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heevr

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine cheevr (character jobz, character range, character uplo, integern, 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)
subroutine dsyevr (character jobz, character range, character uplo, integern, double precision, dimension( lda, * ) a, integer lda, doubleprecision vl, double precision vu, integer il, integer iu, doubleprecision abstol, integer m, double precision, dimension( * ) w, doubleprecision, dimension( ldz, * ) z, integer ldz, integer, dimension( * )isuppz, double precision, dimension( * ) work, integer lwork, integer,dimension( * ) iwork, integer liwork, integer info)
subroutine ssyevr (character jobz, character range, character uplo, integern, real, dimension( lda, * ) a, integer lda, real vl, real vu, integeril, integer iu, real abstol, integer m, real, dimension( * ) w, real,dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz,real, dimension( * ) work, integer lwork, integer, dimension( * )iwork, integer liwork, integer info)
subroutine zheevr (character jobz, character range, character uplo, integern, complex*16, dimension( lda, * ) a, integer lda, double precision vl,double precision vu, integer il, integer iu, double precision abstol,integer m, double precision, dimension( * ) w, complex*16, dimension(ldz, * ) z, integer ldz, integer, dimension( * ) isuppz, complex*16,dimension( * ) work, integer lwork, double precision, dimension( * )rwork, integer lrwork, integer, dimension( * ) iwork, integer liwork,integer info)
Author

NAME

heevr - {he,sy}evr: eig, MRRR

SYNOPSIS

Functions

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 dsyevr (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info)
DSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
subroutine ssyevr (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info)
SSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
subroutine zheevr (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, rwork, lrwork, iwork, liwork, info)
ZHEEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Detailed Description

Function Documentation

subroutine cheevr (character jobz, character range, character uplo, integern, 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 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. Invocations with different choices for
these parameters may result in the computation of slightly different
eigenvalues and/or eigenvectors for the same matrix. The reason for
this behavior is that there exists a variety of algorithms (each
performing best for a particular set of options) with CHEEVR
attempting to select the best based on the various parameters. In all
cases, the computed values are accurate within the limits of finite
precision arithmetic.

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.

This parameter influences the choice of the algorithm and
may alter the computed values.

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

This parameter influences the choice of the algorithm and
may alter the computed values.

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.

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 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 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 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 JOBZ = ’N’, then Z is not referenced.
Note: the user must ensure that at least max(1,M) columns are
supplied in the array Z; if RANGE = ’V’, the exact value of M
is not known in advance and an upper bound must be used.
Supplying N columns is always safe.

LDZ

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

ISUPPZ

ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
The support of the eigenvectors in Z, i.e., the indices
indicating the nonzero elements in Z. The i-th eigenvector
is nonzero only in elements ISUPPZ( 2*i-1 ) through
ISUPPZ( 2*i ). This is an output of 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.
If N <= 1, LWORK >= 1, else LWORK >= 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.
If N <= 1, LRWORK >= 1, else LRWORK >= 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.
If N <= 1, LIWORK >= 1, else LIWORK >= 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 dsyevr (character jobz, character range, character uplo, integern, double precision, dimension( lda, * ) a, integer lda, doubleprecision vl, double precision vu, integer il, integer iu, doubleprecision abstol, integer m, double precision, dimension( * ) w, doubleprecision, dimension( ldz, * ) z, integer ldz, integer, dimension( * )isuppz, double precision, dimension( * ) work, integer lwork, integer,dimension( * ) iwork, integer liwork, integer info)

DSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

Purpose:

DSYEVR computes selected eigenvalues and, optionally, eigenvectors
of a real symmetric matrix A. Eigenvalues and eigenvectors can be
selected by specifying either a range of values or a range of indices
for the desired eigenvalues. Invocations with different choices for
these parameters may result in the computation of slightly different
eigenvalues and/or eigenvectors for the same matrix. The reason for
this behavior is that there exists a variety of algorithms (each
performing best for a particular set of options) with DSYEVR
attempting to select the best based on the various parameters. In all
cases, the computed values are accurate within the limits of finite
precision arithmetic.

DSYEVR first reduces the matrix A to tridiagonal form T with a call
to DSYTRD. Then, whenever possible, DSYEVR calls DSTEMR to compute
the eigenspectrum using Relatively Robust Representations. DSTEMR
computes eigenvalues by the dqds algorithm, while orthogonal
eigenvectors are computed from various ’good’ L D 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.

The desired accuracy of the output can be specified by the input
parameter ABSTOL.

For more details, see DSTEMR’s documentation and:
- Inderjit S. Dhillon and Beresford N. Parlett: ’Multiple representations
to compute orthogonal eigenvectors of symmetric tridiagonal matrices,’
Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
- Inderjit Dhillon and Beresford Parlett: ’Orthogonal Eigenvectors and
Relative Gaps,’ SIAM Journal on Matrix Analysis and Applications, Vol. 25,
2004. Also LAPACK Working Note 154.
- Inderjit Dhillon: ’A new O(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 : DSYEVR calls DSTEMR when the full spectrum is requested
on machines which conform to the ieee-754 floating point standard.
DSYEVR calls DSTEBZ and DSTEIN on non-ieee machines and
when partial spectrum requests are made.

Normal execution of DSTEMR may create NaNs and infinities and
hence may abort due to a floating point exception in environments
which do not handle NaNs and infinities in the ieee standard default
manner.

Parameters

JOBZ

JOBZ is CHARACTER*1
= ’N’: Compute eigenvalues only;
= ’V’: Compute eigenvalues and eigenvectors.

This parameter influences the choice of the algorithm and
may alter the computed values.

RANGE

This parameter influences the choice of the algorithm and
may alter the computed values.

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.

A is DOUBLE PRECISION array, dimension (LDA, N)
On entry, the symmetric matrix A. If UPLO = ’U’, the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If UPLO = ’L’,
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, the lower triangle (if UPLO=’L’) or the upper
triangle (if UPLO=’U’) of A, including the diagonal, is
destroyed.

LDA

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

VL 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| ) ,

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

If high relative accuracy is important, set ABSTOL to
DLAMCH( ’Safe minimum’ ). Doing so will guarantee that
eigenvalues are computed to high relative accuracy when
possible in future releases. The current code does not
make any guarantees about high relative accuracy, but
future releases will. See J. Barlow and J. Demmel,
’Computing Accurate Eigensystems of Scaled Diagonally
Dominant Matrices’, LAPACK Working Note #7, for a discussion
of which matrices define their eigenvalues to high relative
accuracy.

M 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 JOBZ = ’N’, then Z is not referenced.
Note: the user must ensure that at least max(1,M) columns are
supplied in the array Z; if RANGE = ’V’, the exact value of M
is not known in advance and an upper bound must be used.
Supplying N columns is always safe.

LDZ

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

ISUPPZ

ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
The support of the eigenvectors in Z, i.e., the indices
indicating the nonzero elements in Z. The i-th eigenvector
is nonzero only in elements ISUPPZ( 2*i-1 ) through
ISUPPZ( 2*i ). This is an output of DSTEMR (tridiagonal
matrix). The support of the eigenvectors of A is typically
1:N because of the orthogonal transformations applied by DORMTR.
Implemented only for RANGE = ’A’ or ’I’ and IU - IL = N - 1

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK.
If N <= 1, LWORK >= 1, else LWORK >= 26*N.
For optimal efficiency, LWORK >= (NB+6)*N,
where NB is the max of the blocksize for DSYTRD and DORMTR
returned by ILAENV.

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

IWORK

IWORK is INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

LIWORK

LIWORK is INTEGER
The dimension of the array IWORK.
If N <= 1, LIWORK >= 1, else LIWORK >= 10*N.

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

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: Internal error

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

subroutine ssyevr (character jobz, character range, character uplo, integern, real, dimension( lda, * ) a, integer lda, real vl, real vu, integeril, integer iu, real abstol, integer m, real, dimension( * ) w, real,dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz,real, dimension( * ) work, integer lwork, integer, dimension( * )iwork, integer liwork, integer info)

SSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

Purpose:

SSYEVR computes selected eigenvalues and, optionally, eigenvectors
of a real symmetric matrix A. Eigenvalues and eigenvectors can be
selected by specifying either a range of values or a range of indices
for the desired eigenvalues. Invocations with different choices for
these parameters may result in the computation of slightly different
eigenvalues and/or eigenvectors for the same matrix. The reason for
this behavior is that there exists a variety of algorithms (each
performing best for a particular set of options) with SSYEVR
attempting to select the best based on the various parameters. In all
cases, the computed values are accurate within the limits of finite
precision arithmetic.

SSYEVR first reduces the matrix A to tridiagonal form T with a call
to SSYTRD. Then, whenever possible, SSYEVR calls SSTEMR to compute
the eigenspectrum using Relatively Robust Representations. SSTEMR
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.

The desired accuracy of the output can be specified by the input
parameter ABSTOL.

For more details, see SSTEMR’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 : SSYEVR calls SSTEMR when the full spectrum is requested
on machines which conform to the ieee-754 floating point standard.
SSYEVR calls SSTEBZ and SSTEIN on non-ieee machines and
when partial spectrum requests are made.

Normal execution of SSTEMR 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.

This parameter influences the choice of the algorithm and
may alter the computed values.

RANGE

This parameter influences the choice of the algorithm and
may alter the computed values.

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.

A is REAL array, dimension (LDA, N)
On entry, the symmetric matrix A. If UPLO = ’U’, the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If UPLO = ’L’,
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, the lower triangle (if UPLO=’L’) or the upper
triangle (if UPLO=’U’) of A, including the diagonal, is
destroyed.

LDA

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

VL 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 JOBZ = ’N’, then Z is not referenced.
Note: the user must ensure that at least max(1,M) columns are
supplied in the array Z; if RANGE = ’V’, the exact value of M
is not known in advance and an upper bound must be used.
Supplying N columns is always safe.

LDZ

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

ISUPPZ

ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
The support of the eigenvectors in Z, i.e., the indices
indicating the nonzero elements in Z. The i-th eigenvector
is nonzero only in elements ISUPPZ( 2*i-1 ) through
ISUPPZ( 2*i ). This is an output of SSTEMR (tridiagonal
matrix). The support of the eigenvectors of A is typically
1:N because of the orthogonal transformations applied by SORMTR.
Implemented only for RANGE = ’A’ or ’I’ and IU - IL = N - 1

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK.
If N <= 1, LWORK >= 1, else LWORK >= 26*N.
For optimal efficiency, LWORK >= (NB+6)*N,
where NB is the max of the blocksize for SSYTRD and SORMTR
returned by ILAENV.

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

IWORK

IWORK is INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LWORK.

LIWORK

LIWORK is INTEGER
The dimension of the array IWORK.
If N <= 1, LIWORK >= 1, else LIWORK >= 10*N.

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

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: Internal error

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

subroutine zheevr (character jobz, character range, character uplo, integern, complex16, dimension( lda, ) a, integer lda, double precision vl,double precision vu, integer il, integer iu, double precision abstol,integer m, double precision, dimension( * ) w, complex16, dimension(ldz, ) z, integer ldz, integer, dimension( * ) isuppz, complex16,dimension( ) work, integer lwork, double precision, dimension( * )rwork, integer lrwork, integer, dimension( * ) iwork, integer liwork,integer info)

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

Purpose:

ZHEEVR 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. Invocations with different choices for
these parameters may result in the computation of slightly different
eigenvalues and/or eigenvectors for the same matrix. The reason for
this behavior is that there exists a variety of algorithms (each
performing best for a particular set of options) with ZHEEVR
attempting to select the best based on the various parameters. In all
cases, the computed values are accurate within the limits of finite
precision arithmetic.

ZHEEVR first reduces the matrix A to tridiagonal form T with a call
to ZHETRD. Then, whenever possible, ZHEEVR calls ZSTEMR to compute
eigenspectrum using Relatively Robust Representations. ZSTEMR
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.

The desired accuracy of the output can be specified by the input
parameter ABSTOL.

For more details, see ZSTEMR’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 : ZHEEVR calls ZSTEMR when the full spectrum is requested
on machines which conform to the ieee-754 floating point standard.
ZHEEVR calls DSTEBZ and ZSTEIN on non-ieee machines and
when partial spectrum requests are made.

Normal execution of ZSTEMR 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.

This parameter influences the choice of the algorithm and
may alter the computed values.

RANGE

This parameter influences the choice of the algorithm and
may alter the computed values.

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.

A is COMPLEX*16 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 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 JOBZ = ’N’, then Z is not referenced.
Note: the user must ensure that at least max(1,M) columns are
supplied in the array Z; if RANGE = ’V’, the exact value of M
is not known in advance and an upper bound must be used.
Supplying N columns is always safe.

LDZ

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

ISUPPZ

ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
The support of the eigenvectors in Z, i.e., the indices
indicating the nonzero elements in Z. The i-th eigenvector
is nonzero only in elements ISUPPZ( 2*i-1 ) through
ISUPPZ( 2*i ). This is an output of ZSTEMR (tridiagonal
matrix). The support of the eigenvectors of A is typically
1:N because of the unitary transformations applied by ZUNMTR.
Implemented only for RANGE = ’A’ or ’I’ and IU - IL = N - 1

WORK

WORK is COMPLEX*16 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, else LWORK >= 2*N.
For optimal efficiency, LWORK >= (NB+1)*N,
where NB is the max of the blocksize for ZHETRD and for
ZUNMTR as returned by ILAENV.

RWORK

RWORK is DOUBLE PRECISION 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.
If N <= 1, LRWORK >= 1, else LRWORK >= 24*N.

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.
If N <= 1, LIWORK >= 1, else LIWORK >= 10*N.

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:

Author

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