGE=’I’, the index of the
largest eigenvalue to be returned.
1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
Not referenced if RANGE = ’A’ or ’V’.

ABSTOL

ABSTOL is REAL
The absolute error tolerance for the eigenvalues.
An approximate eigenvalue is accepted as converged
when it is determined to lie in an interval [a,b]
of width less than or equal to

ABSTOL + EPS * max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than
or equal to zero, then EPS*|T| will be used in its place,
where |T| is the 1-norm of the tridiagonal matrix obtained
by reducing AB to tridiagonal form.

Eigenvalues will be computed most accurately when ABSTOL is
set to twice the underflow threshold 2*SLAMCH(’S’), not zero.
If this routine returns with INFO>0, indicating that some
eigenvectors did not converge, try setting ABSTOL to
2*SLAMCH(’S’).

See ’Computing Small Singular Values of Bidiagonal Matrices
with Guaranteed High Relative Accuracy,’ by Demmel and
Kahan, LAPACK Working Note #3.

M 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 is REAL array, dimension (N)
The first M elements contain the selected eigenvalues in
ascending order.

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 (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 = (2KD+1)*N + KD*NTHREADS
where KD is the size of the band.
NTHREADS is the number of threads used when
openMP compilation is enabled, otherwise =1.
If JOBZ = ’V’ and N > 1, LWORK must be queried. Not yet available.

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal 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 (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 dsbevx_2stage (character jobz, character range, character uplo,integer n, integer kd, double precision, dimension( ldab, * ) ab,integer ldab, double precision, dimension( ldq, * ) q, integer ldq,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)

DSBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

DSBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors
of a real symmetric band matrix A using the 2stage technique for
the reduction to tridiagonal. Eigenvalues and eigenvectors can
be selected by specifying either a range of values or a range of
indices for the desired eigenvalues.

Parameters

JOBZ

JOBZ is CHARACTER*1
= ’N’: Compute eigenvalues only;
= ’V’: Compute eigenvalues and eigenvectors.
Not available in this release.

RANGE

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

N is INTEGER
The order of the matrix A. N >= 0.

KD is INTEGER
The number of superdiagonals of the matrix A if UPLO = ’U’,
or the number of subdiagonals if UPLO = ’L’. KD >= 0.

AB is DOUBLE PRECISION array, dimension (LDAB, N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = ’U’, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = ’L’, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

On exit, AB is overwritten by values generated during the
reduction to tridiagonal form. If UPLO = ’U’, the first
superdiagonal and the diagonal of the tridiagonal matrix T
are returned in rows KD and KD+1 of AB, and if UPLO = ’L’,
the diagonal and first subdiagonal of T are returned in the
first two rows of AB.

LDAB

LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD + 1.

Q is DOUBLE PRECISION array, dimension (LDQ, N)
If JOBZ = ’V’, the N-by-N orthogonal matrix used in the
reduction to tridiagonal form.
If JOBZ = ’N’, the array Q is not referenced.

LDQ

LDQ is INTEGER
The leading dimension of the array Q. If JOBZ = ’V’, then
LDQ >= max(1,N).

VL 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 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 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 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| ) ,

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 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 is DOUBLE PRECISION array, dimension (N)
The first M elements contain the selected eigenvalues in
ascending order.

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 (LWORK)

LWORK

LWORK is INTEGER
The length of the array WORK. LWORK >= 1, when N <= 1;
otherwise
If JOBZ = ’N’ and N > 1, LWORK must be queried.
LWORK = MAX(1, 7*N, dimension) where
dimension = (2KD+1)*N + KD*NTHREADS + 2*N
where KD is the size of the band.
NTHREADS is the number of threads used when
openMP compilation is enabled, otherwise =1.
If JOBZ = ’V’ and N > 1, LWORK must be queried. Not yet available

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.

IWORK

IWORK is INTEGER array, dimension (5*N)

IFAIL

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:

subroutine ssbevx_2stage (character jobz, character range, character uplo,integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab,real, dimension( ldq, * ) q, integer ldq, real vl, real vu, integer il,integer iu, real abstol, integer m, real, dimension( * ) w, real,dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integerlwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail,integer info)

SSBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

SSBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors
of a real symmetric band matrix A using the 2stage technique for
the reduction to tridiagonal. Eigenvalues and eigenvectors can
be selected by specifying either a range of values or a range of
indices for the desired eigenvalues.

Parameters

JOBZ

JOBZ is CHARACTER*1
= ’N’: Compute eigenvalues only;
= ’V’: Compute eigenvalues and eigenvectors.
Not available in this release.

RANGE

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

N is INTEGER
The order of the matrix A. N >= 0.

KD is INTEGER
The number of superdiagonals of the matrix A if UPLO = ’U’,
or the number of subdiagonals if UPLO = ’L’. KD >= 0.

AB is REAL array, dimension (LDAB, N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = ’U’, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = ’L’, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

LDAB

LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD + 1.

Q is REAL array, dimension (LDQ, N)
If JOBZ = ’V’, the N-by-N orthogonal matrix used in the
reduction to tridiagonal form.
If JOBZ = ’N’, the array Q is not referenced.

LDQ

LDQ is INTEGER
The leading dimension of the array Q. If JOBZ = ’V’, then
LDQ >= max(1,N).

VL 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 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 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 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| ) ,

See ’Computing Small Singular Values of Bidiagonal Matrices
with Guaranteed High Relative Accuracy,’ by Demmel and
Kahan, LAPACK Working Note #3.

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 is REAL array, dimension (N)
The first M elements contain the selected eigenvalues in
ascending order.

Z is REAL array, dimension (LDZ, max(1,M))
If JOBZ = ’V’, then if INFO = 0, the first M columns of Z
contain the orthonormal eigenvectors of the matrix A
corresponding to the selected eigenvalues, with the i-th
column of Z holding the eigenvector associated with W(i).
If an eigenvector fails to converge, then that column of Z
contains the latest approximation to the eigenvector, and the
index of the eigenvector is returned in IFAIL.
If JOBZ = ’N’, then Z is not referenced.
Note: the user must ensure that at least max(1,M) columns are
supplied in the array Z; if RANGE = ’V’, the exact value of M
is not known in advance and an upper bound must be used.

LDZ

LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1, and if
JOBZ = ’V’, LDZ >= max(1,N).

WORK

WORK is REAL array, dimension (LWORK)

LWORK

IWORK

IWORK is INTEGER array, dimension (5*N)

IFAIL

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:

subroutine zhbevx_2stage (character jobz, character range, character uplo,integer n, integer kd, complex16, dimension( ldab, ) ab, integerldab, complex16, dimension( ldq, ) q, integer ldq, double precisionvl, double precision vu, integer il, integer iu, double precisionabstol, integer m, double precision, dimension( * ) w, complex16,dimension( ldz, ) z, integer ldz, complex16, dimension( ) work,integer lwork, double precision, dimension( * ) rwork, integer,dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

ZHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

ZHBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors
of a complex Hermitian band matrix A using the 2stage technique for
the reduction to tridiagonal. Eigenvalues and eigenvectors
can be selected by specifying either a range of values or a range of
indices for the desired eigenvalues.

Parameters

JOBZ

JOBZ is CHARACTER*1
= ’N’: Compute eigenvalues only;
= ’V’: Compute eigenvalues and eigenvectors.
Not available in this release.

RANGE

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

N is INTEGER
The order of the matrix A. N >= 0.

KD is INTEGER
The number of superdiagonals of the matrix A if UPLO = ’U’,
or the number of subdiagonals if UPLO = ’L’. KD >= 0.

AB is COMPLEX*16 array, dimension (LDAB, N)
On entry, the upper or lower triangle of the Hermitian band
matrix A, stored in the first KD+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = ’U’, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = ’L’, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

On exit, AB is overwritten by values generated during the
reduction to tridiagonal form.

LDAB

LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD + 1.

Q is COMPLEX*16 array, dimension (LDQ, N)
If JOBZ = ’V’, the N-by-N unitary matrix used in the
reduction to tridiagonal form.
If JOBZ = ’N’, the array Q is not referenced.

LDQ

LDQ is INTEGER
The leading dimension of the array Q. If JOBZ = ’V’, then
LDQ >= max(1,N).

VL 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 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 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 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 + EPS * max( |a|,|b| ) ,

See ’Computing Small Singular Values of Bidiagonal Matrices
with Guaranteed High Relative Accuracy,’ by Demmel and
Kahan, LAPACK Working Note #3.

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 is DOUBLE PRECISION array, dimension (N)
The first M elements contain the selected eigenvalues in
ascending order.

Z is COMPLEX*16 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*16 array, dimension (LWORK)

LWORK

RWORK

RWORK is DOUBLE PRECISION array, dimension (7*N)

IWORK

IWORK is INTEGER array, dimension (5*N)

IFAIL

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:

Author

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