# stevr - Man Page

stevr: eig, MRRR

## Synopsis

### Functions

subroutine dstevr (jobz, range, n, d, e, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info)
DSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine sstevr (jobz, range, n, d, e, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info)
SSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

## Function Documentation

### subroutine dstevr (character jobz, character range, integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision vl, double precision vu, integer il, integer iu, double precision abstol, integer m, double precision, dimension( * ) w, double precision, dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

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

Purpose:

``` DSTEVR computes selected eigenvalues and, optionally, eigenvectors
of a real symmetric tridiagonal matrix T.  Eigenvalues and
eigenvectors can be selected by specifying either a range of values
or a range of indices for the desired eigenvalues.

Whenever possible, DSTEVR calls DSTEMR to compute the
eigenspectrum using Relatively Robust Representations.  DSTEMR
computes eigenvalues by the dqds algorithm, while orthogonal
eigenvectors are computed from various 'good' L D L^T representations
(also known as Relatively Robust Representations). Gram-Schmidt
orthogonalization is avoided as far as possible. More specifically,
the various steps of the algorithm are as follows. For the i-th
unreduced block of T,
(a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
is a relatively robust representation,
(b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
relative accuracy by the dqds algorithm,
(c) If there is a cluster of close eigenvalues, 'choose' sigma_i
close to the cluster, and go to step (a),
(d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
compute the corresponding eigenvector by forming a
rank-revealing twisted factorization.
The desired accuracy of the output can be specified by the input
parameter ABSTOL.

For more details, see 'A new O(n^2) algorithm for the symmetric
tridiagonal eigenvalue/eigenvector problem', by Inderjit Dhillon,
Computer Science Division Technical Report No. UCB//CSD-97-971,
UC Berkeley, May 1997.

Note 1 : DSTEVR calls DSTEMR when the full spectrum is requested
on machines which conform to the ieee-754 floating point standard.
DSTEVR calls DSTEBZ and DSTEIN on non-ieee machines and
when partial spectrum requests are made.

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

JOBZ

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

RANGE

```          RANGE is CHARACTER*1
= 'A': all eigenvalues will be found.
= 'V': all eigenvalues in the half-open interval (VL,VU]
will be found.
= 'I': the IL-th through IU-th eigenvalues will be found.
For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
DSTEIN are called```

N

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

D

```          D is DOUBLE PRECISION array, dimension (N)
On entry, the n diagonal elements of the tridiagonal matrix
A.
On exit, D may be multiplied by a constant factor chosen
to avoid over/underflow in computing the eigenvalues.```

E

```          E is DOUBLE PRECISION array, dimension (max(1,N-1))
On entry, the (n-1) subdiagonal elements of the tridiagonal
matrix A in elements 1 to N-1 of E.
On exit, E may be multiplied by a constant factor chosen
to avoid over/underflow in computing the eigenvalues.```

VL

```          VL is DOUBLE PRECISION
If RANGE='V', the lower bound of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = 'A' or 'I'.```

VU

```          VU is DOUBLE PRECISION
If RANGE='V', the upper bound of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = 'A' or 'I'.```

IL

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

IU

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

ABSTOL

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

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

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

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

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

M

```          M is INTEGER
The total number of eigenvalues found.  0 <= M <= N.
If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.```

W

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

Z

```          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
If JOBZ = 'V', then if INFO = 0, the first M columns of Z
contain the orthonormal eigenvectors of the matrix A
corresponding to the selected eigenvalues, with the i-th
column of Z holding the eigenvector associated with W(i).
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 ).
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 (and
minimal) LWORK.```

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.  LWORK >= max(1,20*N).

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 (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 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:

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

Definition at line 301 of file dstevr.f.

### subroutine sstevr (character jobz, character range, integer n, real, dimension( * ) d, real, dimension( * ) e, real vl, real vu, integer il, 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)

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

Purpose:

``` SSTEVR computes selected eigenvalues and, optionally, eigenvectors
of a real symmetric tridiagonal matrix T.  Eigenvalues and
eigenvectors can be selected by specifying either a range of values
or a range of indices for the desired eigenvalues.

Whenever possible, SSTEVR 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 L^T representations
(also known as Relatively Robust Representations). Gram-Schmidt
orthogonalization is avoided as far as possible. More specifically,
the various steps of the algorithm are as follows. For the i-th
unreduced block of T,
(a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
is a relatively robust representation,
(b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
relative accuracy by the dqds algorithm,
(c) If there is a cluster of close eigenvalues, 'choose' sigma_i
close to the cluster, and go to step (a),
(d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
compute the corresponding eigenvector by forming a
rank-revealing twisted factorization.
The desired accuracy of the output can be specified by the input
parameter ABSTOL.

For more details, see 'A new O(n^2) algorithm for the symmetric
tridiagonal eigenvalue/eigenvector problem', by Inderjit Dhillon,
Computer Science Division Technical Report No. UCB//CSD-97-971,
UC Berkeley, May 1997.

Note 1 : SSTEVR calls SSTEMR when the full spectrum is requested
on machines which conform to the ieee-754 floating point standard.
SSTEVR 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.```

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
SSTEIN are called```

N

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

D

```          D is REAL array, dimension (N)
On entry, the n diagonal elements of the tridiagonal matrix
A.
On exit, D may be multiplied by a constant factor chosen
to avoid over/underflow in computing the eigenvalues.```

E

```          E is REAL array, dimension (max(1,N-1))
On entry, the (n-1) subdiagonal elements of the tridiagonal
matrix A in elements 1 to N-1 of E.
On exit, E may be multiplied by a constant factor chosen
to avoid over/underflow in computing the eigenvalues.```

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 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).
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 ).
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 (and
minimal) LWORK.```

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.  LWORK >= 20*N.

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 (and
minimal) LIWORK.```

LIWORK

```          LIWORK is INTEGER
The dimension of the array IWORK.  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:

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

Definition at line 303 of file sstevr.f.

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